Independent evaluation and post-training potential [ftip-005C]
✍️source

A training score cannot by itself answer how much reliable capability was obtained. This section fixes an evaluation law, an inference procedure, and a resource account before comparing post-training protocols.

Convention 1. Fixed evaluation interface for protocol comparison [ftip-005D]

Fix an evaluation task $\mathsf T_{\rm ev}$ from Definition [ftip-001O] and a measurable scalar evaluation utility $u_{\mathsf T_{\rm ev}}:\mathcal X_{\mathsf T_{\rm ev}}\times \mathcal O_{\mathsf T_{\rm ev}}\times \Omega _E\to \mathbb R$, the $d=1$ case of Definition [ftip-001T]. Fix its task law $Q_{\rm ev}:=\mu _{\mathsf T_{\rm ev}}$, an inference protocol $\mathsf I_{\rm ev}$ from Definition [ftip-001R], and an inference budget $b\in \mathcal B_{\mathrm {eval}}$. Use the inference-seed space $\Omega _I$ and evaluator-seed space $\Omega _E$ from Notation [ftip-001N]. A declared probability kernel on these measurable spaces $\Lambda _{\rm ev}(d\xi ,d\omega \mid x)$ assigns inference and evaluator randomness to each instance. Together with $Q_{\rm ev}$, it gives the joint evaluation law $\nu _{\rm ev}(dx,d\xi ,d\omega ) =Q_{\rm ev}(dx)\Lambda _{\rm ev}(d\xi ,d\omega \mid x)$, which is fixed independently of training.

Declare a measurable space $\mathcal M$ of executable artifacts. For $M\in \mathcal M$, define the aliases \[ \mathsf {Eval}_b(M,x;\xi ) :=\operatorname {pr}_1\left (\mathsf I_{\rm ev}(M,x,b;\xi )\right ), \qquad U(x,o;\omega ):=u_{\mathsf T_{\rm ev}}(x,o;\omega ). \] Require $\mathsf {Eval}_b$ to be measurable in $(M,x,\xi )$ and to return admissible outcomes on admitted runs.

Each post-training protocol $P$ has a probability space $(\Omega _P,\Sigma _P,\mathbb P_P)$ for its randomness and a measurable artifact map $M_P:\Omega _P\to \mathcal M$ produced from the common base artifact $M_0$. The no-training protocol is $P_0$. The complete evaluation law for $P$ is the product $\mathbb P_P\otimes \nu _{\rm ev}$, expressing independence between protocol randomness and the fixed evaluation draw. For $\zeta \in \Omega _P$, write \[ Z_P(\zeta ,x,\xi ,\omega ) =U\left (x,\mathsf {Eval}_b(M_P(\zeta ),x;\xi );\omega \right ). \] This composite is measurable. Every protocol compared through expected performance, including the baseline $P_0$, must satisfy \[ \int |Z_P|\,d(\mathbb P_P\otimes \nu _{\rm ev})<\infty . \] This is the evaluation domain; finite pointwise utility does not replace the absolute-integrability condition. Fix also a scalar success threshold $u_*\in \mathbb R$.

Definition 2. Expected evaluation performance [ftip-005E]

For a protocol $P$ in the integrable evaluation domain of Convention 1, its expected evaluation performance is the finite real number

\[ \begin {aligned} J_{\rm ev}(P) &=\int Z_P\,d(\mathbb P_P\otimes \nu _{\rm ev}),\\ &=\mathbb E\left [ U\left (X,\mathsf {Eval}_b(M_P,X;\Xi );\Omega \right ) \right ], \\ (X,\Xi ,\Omega )&\sim Q_{\rm ev}(dx)\Lambda _{\rm ev}(d\xi ,d\omega \mid x). \end {aligned} \]

The label $\rm ev$ abbreviates the task, law, inference protocol, budget, utility, and seed law fixed above. Changing any of them changes the estimand. Both positive and negative parts of $Z_P$ have finite expectation; an undefined expectation or an infinite value is outside this real-valued performance domain.

Definition 3. Post-training gain [ftip-005F]

For $P$ and the no-training protocol $P_0$ both satisfying the absolute-integrability condition of Convention 1, the post-training gain is the real-valued difference

\[ \Delta J_{\rm ev}(P) =J_{\rm ev}(P)-J_{\rm ev}(P_0). \]

Both terms use the same evaluator, task law, utility, and inference budget. The subtraction does not compare different model families or evaluation procedures. In particular, $+\infty -(+\infty )$ is not an admitted gain.

Remark 4. Evaluation gain under a fixed task law [ftip-005G]

The policy-evaluation expectation of Section 3.5 of Reinforcement learning: An introduction[sutton2018reinforcement] specializes to an independently fixed task law and inference procedure. The gain is a paired contrast with the same base artifact and evaluator.

A positive value shows a change under this evaluation. It does not alone show that the trained policy acquired a capability under the criterion of Definition [ftip-0007].

Definition 5. Lifecycle cost vector [ftip-005H]

The realized lifecycle cost vector of a protocol is the random vector $C(P)\in \mathbb R_+^7$, measurable on its declared lifecycle-run probability space, that records separately

\[ C(P)= (C_{\rm data},C_{\rm feedback},C_{\rm rollout},C_{\rm env}, C_{\rm update},C_{\rm storage},C_{\rm eval}). \]

A componentwise lifecycle budget is a vector $\mathbf B\in \mathbb R_+^7$. Each cost coordinate has a declared unit, such as tokens, labels, accelerator floating-point operations (FLOPs), sandbox time, bytes, or evaluator calls. A scalar price may be applied later, but the typed vector is retained. Each nonnegative coordinate has a well-defined expectation in $[0,+\infty ]$; finite realized costs need not have finite expectations. A finite componentwise budget excludes protocols with any infinite expected-cost coordinate. When costs include the fixed independent evaluation draw, the run law may include the product law in Convention 1; the accounting record must specify which randomness is averaged.

Remark 6. Lifecycle costs have distinct units [ftip-005I]

The scaling study behind Definition [ftip-001C] reports training tokens and compute separately.

[hoffmann2022training, sec. 3] studies their allocation. Training tokens and compute are therefore separate cost coordinates.

Agentic and replay protocols also incur verifier, environment, storage, and selection work. A training step, a generated rollout, and one second of wall time are therefore different units.

The vector permits later cost models without hiding which conversion rates or hardware assumptions they use.

Definition 7. Admissible protocol class [ftip-005J]

For a base artifact $M_0$, an admissible protocol class $\mathfrak P(M_0)$ is a declared set of type-correct training procedures. Membership fixes which data sources, feedback channels, model changes, environment interfaces, and update rules are permitted. For the expected performance comparison, every member also satisfies the measurable generation and integrable evaluation domain of Convention 1, and carries the measurable nonnegative cost vector of Definition 5. The baseline $P_0$ must satisfy the evaluation domain even if it is not feasible at the chosen cost budget.

The class is part of the research question. Enlarging it can only enlarge the set of attainable outcomes, but may make a comparison less informative.

Definition 8. Trust contract [ftip-005K]

A trust contract $\mathsf {Trust}(P)$ is a predicate requiring the protocol to respect declared provenance, information, and accounting boundaries. At minimum it specifies evaluation independence, behavior-policy stamps for reused rollouts, environment and verifier versions, and complete lifecycle-cost reporting.

The predicate is evidence to be checked. It is not a statement that a protocol is safe, aligned, or free from distribution shift.

Remark 9. Allowed procedures and admissible evidence [ftip-005L]

The protocol class and trust contract impose distinct restrictions. The class states what may be attempted; the contract states what evidence an attempt must retain before it can enter the comparison.

Their joint specification is a proposed evaluation framework.

A supremum over an unspecified class or an unaudited evaluation boundary has no stable empirical interpretation.

Definition 10. Costed post-training potential [ftip-005M]

For a scalar evaluation utility, the costed post-training potential of $M_0$ under a finite componentwise budget $\mathbf B\in \mathbb R_+^7$ is the extended-real supremum

\[ \Phi (M_0,\mathbf B) =\sup _{P\in \mathfrak P(M_0)} \left \{ J_{\rm ev}(P): \mathsf {Trust}(P),\ \mathbb E[C(P)]\leq \mathbf B \right \}. \]

Here $\mathfrak P(M_0)$ has the evaluation domain of Definition 7, so each $J_{\rm ev}(P)$ is real. The expected-cost inequality is componentwise in $[0,+\infty ]^7$; it cannot hold against a finite budget if any coordinate is infinite. Define $\sup \varnothing =-\infty $, and use $+\infty $ when feasible performance values are unbounded above. Thus $\Phi $ takes values in $\overline {\mathbb R}=\mathbb R\cup \{-\infty ,+\infty \}$.

The potential is finite real exactly when its feasible performance set is nonempty and bounded above. For example, a common finite upper bound on utility and at least one feasible protocol suffice. This does not guarantee that any protocol attains the supremum. Ordinary differences, ratios, or derivatives of potential values require finite real operands and any further regularity hypotheses needed by that operation. The value is conditional on every object named in the display; it is not an intrinsic constant of the model.

Remark 11. Potential depends on the intervention and budget [ftip-005N]

The potential $\Phi $ describes how evaluation performance varies with the base artifact, admissible information, protocol family, and resource budget. The definition supplies no scaling exponent and no guarantee that a maximizing protocol exists.

Empirical studies provide conditional points or bounds only after their protocol, evaluator, and costs are embedded in this interface. An observed benchmark maximum need not equal a universal ceiling.

Definition 12. feasible objective set [boyd2004convex, Section 4.7.4] [ftip-005O]

For a multi-objective optimization problem with vector objective $f(x)$, the feasible objective set is the collection of vectors $f(x)$ attained by feasible choices $x$.

In a post-training application, the feasible choice is a trusted, budget-feasible protocol and the objective vector contains independently declared evaluation outcomes.

Definition 13. weighted-sum scalarization [boyd2004convex, Section 4.7.5] [ftip-005P]

For objective vector $v\in \mathbb R^m$ and nonnegative weights $w\in \mathbb R_+^m$, the weighted-sum scalarization assigns the score $w^\top v$. Optimizing this score selects a point according to the declared weights.

Changing $w$ changes the research question. A single weighted score can hide tradeoffs among evaluation groups or task families.

Definition 14. Pareto-optimal objective vector [boyd2004convex, Section 4.7.4] [ftip-005Q]

A feasible vector $v$ is Pareto optimal when there is no feasible $v'$ with $v'_j\geq v_j$ in every coordinate and a strict inequality in at least one coordinate. The Pareto frontier is the set of such vectors.

This order preserves visible tradeoffs. It does not choose one point on the frontier.

Definition 15. worst-group risk [sagawa2020distributionally, Section 2] [ftip-005R]

Given predefined groups with risks $L_g$, the worst-group risk is $\max _g L_g$. For utilities, the corresponding robust score is $\min _g J_g$.

A post-training comparison must declare the groups and their evaluation laws before using this score. It is different from an average and from Pareto dominance.

Example 16. Three attainable protocols with different scalar, Pareto, and worst-group summaries [ftip-005S]

Three attainable utility vectors separate the answers returned by scalarization, Pareto comparison, and a worst-group summary.

$\text{[math]}$

For scalar weights $(0.8,0.2)$, the scores are $S(A)=0.80$, $S(B)=0.70$, and $S(C)=0.56$, so $A$ wins. The worst-group scores are $0.4$, $0.7$, and $0.5$, so $B$ wins. No point dominates another coordinatewise, hence the Pareto summary retains all three.

The scalar, Pareto, and worst-group summaries were defined in Definition 13--Definition 15. Their non-equivalence is explicit in these three points, while the choice of a social or evaluation rule remains open.

References

Reference. Training compute-optimal large language models [hoffmann2022training]

2022
Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, Tom Hennigan, Eric Noland, Katie Millican, George van den Driessche, Bogdan Damoc, Aurelia Guy, Simon Osindero, Karen Simonyan, Erich Elsen, Jack W. Rae, Oriol Vinyals, Laurent Sifre
https://arxiv.org/abs/2203.15556

@article{hoffmann2022training,
  title = {Training Compute-Optimal Large Language Models},
  author = {Hoffmann, Jordan and Borgeaud, Sebastian and Mensch, Arthur and Buchatskaya, Elena and Cai, Trevor and Rutherford, Eliza and de Las Casas, Diego and Hendricks, Lisa Anne and Welbl, Johannes and Clark, Aidan and Hennigan, Tom and Noland, Eric and Millican, Katie and van den Driessche, George and Damoc, Bogdan and Guy, Aurelia and Osindero, Simon and Simonyan, Karen and Elsen, Erich and Rae, Jack W. and Vinyals, Oriol and Sifre, Laurent},
  year = {2022},
  url = {https://arxiv.org/abs/2203.15556},
  journal = {arXiv preprint arXiv:2203.15556}
}

Reference. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization [sagawa2020distributionally]

2020
Shiori Sagawa, Pang Wei Koh, Tatsunori B. Hashimoto, Percy Liang
https://arxiv.org/abs/1911.08731

@inproceedings{sagawa2020distributionally,
  title = {Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization},
  author = {Sagawa, Shiori and Koh, Pang Wei and Hashimoto, Tatsunori B. and Liang, Percy},
  booktitle = {International Conference on Learning Representations},
  year = {2020},
  url = {https://arxiv.org/abs/1911.08731}
}

Reference. Reinforcement learning: An introduction [sutton2018reinforcement]

2018
Richard S. Sutton, Andrew G. Barto
http://incompleteideas.net/book/the-book-2nd.html

@book{sutton2018reinforcement,
  title = {Reinforcement Learning: An Introduction},
  author = {Sutton, Richard S. and Barto, Andrew G.},
  publisher = {MIT Press},
  edition = {2},
  year = {2018},
  url = {http://incompleteideas.net/book/the-book-2nd.html}
}

Reference. Convex optimization [boyd2004convex]

2004
Stephen Boyd, Lieven Vandenberghe
https://web.stanford.edu/~boyd/cvxbook/

@book{boyd2004convex,
  title = {Convex Optimization},
  author = {Boyd, Stephen and Vandenberghe, Lieven},
  publisher = {Cambridge University Press},
  year = {2004},
  url = {https://web.stanford.edu/~boyd/cvxbook/}
}

Context

Evaluation, evidence, and finite limits [ftip-00JB]

The training and agent mechanisms described earlier produce an artifact whose capability must be evaluated under a declared task law, inference procedure and resource account. This chapter fixes that comparison, studies what survives a change of evaluation, and examines discovery probabilities, support, proxy error and the information available through feedback.

Finite results and counterexamples establish consequences under specific probability laws and computational assumptions. They provide tools for the model-lineage question, where their assumptions must cover an evolving research and training process. The architecture analysis uses the same evaluation framework for comparisons that depend on a model implementation or optimizer.

Independent evaluation and post-training potential [ftip-005C]
✍️source

Convention. Fixed evaluation interface for protocol comparison [ftip-005D]

Definition. Expected evaluation performance [ftip-005E]

For a protocol $P$ in the integrable evaluation domain of Convention [ftip-005D], its expected evaluation performance is the finite real number

Definition. Post-training gain [ftip-005F]

For $P$ and the no-training protocol $P_0$ both satisfying the absolute-integrability condition of Convention [ftip-005D], the post-training gain is the real-valued difference

\[ \Delta J_{\rm ev}(P) =J_{\rm ev}(P)-J_{\rm ev}(P_0). \]

Remark. Evaluation gain under a fixed task law [ftip-005G]

A positive value shows a change under this evaluation. It does not alone show that the trained policy acquired a capability under the criterion of Definition [ftip-0007].

Definition. Lifecycle cost vector [ftip-005H]

The realized lifecycle cost vector of a protocol is the random vector $C(P)\in \mathbb R_+^7$, measurable on its declared lifecycle-run probability space, that records separately

\[ C(P)= (C_{\rm data},C_{\rm feedback},C_{\rm rollout},C_{\rm env}, C_{\rm update},C_{\rm storage},C_{\rm eval}). \]

A componentwise lifecycle budget is a vector $\mathbf B\in \mathbb R_+^7$. Each cost coordinate has a declared unit, such as tokens, labels, accelerator floating-point operations (FLOPs), sandbox time, bytes, or evaluator calls. A scalar price may be applied later, but the typed vector is retained. Each nonnegative coordinate has a well-defined expectation in $[0,+\infty ]$; finite realized costs need not have finite expectations. A finite componentwise budget excludes protocols with any infinite expected-cost coordinate. When costs include the fixed independent evaluation draw, the run law may include the product law in Convention [ftip-005D]; the accounting record must specify which randomness is averaged.

Remark. Lifecycle costs have distinct units [ftip-005I]

The scaling study behind Definition [ftip-001C] reports training tokens and compute separately.

[hoffmann2022training, sec. 3] studies their allocation. Training tokens and compute are therefore separate cost coordinates.

Agentic and replay protocols also incur verifier, environment, storage, and selection work. A training step, a generated rollout, and one second of wall time are therefore different units.

The vector permits later cost models without hiding which conversion rates or hardware assumptions they use.

Definition. Admissible protocol class [ftip-005J]

For a base artifact $M_0$, an admissible protocol class $\mathfrak P(M_0)$ is a declared set of type-correct training procedures. Membership fixes which data sources, feedback channels, model changes, environment interfaces, and update rules are permitted. For the expected performance comparison, every member also satisfies the measurable generation and integrable evaluation domain of Convention [ftip-005D], and carries the measurable nonnegative cost vector of Definition [ftip-005H]. The baseline $P_0$ must satisfy the evaluation domain even if it is not feasible at the chosen cost budget.

The class is part of the research question. Enlarging it can only enlarge the set of attainable outcomes, but may make a comparison less informative.

Definition. Trust contract [ftip-005K]

The predicate is evidence to be checked. It is not a statement that a protocol is safe, aligned, or free from distribution shift.

Remark. Allowed procedures and admissible evidence [ftip-005L]

Their joint specification is a proposed evaluation framework.

A supremum over an unspecified class or an unaudited evaluation boundary has no stable empirical interpretation.

Definition. Costed post-training potential [ftip-005M]

For a scalar evaluation utility, the costed post-training potential of $M_0$ under a finite componentwise budget $\mathbf B\in \mathbb R_+^7$ is the extended-real supremum

\[ \Phi (M_0,\mathbf B) =\sup _{P\in \mathfrak P(M_0)} \left \{ J_{\rm ev}(P): \mathsf {Trust}(P),\ \mathbb E[C(P)]\leq \mathbf B \right \}. \]

Here $\mathfrak P(M_0)$ has the evaluation domain of Definition [ftip-005J], so each $J_{\rm ev}(P)$ is real. The expected-cost inequality is componentwise in $[0,+\infty ]^7$; it cannot hold against a finite budget if any coordinate is infinite. Define $\sup \varnothing =-\infty $, and use $+\infty $ when feasible performance values are unbounded above. Thus $\Phi $ takes values in $\overline {\mathbb R}=\mathbb R\cup \{-\infty ,+\infty \}$.

Remark. Potential depends on the intervention and budget [ftip-005N]

Definition. feasible objective set [boyd2004convex, Section 4.7.4] [ftip-005O]

For a multi-objective optimization problem with vector objective $f(x)$, the feasible objective set is the collection of vectors $f(x)$ attained by feasible choices $x$.

In a post-training application, the feasible choice is a trusted, budget-feasible protocol and the objective vector contains independently declared evaluation outcomes.

Definition. weighted-sum scalarization [boyd2004convex, Section 4.7.5] [ftip-005P]

Changing $w$ changes the research question. A single weighted score can hide tradeoffs among evaluation groups or task families.

Definition. Pareto-optimal objective vector [boyd2004convex, Section 4.7.4] [ftip-005Q]

This order preserves visible tradeoffs. It does not choose one point on the frontier.

Definition. worst-group risk [sagawa2020distributionally, Section 2] [ftip-005R]

Given predefined groups with risks $L_g$, the worst-group risk is $\max _g L_g$. For utilities, the corresponding robust score is $\min _g J_g$.

A post-training comparison must declare the groups and their evaluation laws before using this score. It is different from an average and from Pareto dominance.

Example. Three attainable protocols with different scalar, Pareto, and worst-group summaries [ftip-005S]

Three attainable utility vectors separate the answers returned by scalarization, Pareto comparison, and a worst-group summary.

$\text{[math]}$

The scalar, Pareto, and worst-group summaries were defined in Definition [ftip-005P]--Definition [ftip-005R]. Their non-equivalence is explicit in these three points, while the choice of a social or evaluation rule remains open.

Evaluation transport and matched protocols [ftip-00F9]

This section compares post-training artifacts only through declared evaluation laws, inference protocols, utilities, seeds, and budgets. It proves finite transport statements locally and treats long-horizon research-state records as empirical evidence, not as theorems.

Evaluation laws and couplings [ftip-00FA]

Convention. A named evaluation experiment [ftip-00FB]

A named evaluation experiment records the artifact, task law, inference protocol, evaluator, seed kernel, budget, and sampling unit. Two reported scores are comparable only after these coordinates and their intended target have been stated.

Definition. Evaluation law [ftip-00FC]

For a fixed evaluation interface from Convention [ftip-005D], an evaluation law is a probability measure $\mathsf Q$ on the finite outcome space $\mathcal Z=\mathcal X_{\mathsf T_{\rm ev}}\times \mathcal O_{\mathsf T_{\rm ev}}$ with random pair $(X,O)\sim \mathsf Q$. Its score for artifact $M$ is a bounded measurable map $s_M:\mathcal Z\to [0,1]$. Write $J_{\mathsf Q}(M)=\mathbb E_{\mathsf Q}[s_M(X,O)]$.

Definition. Matched evaluation pair [ftip-00FD]

A matched evaluation pair is $(\mathsf Q,\mathsf I,b,\Lambda )$ and $(\mathsf Q',\mathsf I',b',\Lambda ')$ together with a coupling $(Z,Z')$ whose marginals are the two laws. The pair is matched on the declared coordinates when the task, utility, evaluator version, and sampling unit are shared; any changed coordinate is named.

Definition. Paired finite score estimator [ftip-00FE]

For iid coupled draws $(Z_j,Z'_j)_{j=1}^m$, define $\widehat \Delta _m=m^{-1}\sum _{j=1}^m(s_M(Z_j)-s_{M'}(Z'_j))$. The pairing is part of the estimator specification; it does not silently identify the two evaluation laws.

Theorem. Unbiasedness of the paired score estimator [ftip-00FF]

Under the iid coupling in Definition [ftip-00FE], $\mathbb E[\widehat \Delta _m]=J_{\mathsf Q}(M)-J_{\mathsf Q'}(M')$.

Proof.

Linearity of expectation gives the displayed identity.

\[ \mathbb E[\widehat \Delta _m] =m^{-1}\sum _j\left (\mathbb E[s_M(Z_j)]- \mathbb E[s_{M'}(Z'_j)]\right ). \]

The coupling fixes the joint sampling but does not change either marginal, so the two terms are the stated expectations.

Example. Common seeds can reduce paired variance [ftip-00G0]

If $s_M(Z)=s_{M'}(Z')$ almost surely under a common-seed coupling, then $\widehat \Delta _m=0$ for every sample even when independent sampling would have nonzero variance. This is a variance statement, not evidence that the two artifacts have equal behavior on a different law.

Remark. A coupling does not merge estimands [ftip-00G1]

The result in Theorem [ftip-00FF] follows from its displayed hypotheses. A shared seed can make a paired comparison precise while the changed task law, evaluator, or inference budget still changes the estimand. No causal effect follows without a declared intervention and matched coordinates.

Utility transport [ftip-00G2]

Definition. Bounded utility perturbation [ftip-00G3]

On a common finite outcome law $\mathsf Q$, let each artifact $M$ have score maps $s_M,t_M:\mathcal Z\to [0,1]$. They have perturbation radius $\varepsilon =\sup _{M,\,z\in \operatorname {supp}(\mathsf Q)} |s_M(z)-t_M(z)|$ when $0\leq \varepsilon \leq 1$; write $J_s(M)=\mathbb E_{\mathsf Q}[s_M]$ and $J_t(M)=\mathbb E_{\mathsf Q}[t_M]$.

Theorem. Direct utility transport bound [ftip-00G4]

Under Definition [ftip-00G3], every artifact $M$ satisfies $|J_s(M)-J_t(M)|\leq \varepsilon $.

Proof.

Pointwise bounds give $-\varepsilon \leq s_M(z)-t_M(z)\leq \varepsilon $. Taking expectations preserves both inequalities.

Corollary. Transporting an artifact comparison [ftip-00G5]

If two artifacts $M_1,M_0$ are scored by both $s$ and $t$ under the same law and $\lVert s-t\rVert _{\infty ,\mathsf Q}\leq \varepsilon $, then the gain difference obeys $|(J_s(M_1)-J_s(M_0))-(J_t(M_1)-J_t(M_0))| \leq 2\varepsilon $. This is a finite utility perturbation bound, not a training or distribution-shift theorem.

Example. Equal aggregate score can hide a slice reversal [ftip-00G6]

On equally likely outcomes $a,b$, take score vectors $(s_{M_1}(a),s_{M_1}(b))=(1,0)$ and $(s_{M_0}(a),s_{M_0}(b))=(0,1)$; a second score map swaps coordinates. Both artifacts have aggregate score $1/2$ under both maps, but their per-outcome ordering reverses. Aggregate transport alone does not identify localized behavior.

Remark. Common domains are a transport hypothesis [ftip-00G7]

The bound in Theorem [ftip-00G4] requires the same finite outcome support. Changing prompts, parsers, evaluators, or inference budgets is a new law and must be recorded rather than hidden inside a score perturbation.

Prompt and research-state transport [ftip-00G8]

Definition. Research-state observable [ftip-00G9]

For a finite record space $\mathcal R$, an observable is a map $h:\mathcal R\to \mathcal H$. A summary-only evaluator sees $h(r)$ and not the underlying record $r$; the omitted coordinates are unavailable to that evaluator.

Definition. Compaction kernel [ftip-00GA]

A compaction kernel is a randomized map $K(dh\mid r)$ from records in $\mathcal R$ to summaries in $\mathcal H$. A deterministic summary is the special case $K(dh\mid r)=\delta _{h(r)}$.

Theorem. Summary-only indistinguishability [ftip-00GB]

Let $r,r'\in \mathcal R$ induce the same summary law under the kernel in Definition [ftip-00GA]. Any randomized decision rule that receives only that summary has the same output law in the two worlds.

Proof.

The output law is the pushforward of the common summary law through the same decision kernel. Pushforwards of equal finite measures are equal.

Example. An omitted feasibility constraint [ftip-00GC]

Two research records can share every summary score while one contains a withdrawn lemma and the other contains a feasible proof plan. A summary-only selector therefore chooses identically, although the correct next action differs. This is a finite witness to information loss, not a claim about a particular model.

Remark. Long-horizon records are empirical transport evidence [ftip-00GD]

The case study in [li2026longhorizon], Sections 4--7, reports file-based memory, human steering, and research-state failures under one system and task. It motivates explicit observables and omitted constraints; it does not supply the finite theorem in Theorem [ftip-00GB] or a general model capability law.

Evaluation cells and budget accounting [ftip-00GE]

Definition. Evaluation cell [ftip-00GF]

An evaluation cell is $C=(M,\mathsf Q,\mathsf I,b,\Lambda ,v)$, where $M$ is an artifact, $\mathsf Q$ a task law, $\mathsf I$ an inference protocol, $b$ a budget, $\Lambda $ a seed kernel, and $v$ an evaluator version. A cell comparison may differ in named coordinates only.

Lemma. Additive evaluation cost [ftip-00GG]

For a finite evaluation plan with disjoint stages of costs $c_1,\ldots ,c_k\in \mathbb R_{\geq 0}$, total cost is $c=\sum _{i=1}^k c_i$. Appending a stage increases cost by its declared amount and cannot preserve a budget $B$ unless the remaining slack covers it.

Proof.

The statement is the associativity and monotonicity of finite sums.

Theorem. Budget-preserving matched comparison [ftip-00GH]

Let two evaluation plans use the same declared stages except for a substitution whose cost is at most the replaced stage cost. If their common prefix cost is $c_0$ and both suffix costs fit $B-c_0$, then both cells in Definition [ftip-00GF] are admissible under budget $B$.

Proof.

Apply Lemma [ftip-00GG] to the common prefix and each suffix. The assumed inequalities give total cost at most $B$ in each plan.

Example. Cost-balanced evaluation cells [ftip-00GI]

A paired run can spend $B=100$ units as 40 units of inference, 20 of evaluator calls, and 40 of audit. Replacing 10 inference units by 10 audit units preserves the budget, but changes the cell and must not be described as the same inference protocol.

Remark. Execution and evaluator confounds [ftip-00GJ]

A score change can arise from the artifact, prompt law, inference budget, parser, evaluator version, or execution failure. A matched-cell report names these coordinates before interpreting a paired gain.

Distribution shift and ranking reversals [ftip-00GK]

Example. Train and evaluation laws can reverse a ranking [ftip-00GL]

Let $\mathsf Q_{\rm tr}$ put masses $.9,.1$ on $a,b$, and let $\mathsf Q_{\rm ev}$ swap them. Artifacts $M_1,M_0$ with score vectors $(s_{M_1}(a),s_{M_1}(b))=(1,0)$ and $(s_{M_0}(a),s_{M_0}(b))=(0,1)$ rank oppositely under the two laws. A training score is not an evaluation transport certificate.

Theorem. Total-variation transport bound [ftip-00GM]

For any score $s:\mathcal Z\to [0,1]$ and finite laws $\mathsf Q,\mathsf Q'$, $|\mathbb E_{\mathsf Q}s-\mathbb E_{\mathsf Q'}s|\leq \lVert \mathsf Q-\mathsf Q'\rVert _{\rm TV}$, where $\lVert \mathsf Q-\mathsf Q'\rVert _{\rm TV}:=\frac 12\sum _{z\in \mathcal Z} |\mathsf Q(z)-\mathsf Q'(z)|$.

Proof.

Expand the finite expectation difference and use $|s(z)|\leq 1$. The positive and negative parts are bounded by the total variation mass, giving the displayed inequality.

Example. A sharp two-point transport bound [ftip-00GN]

On $\mathcal Z=\{a,b\}$, let $s(a)=1,s(b)=0$, and let $\mathsf Q(a)=1$, $\mathsf Q'(a)=1-\eta $, and $\mathsf Q'(b)=\eta $. The score difference and total variation distance are both $\eta $.

Remark. What transport does not identify [ftip-00GO]

The bounds above transport a declared score across a declared law. They do not identify why an artifact changed, establish support expansion, prove generalization, or order different evaluators whose outcome spaces differ.

Example. A minimal evaluation trace [ftip-00GP]

A reproducible record is $(M,\mathsf Q,\mathsf I,b,\Lambda ,v,m, \widehat \Delta _m,c)$, listing the cells, sample count, paired estimate, and cost. Replaying this tuple reproduces the estimator target only when the recorded laws and versions remain available.

Remark. Execution controls for a matched evaluation [ftip-00GQ]

A matched evaluation requires execution gates, contamination checks, grader calibration, and an audit of the result before it is committed. These controls make the comparison auditable; they do not by themselves establish capability acquisition.

Analytical questions and falsifiers [ftip-005T]

Successful support, verifier error, and entropy statistics distinguish properties that a benchmark summary can combine. Finite examples and counterexamples distinguish these quantities.

Definition. Successful-support probability [ftip-005U]

Fix the evaluation interface and success threshold $u_*$ from Convention [ftip-005D]. For a protocol $P$ and instance $x\in \mathcal X_{\mathsf T_{\rm ev}}$, its successful-support probability is

\[ p_P(x) =\Pr \left [ U\left (x,\mathsf {Eval}_b(M_P,x;\Xi );\Omega \right )\geq u_* \right ]. \]

The probability includes protocol, inference, environment, and evaluator randomness: $M_P$ is drawn from the protocol, while $(\Xi ,\Omega )$ is drawn from $\Lambda _{\rm ev}(\cdot \mid x)$ independently of training. The quantity is instance-specific and budget-specific.

Definition. Successful-support coverage [ftip-005V]

For threshold $0<\alpha \leq 1$, the successful-support coverage of protocol $P$ under evaluation law $Q_{\rm ev}$ is

\[ \operatorname {Cov}_\alpha (P) =Q_{\rm ev}\left (\{x:p_P(x)\geq \alpha \}\right ). \]

Coverage records how much task mass has at least the declared success probability. It does not average the probabilities above or below the threshold.

Remark. Task-level success and coverage [ftip-005W]

Increasing probability mass on known solutions need not increase coverage of the task family. Pass-at-$k$ and related support-sensitive measurements motivate this distinction. Sections 3--5 of Does reinforcement learning really incentivize reasoning capacity in LLMs beyond the base model?[yue2025does] and Sections 2--3 of Understanding r1-zero-like training: A critical perspective[liu2025understanding] compare post-training behavior with base-model support and discuss why response accuracy alone may overstate what RL creates beyond a base model.

The threshold $\alpha $, inference budget, task law, and success predicate must all be reported. Finite samples estimate this quantity; they do not reveal the unobserved tail without assumptions.

Definition. Uniform task-family acquisition witness [ftip-005X]

Fix a finite, independently selected family $G\subseteq \mathcal X_{\mathsf T_{\rm ev}}$, thresholds $0\leq \alpha <\beta \leq 1$, and the common evaluation interface of Convention [ftip-005D]. Protocols $P_0$ and $P_1$ have a uniform task-family acquisition witness on $G$ when

\[ p_{P_0}(x)\leq \alpha \quad \hbox {and}\quad p_{P_1}(x)\geq \beta \qquad \text {for every }x\in G. \]

For each $x$, the successful event in Definition [ftip-005U] supplies the fixed-budget transfer event in Definition [ftip-0007]. The displayed inequalities specialize that earlier acquisition witness with $\varepsilon _0=\alpha $ and $\varepsilon _1=\beta $, then require the specialization uniformly over $G$.

Remark. Acquisition evidence across a fixed task family [ftip-005Y]

The proposed family-level criterion is a demanding paired comparison. The task family and evaluator are fixed independently, and both protocols use the same inference budget. It is stronger than one acquisition witness because the threshold change must hold for every member of $G$.

A theorem may strengthen the witness with confidence bounds, transfer to a predeclared related family, or a lower bound on newly successful support. Each strengthening requires additional sampling or structural assumptions.

Definition. Verifier error rates [ftip-005Z]

Let $S\in \{0,1\}$ be an independently defined success label and $A\in \{0,1\}$ the verifier's accept decision on the same outcome. The false-accept and false-reject rates are

\[ \eta _+=\Pr (A=1\mid S=0), \qquad \eta _-=\Pr (A=0\mid S=1). \]

Both rates are conditional on the task and outcome distribution used for the audit. They are undefined when the conditioning event has probability zero.

Remark. Verifier error under an independent audit law [ftip-0061]

Verifier error is measured against an independent success label because a training reward cannot audit itself. The verifier and evidence interfaces are in Definition [ftip-0046]--Definition [ftip-0049]. DeepSeek-Prover-V2 [ren2025deepseekproverv2, Section 2.3, ``Reinforcement Learning''] supplies a proof-checking instance. The false-accept/false-reject decomposition is a binary measurement model.

The rates may vary by task, policy, trajectory length, and adversarial pressure. A low average error rate need not prevent reward hacking on the subpopulation favored by optimization.

Example. Reward hacking under an incomplete checker [ftip-0060]

Shifting probability mass toward a checker exploit raises proxy reward while reducing independently judged utility.

$\text{[math]}$

Let the three outputs be $y_0=\text {41}$, $y_1=\text {42}$, and $y_2=\text {ignore; 42; done}$. The proxy accepts $y_1,y_2$; the independent evaluator accepts only $y_1$. Moving their probabilities from $(0.5,0.4,0.1)$ to $(0.1,0.1,0.8)$ changes expected proxy reward from $0.5$ to $0.9$, but true utility from $0.4$ to $0.1$.

Reward exploitation in formal verification is discussed in [ren2025deepseekproverv2, Section 3.2]. The constructed checker exposes one failure mechanism. It provides no basis for treating every executable verifier as incomplete or every proxy improvement as reward hacking.

Definition. Global policy entropy [ftip-0062]

Fix a finite action space, a finite horizon $T$, and a declared distribution $\mu _t$ over public histories at each position. The global policy entropy is the weighted average

\[ H_{\rm global}(\pi ) =\frac 1T\sum _{t=0}^{T-1} \mathbb E_{\mathsf H_t\sim \mu _t} \left [-\sum _a\pi (a\mid \mathsf H_t) \log \pi (a\mid \mathsf H_t)\right ]. \]

The history distribution and action alphabet are part of the statistic.

We use the convention $0\log 0=0$ in this and the following position-entropy formula.

Definition. Token-position entropy [ftip-0063]

Let $T_{\rm tok}$ be the random number of generated tokens. For a one-based token position $t$ satisfying $\Pr (T_{\rm tok}\geq t)>0$, condition on $T_{\rm tok}\geq t$ and let $\xi _t$ be the token. The token-position entropy is

\[ H_t^{\rm pos} =-\sum _{v\in \mathcal V} \Pr (\xi _t=v\mid T_{\rm tok}\geq t) \log \Pr (\xi _t=v\mid T_{\rm tok}\geq t). \]

This is the entropy of a position marginal. It is generally different from the history-conditioned entropy averaged in Definition [ftip-0062].

Definition. selected-token surprisal [shannon1948mathematical, Part I, Section 6] [ftip-0064]

For a sampled token $\xi _t$ under next-token law $\pi (\,\cdot \mid \mathsf H_t)$, its surprisal is

\[-\log \pi (\xi _t\mid \mathsf H_t).\]

It is a random value attached to the selected token. Its conditional expectation over the token equals the finite Shannon entropy of the next-token law.

Remark. Why the entropy statistics are not interchangeable [ftip-0065]

Shannon's finite entropy A mathematical theory of communication[shannon1948mathematical] can be aggregated over different random objects. Global and position-level entropy statistics use different laws. Selected-token surprisal is a sample statistic, while the other two are expectations under declared history or position laws.

A training intervention can raise one quantity while lowering another. Later claims about exploration or collapse must name the statistic, sampling law, horizon, and conditioning event.

Definition. empirical squared-ratio excess [miao2026when, Theorem 2] [ftip-0066]

For token-level current-to-behavior ratios $r_1,\ldots ,r_T$, DGG defines

\[\widehat \chi ^2=\frac 1T\sum _{i=1}^T(r_i^2-1).\]

The source relates this statistic to an upper bound on language-model-head gradient energy through a batch-dependent constant. In a finite batch $\widehat \chi ^2$ can be negative, so it is not automatically a nonnegative empirical Pearson divergence. The result also gives no converse from a small batch gradient to small policy shift.

Example. Lifecycle compute allocation and a finite rollout observation bound [ftip-0067]

A componentwise lifecycle allocation separates resource accounting from the finite number of rollouts available for observation.

$\text{[math]}$

In the coordinate order of Definition [ftip-005H], declare a lifecycle budget vector $\mathbf B$. Its rollout coordinate permits $N$ attempts. If each attempt has declared success probability $p$, then \[ \Pr (\text {at least one success})=1-(1-p)^N\leq Np. \] Only the rollout coordinate enters this calculation; no sum or conversion among heterogeneous coordinates is defined.

The finite bound is the union bound applied to Bernoulli rollouts; finite-horizon success probabilities are standard policy-evaluation objects in [sutton2018reinforcement, Section 3.5]. Independence is needed only for the displayed exact expression. The budget vector is illustrative: neither empirical optimality nor interchangeability of heterogeneous units is inferred from it.

Remark. Finite questions and necessary assumptions [ftip-0068]

Finite analysis raises three questions: how conditional success rates bound long-horizon success, how replay-ratio concentration controls update error, and how trajectory extrapolation with periodic recovery affects evaluation regret. A cost comparison also depends on rollout, update, environment, storage, and evaluation work.

Finite counterexamples show why these questions need additional assumptions. Finite policies can separate the entropy statistics above. Two verifiers can agree on all observed training records yet differ on an unobserved adversarial region. Two parameter paths can share early low-rank summaries and diverge after a new successful rollout. These examples mark where additional assumptions are unavoidable.

The DGG and NExt papers motivate reuse and trajectory-compression hypotheses but do not establish these general results. Bounds on update error and extrapolation regret require assumptions relating the monitored quantities to the corresponding outcomes.

Finite consequences and obstructions [ftip-0075]

This section turns parts of the preceding setup into finite mathematical statements. The results concern declared probability spaces, finite decision sets, recorded feedback, or explicitly stated matrix models. They do not by themselves establish a scaling law for language-model capability.

We begin with discovery and support, then study proxy rewards and verifiers. The third subsection asks what can be learned from a fixed feedback transcript. The last subsection studies the DGG inequalities before introducing trajectory-compression diagnostics and counterexamples.

Finite discovery and support [ftip-0076]

A successful continuation can have positive probability and still be difficult to observe within a finite rollout budget. This subsection separates the probability of observing a success from support, coverage, and capability acquisition.

Convention. Repeated attempts and the discovery event [ftip-0077]

Fix a task instance, evaluator, success threshold, and a joint law for $B\geq 1$ attempts. Let $E_i$ be the event that attempt $i$ succeeds. The discovery event by attempt $B$ is

\[ D_B=\bigcup _{i=1}^{B}E_i. \]

Write $p_i=\Pr (E_i)$. Independence is an additional property of the declared joint law; it is not implied by repeated decoding, a shared model, or a common environment. In the independent and identically distributed case, write $p_i=p$. When the attempt law is the one fixed in Definition [ftip-005U], this $p$ is the corresponding successful-support probability.

Theorem. Discovery under independent attempts [ftip-0078]

Assume the events $E_1,\ldots ,E_B$ of Convention [ftip-0077] are independent. Then

\[ \Pr (D_B)=1-\prod _{i=1}^{B}(1-p_i). \]

In particular, independent attempts with a common success probability $p$ satisfy

\[ \Pr (D_B)=1-(1-p)^B. \]

Proof.

The complement of discovery is $D_B^{\mathsf c}=\bigcap _{i=1}^{B}E_i^{\mathsf c}$. Independence gives $\Pr (D_B^{\mathsf c})=\prod _i\Pr (E_i^{\mathsf c}) =\prod _i(1-p_i)$. Taking complements proves the first identity; substituting $p_i=p$ proves the specialization.

This finite statement follows from the displayed hypotheses. The calculation in Example [ftip-0067] is its ten-attempt numerical preview.

Corollary. A rollout budget for target discovery probability [ftip-0079]

This is a finite corollary of Theorem [ftip-0078], obtained from its displayed hypotheses.

Let $0<\delta <1$. Under the independent and identically distributed setup of Theorem [ftip-0078], suppose first that $0<p<1$. The least positive integer budget whose discovery failure probability is at most $\delta $ is

\[ B_{\min } =\left \lceil \frac {\log \delta }{\log (1-p)}\right \rceil . \]

If $p=0$, no finite positive budget reaches failure probability below one. If $p=1$, one attempt suffices.

Proof.

For $0<p<1$, Theorem [ftip-0078] gives failure probability $(1-p)^B$. The inequality $(1-p)^B\leq \delta $ is equivalent to $B\geq \frac {\log \delta }{\log (1-p)}$ because both logarithms are negative. The least integer solution is the displayed ceiling. The two boundary cases follow directly from $(1-p)^B$.

Lemma. Discovery bounds from conditional success rates [ftip-007A]

Put $D_0=\varnothing $. Whenever $\Pr (D_{i-1}^{\mathsf c})>0$, define the surviving conditional success rate

\[ q_i=\Pr (E_i\mid D_{i-1}^{\mathsf c}). \]

If these rates are defined through step $B$, then

\[ \Pr (D_B^{\mathsf c})=\prod _{i=1}^{B}(1-q_i). \]

Consequently, if $0\leq \underline q\leq q_i\leq \overline q\leq 1$ for every surviving step, then

\[ 1-(1-\underline q)^B \leq \Pr (D_B) \leq 1-(1-\overline q)^B. \]

Proof.

The chain rule gives $\Pr (D_i^{\mathsf c})=\Pr (D_{i-1}^{\mathsf c})(1-q_i)$. Iteration proves the product identity, and the coordinatewise bounds on $1-q_i$ give the two discovery bounds.

No independence assumption is used. The conditional rates may change with the earlier failures, an adaptive decoder, or a changing environment state.

This finite statement follows from the displayed hypotheses.

Remark. Discovery is neither acquisition nor coverage [ftip-007B]

The discovery event concerns whether a declared sampling procedure observes at least one success. Increasing its probability can be an elicitation effect in the sense of Definition [ftip-0005]: more attempts expose behaviour that already had positive probability. It is not by itself the independent before--after comparison required by Definition [ftip-0007].

Successful-support coverage in Definition [ftip-005V] integrates an instance-level threshold over a task law. A large discovery probability on one instance therefore does not establish broad coverage or the uniform task-family witness of Definition [ftip-005X].

Theorem. Finite exponential tilting preserves support [ftip-007C]

Fix a prompt $x$, a finite response set $\mathcal Y$, a reference policy $\pi _{\mathrm {ref}}$, a finite real reward $r(x,y)$, and $\beta >0$. Let $\pi _r$ be the normalized exponential tilt of Definition [ftip-003O]. Then, for every $y\in \mathcal Y$,

\[ \pi _r(y\mid x)>0 \quad \Longleftrightarrow \quad \pi _{\mathrm {ref}}(y\mid x)>0. \]

Proof.

The multiplier $\exp (r(x,y)/\beta )$ is finite and strictly positive. The finite normalizer $Z_r(x)$ is also strictly positive. Multiplication by the first quantity and division by the second therefore preserve whether the reference mass is zero or positive.

The source policy form appears as equation (4) in [rafailov2023direct, Section 4]; the displayed result is the finite corollary derived here. Infinite rewards, non-normalizable response spaces, approximate optimization, and changes to the generation mechanism lie outside the statement.

Remark. Formal support and operational discoverability [ftip-007D]

Support preservation is a statement about exact positive probability. A response can remain in mathematical support while its mass becomes too small to observe within the available rollout budget. The identities in Theorem [ftip-0078] and Corollary [ftip-0079] quantify this distinction for declared independent attempts.

Top-$k$ truncation, nucleus sampling, finite numerical precision, context limits, and parser constraints can also remove an operational route even when the underlying softmax law assigns it positive mass. Claims about elicitation must therefore name both the policy law and the executable inference procedure.

Example. Counterexample: Equal one-shot success, unequal successful coverage [ftip-007E]

Let the evaluation law be uniform on two instances $x_1,x_2$. Consider protocols $P$ and $P'$ with successful-support probabilities

\[ \bigl (p_P(x_1),p_P(x_2)\bigr ) =\left (\frac 12,\frac 12\right ), \qquad \bigl (p_{P'}(x_1),p_{P'}(x_2)\bigr ) =(1,0). \]

Both protocols have mean one-shot success $1/2$. At threshold $\alpha =1/2$, however, Definition [ftip-005V] gives

\[ \operatorname {Cov}_{1/2}(P)=1, \qquad \operatorname {Cov}_{1/2}(P')=\frac 12. \]

Thus average success does not determine successful-support coverage. The example is finite and uses the same task law and threshold for both protocols; it does not compare their training costs or establish acquisition.

Proxy reward and verifier fidelity [ftip-007F]

A proxy can guide an update or rank sampled candidates without agreeing with the independently declared utility of the resulting output. This subsection separates pointwise proxy error from average error under a shifted law. It then distinguishes repeated false-accept events from the output of a candidate selector. Pointwise approximation and explicit probability assumptions support different guarantees; an average audit number alone supplies neither.

Definition. Uniform proxy approximation [ftip-007G]

Let $\mathcal Y_0$ be a nonempty finite set of candidate outcomes. Let $U:\mathcal Y_0\to \mathbb R$ be an independently declared utility and $\widehat U:\mathcal Y_0\to \mathbb R$ a proxy score. The proxy is a uniform approximation to $U$ on $\mathcal Y_0$ with error $\varepsilon $ when $\varepsilon \geq 0$ and

\[ \sup _{y\in \mathcal Y_0}\lvert \widehat U(y)-U(y)\rvert \leq \varepsilon . \]

The candidate set is part of the claim. Approximation on training outcomes, on one policy's support, or in expectation is not uniform approximation on a larger set reached after optimization.

Theorem. Uniform proxy error gives two-epsilon selection regret [ftip-007H]

Under the conditions of Definition [ftip-007G], choose

\[ y^\star \in \operatorname *{arg\,max}_{y\in \mathcal Y_0}U(y), \qquad \widehat y\in \operatorname *{arg\,max}_{y\in \mathcal Y_0}\widehat U(y). \]

Then the utility regret of selecting by the proxy satisfies

\[ 0\leq U(y^\star )-U(\widehat y)\leq 2\varepsilon . \]

Proof.

Optimality of $\widehat y$ for the proxy and the two pointwise error bounds give

\[ \begin {aligned} U(y^\star ) &\leq \widehat U(y^\star )+\varepsilon \\ &\leq \widehat U(\widehat y)+\varepsilon \\ &\leq U(\widehat y)+2\varepsilon . \end {aligned} \]

Optimality of $y^\star $ for $U$ gives the lower bound.

This finite statement follows from the displayed hypotheses.

Remark. The two-epsilon theorem is a decision bound [ftip-007I]

The bound in Theorem [ftip-007H] follows directly from the uniform approximation condition. It compares two maximizers on one fixed finite candidate set. It does not describe how either score was learned or how a policy changes when that score becomes a training objective.

The result is useful because its transfer assumption is visible. To apply it after an update or a larger search, the uniform bound must hold on the outcomes that the new procedure can actually select. The verifier evidence of Definition [ftip-0047] can support such an audit, but reproducible evidence alone does not establish the bound.

Example. Counterexample: Small average proxy error fails after distribution shift [ftip-007J]

Let the finite decision set be $\mathcal Y_0=\{y_{\rm safe},y_{\rm exploit}\}$. Define its utility and proxy utility by the following table.

\[ \begin {array}{c|cc} &y_{\rm safe}&y_{\rm exploit}\\ \hline U&1&0\\ \widehat U&1&2. \end {array} \]

For $0<\delta <1$, let a reference law $P_\delta $ assign probability $\delta $ to $y_{\rm exploit}$. Its mean absolute proxy error is

\[ \mathbb E_{P_\delta }\lvert \widehat U-U\rvert =2\delta , \]

which tends to zero with $\delta $. Nevertheless, proxy maximization selects $y_{\rm exploit}$ and incurs utility regret $1$. Under the shifted law concentrated on that selected outcome, the mean error is $2$. Thus no vanishing selection-regret bound can depend only on average error under the reference law.

This two-outcome construction isolates the distribution-shift warning in Remark [ftip-0061]. It does not claim that a particular learned verifier has this error profile; Example [ftip-0060] gives a source-linked checker instance with the same proxy-versus-utility separation.

Definition. Repeated verifier audit [ftip-007K]

A repeated verifier audit of size $N\geq 1$ records pairs $(S_i,A_i)\in \{0,1\}^2$ for candidates $1\leq i\leq N$. Here $S_i$ is an independently defined success label and $A_i$ is the verifier decision, as in Definition [ftip-005Z]. Define

\[ C_i^{\rm fa}=\{S_i=0,\ A_i=1\}, \qquad \mathcal E_N^{\rm fa}=\bigcup _{i=1}^N C_i^{\rm fa}. \]

The event $\mathcal E_N^{\rm fa}$ says that at least one invalid candidate was accepted. It does not say which candidate a downstream selector returns. Write $\phi _i=\Pr (C_i^{\rm fa})$. When $\iota _i=\Pr (S_i=0)>0$, the conditional false-accept rate $\eta _{+,i}=\Pr (A_i=1\mid S_i=0)$ is defined and $\phi _i=\iota _i\eta _{+,i}$.

Theorem. Independent false accepts amplify with candidate count [ftip-007L]

In the repeated audit of Definition [ftip-007K], suppose the pairs $(S_i,A_i)$ are independent and identically distributed. Assume

\[ \iota =\Pr (S_i=0)>0, \qquad \eta _+=\Pr (A_i=1\mid S_i=0). \]

Then the probability that at least one accepted invalid candidate exists among the $N$ audited candidates is

\[ \Pr (\mathcal E_N^{\rm fa})=1-(1-\iota \eta _+)^N. \]

It is strictly increasing in $N$ when $0<\iota \eta _+<1$, and it converges to $1$ when $\iota \eta _+>0$.

Proof.

Each $C_i^{\rm fa}$ has probability $\iota \eta _+$. Independence of the audited pairs makes the events $C_i^{\rm fa}$ independent. Therefore

\[ \Pr \left ((\mathcal E_N^{\rm fa})^c\right ) =\Pr \left (\bigcap _{i=1}^N (C_i^{\rm fa})^c\right ) =\prod _{i=1}^N(1-\iota \eta _+) =(1-\iota \eta _+)^N. \]

Taking complements gives the equality. The monotonicity and limit follow from the elementary powers of $1-\iota \eta _+$.

This finite statement follows from the displayed hypotheses.

Theorem. A false-accept union bound without independence [ftip-007M]

For any joint law of a repeated verifier audit,

\[ \Pr (\mathcal E_N^{\rm fa})\leq \sum _{i=1}^N \phi _i. \]

If $\iota _i=\Pr (S_i=0)>0$ for every $i$, this becomes

\[ \Pr (\mathcal E_N^{\rm fa})\leq \sum _{i=1}^N \iota _i\eta _{+,i}. \]

Proof.

Apply the union bound to $\mathcal E_N^{\rm fa}=\bigcup _i C_i^{\rm fa}$. The factorization $\phi _i=\iota _i\eta _{+,i}$ follows from conditional probability whenever $\iota _i>0$. If $\iota _i=0$, then $\phi _i=0$ while the corresponding conditional rate is undefined, so the first display remains the unconditional statement.

Without a dependence assumption, matching marginal error rates do not give the equality in Theorem [ftip-007L]; the events $C_i^{\rm fa}$ could coincide.

This finite statement follows from the displayed hypotheses.

Example. Weight updates and candidate selection are different proxy operators [ftip-007N]

A proxy score can enter a training operator or a selection operator. In the first route it changes the policy; in the second it ranks a finite sample from a policy whose weights remain fixed.

$\text{[math]}$

The lifecycle accounting of Definition [ftip-005H] places the first route's rollout and update work in separate coordinates. The second route incurs rollout and evaluation work but no parameter update. Consequently, the event $\mathcal E_N^{\rm fa}$ in Definition [ftip-007K] concerns the candidate set presented to selection; it is not a claim about the selected output or a training-time policy change.

Remark. What verifier error rates do not identify [ftip-007O]

The rates in Definition [ftip-005Z] are properties of a declared audit law. They do not identify where errors occur within the task family, how error events depend across repeated candidates, or which accepted candidate a selector returns. Theorem [ftip-007L] adds identical marginals and independence; Theorem [ftip-007M] retains only the union bound.

Nor do these rates determine the effect of optimizing against the verifier. That effect depends on how the induced policy or selection law moves mass toward particular outcomes. The incomplete-checker example Example [ftip-0060] demonstrates one such movement, while Example [ftip-007J] shows why an average reference-law error cannot rule it out. These are local finite statements. Any claim about a named verifier still requires its own audit distribution, evidence, and transfer argument.

The deterministic regret lemma Theorem [ftip-007H], the independent-audit identity Theorem [ftip-007L], and the union bound Theorem [ftip-007M] follow from uniform approximation, independent trials, and subadditivity, respectively. Their hypotheses need not hold for a verifier chosen only for its empirical performance.

Feedback identifiability and fixed records [ftip-007P]

A training protocol can respond only to distinctions present in its observations. The following finite setup makes that restriction precise before considering stronger, model-specific limits on preference data.

Definition. Finite declared feedback protocol [ftip-007Q]

Fix a horizon $N\geq 0$, finite action sets $\mathcal A_n$, finite feedback sets $\mathcal F_n$, and a finite internal-seed set $\mathcal U$ with a declared law $\lambda $. A finite declared protocol $\mathsf P$ consists of selection maps

\[ a_n:\mathcal U\times \prod _{j<n}(\mathcal A_j\times \mathcal F_j) \longrightarrow \mathcal A_n \qquad (0\leq n<N). \]

An action may specify the task, rollout record, feedback request, or sampling choice made at that round. The protocol can depend on earlier exposed feedback and its declared seed, but it cannot depend on a latent world field that has not entered those arguments.

Definition. Finite feedback world [ftip-008C]

For the action and feedback sets of Definition [ftip-007Q], a finite feedback world $w$ supplies, for each round $0\leq n<N$, a probability kernel $K_{n,w}$ from a generic action--feedback history $(a_0,f_0,\ldots ,a_n)$ ending in $a_n\in \mathcal A_n$ to $\mathcal F_n$. Thus, whenever $a_j\in \mathcal A_j$ and $f_j\in \mathcal F_j$,

\[ K_{n,w}(\,\cdot \mid a_0,f_0,\ldots ,a_n) \in \Delta (\mathcal F_n). \]

Two worlds may agree on every feedback kernel queried by a protocol while differing in latent utility, unrequested labels, or unobserved environment facts. Those latent fields are not feedback until a declared query exposes them.

Definition. Observed feedback transcript [ftip-008D]

Fix a finite declared protocol $\mathsf P$ from Definition [ftip-007Q] and a finite feedback world $w$ from Definition [ftip-008C]. Draw $U\sim \lambda $ and, for $0\leq n<N$, set

\[ \begin {aligned} A_n&=a_n(U,A_0,F_0,\ldots ,A_{n-1},F_{n-1}),\\ F_n&\sim K_{n,w}(\,\cdot \mid A_0,F_0,\ldots ,A_n). \end {aligned} \]

The resulting observed feedback transcript is

\[ T_w^{\mathsf P}=(U,A_0,F_0,\ldots ,A_{N-1},F_{N-1}). \]

All protocol randomness is included in $U$. Immutable rollout records and typed feedback events may be components of the finite action and feedback sets, as in Definition [ftip-0051] and Definition [ftip-0053].

Definition. Observational equivalence for a declared protocol [ftip-007R]

Let $\mathcal T_{\mathsf P}$ be the finite set of possible transcripts for a declared protocol $\mathsf P$. Two feedback worlds $w_0,w_1$ are observationally equivalent for $\mathsf P$, written $w_0\equiv _{\mathsf P}w_1$, when

\[ \Pr (T_{w_0}^{\mathsf P}=t)=\Pr (T_{w_1}^{\mathsf P}=t) \qquad \text {for every }t\in \mathcal T_{\mathsf P}. \]

The relation is protocol-relative. Another protocol may issue a different feedback request and thereby separate the same worlds. Equality only on the realized transcript is weaker than this definition, which compares the whole finite transcript law.

Theorem. Post-training cannot distinguish observationally equivalent worlds [ftip-007S]

For a finite random variable $X$, write $\operatorname {Law}(X)$ for its probability law. If $h$ maps the value space of $X$ to a finite set $\mathcal Y$, its pushforward law $h_{\#}\operatorname {Law}(X)$ is characterized by \[ \bigl (h_{\#}\operatorname {Law}(X)\bigr )(\{y\}) =\Pr (h(X)=y),\qquad y\in \mathcal Y. \]

Let $w_0\equiv _{\mathsf P}w_1$. For any finite output set $\mathcal Y$ and any readout $h:\mathcal T_{\mathsf P}\to \mathcal Y$,

\[ h_{\#}\operatorname {Law}(T_{w_0}^{\mathsf P}) =h_{\#}\operatorname {Law}(T_{w_1}^{\mathsf P}). \]

Thus a final artifact stamp, update decision, or test chosen solely from the declared protocol's transcript has the same distribution in both worlds.

Proof.

For every $y\in \mathcal Y$, finiteness gives

\[ \begin {aligned} \Pr \left (h(T_{w_0}^{\mathsf P})=y\right ) &=\sum _{\substack {t\in \mathcal T_{\mathsf P}\\h(t)=y}} \Pr (T_{w_0}^{\mathsf P}=t)\\ &=\sum _{\substack {t\in \mathcal T_{\mathsf P}\\h(t)=y}} \Pr (T_{w_1}^{\mathsf P}=t) =\Pr \left (h(T_{w_1}^{\mathsf P})=y\right ). \end {aligned} \]

The middle equality is observational equivalence from Definition [ftip-007R]. These point probabilities determine the two pushforward laws.

This finite statement follows from the displayed hypotheses.

Remark. A no-free-feedback result, not a no-learning result [ftip-007T]

Theorem [ftip-007S] says that the declared observations supply no information that distinguishes $w_0$ from $w_1$. It does not say that the output must equal the initial model, that the parameters cannot change, or that performance cannot improve in both worlds. Pretraining, inductive bias, computation on the observed records, and generalization may still produce an improvement shared by the two worlds.

Calling the theorem a no-learning result would therefore erase the central condition: only world-dependent conclusions unavailable from the common transcript are ruled out.

Corollary. Replaying a fixed pool supplies no new feedback information [ftip-007U]

Fix a realized finite replay pool $d$ as in Definition [ftip-0056]. Let $\mathcal V$ be a finite seed set, and let $V\in \mathcal V$ be a replay-and-update seed with the same declared law in two feedback worlds and whose law does not depend on the world. If a replay-only procedure makes no new environment or feedback query, then its output has the form $Y=h(d,V)$. The law of $Y$ is the same in the two worlds.

Proof.

For every output $y$,

\[ \Pr (h(d,V)=y)= \sum _{\substack {v\in \mathcal V\\h(d,v)=y}}\Pr (V=v). \]

The fixed pool, seed law, and readout are identical in the two worlds, so the displayed sum is identical. Equivalently, this is the pushforward argument of Theorem [ftip-007S] applied after conditioning on the realized pool.

This finite statement follows from the displayed hypotheses.

Remark. Reuse can change optimization without enlarging evidence [ftip-007V]

A replay schedule can reweight records, reduce optimization error on the fixed pool, or produce a different update proposal in the sense of Definition [ftip-006C]. Those are genuine computational effects. They do not add a label, preference, verifier result, or environment transition to the fixed evidence. Fresh optimizer randomness is likewise not feedback about the latent world.

This distinction prevents sample reuse from being counted as feedback acquisition. It does not imply that replay is useless; it isolates the source of any benefit as further computation on already acquired records.

Example. Counterexample: An off-query utility reversal [ftip-007W]

Let the response set be $\{a,b\}$. A one-round protocol queries only $x_0$; both worlds return the preference $a\succ b$. The protocol therefore has the same transcript law in both worlds. Fix a deterministic transcript-to-policy readout that chooses $a$ also at an unqueried $x_1$. The worlds agree on every observed field but reverse utility at $x_1$:

$\text{[math]}$

The transcript laws are identical, so Theorem [ftip-007S] applies, while the deterministic readout returns the same output in both worlds. That output is optimal in $w_+$ and suboptimal in $w_-$ on $x_1$. The example does not show that generalization always fails. It shows that a claim about unqueried utility requires an assumption connecting observed feedback to that utility.

Remark. From finite indistinguishability to preference-data limits [ftip-007X]

[zhao2025limits, Theorems 3.3--3.5] study a more structured post-training model. They give ordinal-preference distortion lower bounds, including a lower bound under Bradley--Terry noise with linear scores, and a positive result using a limited number of cardinal queries. These results depend on the routing model, utility class, and query budget developed in § [ftip-008X].

Theorem [ftip-007S] gives a finite pushforward theorem, while Example [ftip-007W] gives a counterexample to identification from an incomplete transcript. They are neither proofs nor special cases of the paper's distortion results.

Replay monitors and trajectory compression [ftip-007Y]

Reusing a rollout and jumping along a predicted checkpoint path save different kinds of work. The first changes how often recorded feedback enters an update. The second replaces some realized training steps by a parameter forecast. This subsection records what their proposed monitors establish and, separately, what remains unmeasured.

Theorem. DGG head gradient and shared-weight occurrence sum [ftip-007Z]

Fix a sampled history and an active token $i$ in the interior of the unclipped branch of the GRPO objective. Consider a finite directed acyclic computation graph that is differentiable at the parameter point in question. Let $a_i$ be the sampled token, $\widehat A_i$ its fixed normalized advantage, and $p_i=\operatorname {softmax}(z_i)$ the current distribution on a finite vocabulary $\mathcal V$. With fixed behavior probability $b_i>0$, set \[ r_i=\frac {p_i(a_i)}{b_i}, \qquad \mathcal L_i=r_i\widehat A_i, \qquad E_i=r_i\widehat A_i(e_{a_i}-p_i). \] Here $e_{a_i}\in \mathbb R^{|\mathcal V|}$ is the sampled-token basis vector. The history and sampling decisions are held fixed during differentiation.

Suppose the output head $W_{\rm lm}\in \mathbb R^{|\mathcal V|\times d_{\rm model}}$ is untied: its only path to $\mathcal L_i$ is through $z_i=W_{\rm lm}h_{L,i}$, and $h_{L,i}\in \mathbb R^{d_{\rm model}}$ is independent of $W_{\rm lm}$. Then \[ G_i^{\rm lm}=\nabla _{W_{\rm lm}}\mathcal L_i =E_i h_{L,i}^{\mathsf T}. \]

Let $W_{\rm int}\in \mathbb R^{m\times d}$ be a shared intermediate weight. Index by a finite set $\mathcal O_i$ every occurrence of this weight that can affect $z_i$, assuming that all such uses have the form $y_o=W_{\rm int}x_o$, with $x_o\in \mathbb R^d$. First replace these uses by independent copies $W_o$ and evaluate them all at $W_o=W_{\rm int}$. Let $J_{io}\in \mathbb R^{|\mathcal V|\times m}$ be the downstream Jacobian from node $y_o$ to $z_i$ in this graph, with the other weight copies fixed. Define the contribution of occurrence $o$ by \[ H_{io}=\nabla _{W_o}\mathcal L_i =(J_{io}^{\mathsf T}E_i)x_o^{\mathsf T}. \] On tying the copies, the total derivative is \[ G_i^{\rm int}=\nabla _{W_{\rm int}}\mathcal L_i =\sum _{o\in \mathcal O_i}H_{io}. \] The head gradient and each occurrence contribution have rank at most one; the total shared-weight gradient need not.

For a layer used once per position in a causal network, the sum includes every earlier position whose output affects token $i$. Reuse across depth adds further occurrences. Tying the head to embeddings or other blocks also requires their contributions; the displayed head identity assumes that such tying is absent.

Proof.

Softmax differentiation gives $\nabla _{z_i}\mathcal L_i=E_i$. The outer-product rule gives the untied head identity. In the graph with independent copies, $x_o$ does not depend on its own $W_o$; all downstream paths from $y_o$ are included in $J_{io}$. The chain rule therefore gives $\nabla _{W_o}\mathcal L_i=(J_{io}^{\mathsf T}E_i)x_o^{\mathsf T}$. Finally, the derivative of the diagonal map $W_{\rm int}\mapsto (W_o=W_{\rm int})_{o\in \mathcal O_i}$ adds these partial derivatives.

Clipped or inactive terms require their own derivative or mask. [miao2026when, Section 4.2.1, Proposition 1, and Appendix A.1] derives the intermediate outer product through one local application. For a shared weight, that calculation gives $H_{io}$; identifying it with the total $G_i^{\rm int}$ omits the other occurrences. The sum above supplies the chain rule needed for shared parameters.

Theorem. DGG occurrence-to-head gradient-energy bound [ftip-0080]

Use the per-token quantities and untied head of Theorem [ftip-007Z], and fix one occurrence $o\in \mathcal O_i$. Assume $d_{\rm model}\geq 1$ and positive constants $\alpha _{\min }$, $\beta _{\max }$, and $C$ satisfy \[ \|h_{L,i}\|_2^2\geq \alpha _{\min }d_{\rm model}, \qquad \|x_o\|_2^2\leq \beta _{\max }d_{\rm model}, \] and the two logit-sensitivity bounds \[ \mathbb E_{a\sim p_i} \|(J_{io})_{a,:}\|_2^2\leq C, \qquad \|(J_{io})_{a_i,:}\|_2^2\leq C. \] If $\widehat A_i\neq 0$ and $p_i(a_i)<1$, then \[ \frac {\|H_{io}\|_F^2}{\|G_i^{\rm lm}\|_F^2} \leq \frac {\mathcal C_{\rm occ}} {(1-p_i(a_i))^2}, \qquad \mathcal C_{\rm occ} =\frac {4\beta _{\max }C}{\alpha _{\min }}. \]

Proof.

The sampled-token coordinate of $E_i$ and the lower activation bound give \[ \|G_i^{\rm lm}\|_F^2 =\|E_i\|_2^2\|h_{L,i}\|_2^2 \geq r_i^2\widehat A_i^2 (1-p_i(a_i))^2 \alpha _{\min }d_{\rm model}. \] Write $J_{io}^{\mathsf T}E_i$ as $r_i\widehat A_i$ times the difference between the sampled row and the $p_i$-weighted mean row. The squared-norm inequality $\|u-v\|_2^2\leq 2\|u\|_2^2+2\|v\|_2^2$, Jensen's inequality for that mean, and the two row-energy bounds give $\|J_{io}^{\mathsf T}E_i\|_2^2\leq 4r_i^2\widehat A_i^2C$. Hence \[ \|H_{io}\|_F^2 \leq 4r_i^2\widehat A_i^2C\, \beta _{\max }d_{\rm model}. \] The source's softmax probabilities make $r_i>0$; the nonzero-advantage and nonunit-probability hypotheses make the head lower bound positive. Division and cancellation give the displayed ratio.

This bounds one occurrence contribution, not the total derivative of a shared intermediate weight. For the latter, the sum in Theorem [ftip-007Z] gives only \[ \|G_i^{\rm int}\|_F \leq \sum _{o\in \mathcal O_i} \|J_{io}^{\mathsf T}E_i\|_2\|x_o\|_2. \] If the same displayed hypotheses hold for every occurrence, the triangle inequality yields the ratio bound $|\mathcal O_i|^2\mathcal C_{\rm occ}/(1-p_i(a_i))^2$ for $\|G_i^{\rm int}\|_F^2/\|G_i^{\rm lm}\|_F^2$. Controlling only the application at position $i$ does not establish this all-occurrence hypothesis or the source's claimed bound for the total gradient. The contributing positions, reuse pattern, and Jacobians depend on the architecture. Neither bound establishes a small gradient or an evaluation-score comparison.

The activation and row-energy hypotheses and the local inequality follow the calculation in [miao2026when, Section 4.2.1, Lemma 1, Assumption 1, Theorem 1, and Appendix A.3], with the differentiated object restricted to an occurrence contribution. They do not prove the source's total shared-weight claim under its local hypotheses.

Theorem. DGG finite-batch head-gradient inequality [ftip-0081]

For $T\geq 1$ active, unclipped tokens with fixed sampled histories and the untied output head of Theorem [ftip-007Z], let \[ G^{\rm lm}=\frac 1T\sum _{i=1}^T G_i^{\rm lm}, \qquad \overline {r^2}=\frac 1T\sum _{i=1}^T r_i^2, \] and set \[ c_{\max }= \max _{1\leq i\leq T} \widehat A_i^2 \|e_{a_i}-\pi _\theta (\mathord \cdot \mid h_{L,i})\|_2^2 \|h_{L,i}\|_2^2. \] Then \[ \|G^{\rm lm}\|_F^2\leq c_{\max }\overline {r^2} =c_{\max }(1+\widehat \chi ^2), \] where $\widehat \chi ^2$ is the statistic of Definition [ftip-0066]. If $c_{\max }>0$, rearrangement gives the source's equivalent direction \[ \widehat \chi ^2\geq \frac {\|G^{\rm lm}\|_F^2}{c_{\max }}-1. \]

Proof.

Convexity of the squared Frobenius norm and the factorization in Theorem [ftip-007Z] give \[ \left \|\frac 1T\sum _iG_i^{\rm lm}\right \|_F^2 \leq \frac 1T\sum _i\|G_i^{\rm lm}\|_F^2 \leq \frac {c_{\max }}T\sum _i r_i^2. \] The identity $\overline {r^2}=1+\widehat \chi ^2$ follows directly from Definition [ftip-0066]; division by positive $c_{\max }$ gives the final form.

The inequality points from observed head-gradient energy to a lower bound on this finite-batch squared-ratio statistic. It is not a converse: cancellation can hide nonunit ratios. The active cancellation example in Example [ftip-0084] states its clipping interval explicitly. The proof uses no intermediate-weight bound.

The finite-batch inequality follows the calculation in [miao2026when, Section 4.2.2, Lemma 2, Theorem 2, and Appendix B], under the untied-head and fixed-history assumptions stated above.

Remark. DGG monitors update geometry, not evaluation safety [ftip-0082]

The identities in Theorem [ftip-007Z]--Theorem [ftip-0081] concern gradients, importance ratios, activations, and occurrence-specific logit Jacobians. The intermediate bound controls a local contribution; a shared-weight bound needs control over all contributing occurrences. None of their hypotheses mentions the fixed independent-evaluation functional $J_{\rm ev}$ of Definition [ftip-005E]. They therefore cannot imply that accepting an update preserves $J_{\rm ev}$, or that rejecting one would have prevented a decrease.

DGG adds an empirical policy on top of those identities: it monitors the increment in head-gradient energy, standardizes that increment against a trailing window, and rejects some reused updates before the optimizer step [miao2026when, Section 5 and Algorithm 1]. Its reported experiments relate that policy to observed training stability. They do not turn the Z-score into a calibrated test of independent-evaluation safety.

Example. Counterexample: empirical squared-ratio excess can be negative [ftip-0083]

Take one observed action $a$ with $\pi _{\rm old}(a)=1/2$ and $\pi _\theta (a)=1/4$. The batch contains only that action, so $T=1$ and $r_1=1/2$. Hence \[ \widehat \chi ^2=r_1^2-1=-\frac 34. \] Both policies can be completed on a two-action space by assigning their remaining mass to the other action.

Thus the finite-batch statistic in Definition [ftip-0066] need not share the nonnegativity of the population Pearson divergence. Theorem Theorem [ftip-0081] remains valid: its right side contains $1+\widehat \chi ^2=1/4=\overline {r^2}$.

Example. Counterexample: active batch cancellation can hide nonunit ratios [ftip-0084]

Fix $\epsilon _{\rm clip}\in (0,1)$ and $1<R<1+\epsilon _{\rm clip}$. On a two-token vocabulary, let both observed tokens have $a_i=1$, current probability $\pi _\theta (1)=1/2$, and behavior probability $\pi _{\rm old}(1)=1/(2R)$. Take scalar hidden states $h_{L,1}=h_{L,2}=1$ and fixed normalized advantages $\widehat A_1=1$, $\widehat A_2=-1$. These may be selected token terms from different rollout members. An untied head with both logits zero realizes the current probabilities. Both terms lie strictly inside the clipping interval of Definition [ftip-003F], so the clipped surrogate locally agrees with $\mathcal L_i=r_i\widehat A_i$. Thus $r_1=r_2=R$, and Theorem [ftip-007Z] gives \[ G_1^{\rm lm}=R(1/2,-1/2)^{\mathsf T}, \qquad G_2^{\rm lm}=-G_1^{\rm lm}. \] Consequently $G^{\rm lm}=0$, while $\widehat \chi ^2=R^2-1>0$. For this fixed clipping rule, the statistic in this construction is bounded above by $(1+\epsilon _{\rm clip})^2-1$.

For the raw surrogate $\mathcal L_i=r_i\widehat A_i$ considered as a separate objective, the same algebra works at every $R>1$ and yields arbitrarily large squared-ratio excess. This does not extend the active construction to arbitrary $R$ under clipping: when $R>1+\epsilon _{\rm clip}$, the positive-advantage term is locally constant, while the negative term remains active. Their clipped gradients are then $0$ and $-R(1/2,-1/2)^{\mathsf T}$, with nonzero mean $(-R/4,R/4)^{\mathsf T}$.

The example does not contradict Theorem [ftip-0081]: that theorem upper-bounds batch-gradient energy by a ratio moment. It supplies no lower bound on the gradient and no converse from zero batch-gradient energy to ratios equal to one. The arbitrary-ratio version concerns only the separate raw surrogate.

Example. Counterexample: a Z-score gate is not an evaluation-safety certificate [ftip-0085]

Let two consecutive proposed reused updates have the same monitored energy, $g_{t-1}=g_t=1$. With trailing-window increment mean $\mu _t=0$, any positive scale $\sigma _t+\varepsilon $, and any positive threshold, DGG's score is \[ z_t=\frac {(g_t-g_{t-1})-\mu _t}{\sigma _t+\varepsilon }=0, \] so the gate accepts the proposal. Now choose a fixed evaluation interface for which the artifact before the accepted update has performance $J_{\rm ev}=1$ and the artifact after it has performance $J_{\rm ev}=0$.

This is consistent because the gate rule of Definition [ftip-005A] imposes no mathematical relation between its scalar monitor and $J_{\rm ev}$. A safety claim would need an additional assumption connecting proposed updates to the independent evaluation law; the Z-score calculation alone cannot provide it.

Definition. Leading squared singular-energy share [ftip-0086]

Let $\Delta W\in \mathbb R^{m\times n}$ be nonzero, set $q=\min (m,n)$, and order its singular values as $\sigma _1\geq \cdots \geq \sigma _q\geq 0$. Its leading squared singular-energy share is \[ \rho _1(\Delta W) =\frac {\sigma _1(\Delta W)^2}{\sum _{j=1}^q\sigma _j(\Delta W)^2} =\frac {\sigma _1(\Delta W)^2}{\|\Delta W\|_F^2}. \] The nonzero hypothesis prevents an undefined $0/0$; the value lies in $[1/q,1]$.

NExt instead reports $E_1=\sigma _1/\sum _j\sigma _j$ in Section 3.2 Low-rank optimization trajectories modeling for LLM RLVR acceleration[chen2026lowrank]. That nuclear-share statistic and $\rho _1$ answer different questions. The squared share measures contribution to squared Frobenius norm.

Definition. Leading spectral gap [ftip-0087]

For $\Delta W\in \mathbb R^{m\times n}$ with $q=\min (m,n)\geq 2$, its leading spectral gap is \[ \operatorname {gap}_1(\Delta W) =\sigma _1(\Delta W)-\sigma _2(\Delta W). \] A positive gap makes the leading left and right one-dimensional singular subspaces unique, although each chosen singular vector still has an arbitrary sign. When the gap is zero, a single leading vector is not intrinsic.

The quantity is undefined when $q=1$, since there is no second singular value. Projector comparisons require a positive gap at every compared checkpoint.

Definition. Leading-subspace projector drift [ftip-0088]

Let $\Delta W_t$ and $\Delta W_s$ be nonzero matrices of the same shape, each with positive leading spectral gap. If $u_1(t)$ and $u_1(s)$ are unit leading left singular vectors, define their leading-subspace projector drift by \[ d_{\rm proj}(t,s) =\|\Pi _t^{\rm svd}-\Pi _s^{\rm svd}\|_F, \qquad \Pi _t^{\rm svd}=u_1(t)u_1(t)^{\mathsf T}, \quad \Pi _s^{\rm svd}=u_1(s)u_1(s)^{\mathsf T}. \] The definition is independent of both singular-vector signs and takes values in $[0,\sqrt {2}]$.

This is an additional proposed trajectory diagnostic; NExt does not define it.

A large value of $\rho _1$ from Definition [ftip-0086] says that one singular mode dominates a particular difference matrix. It does not say that the dominant subspace remains fixed across checkpoints; $d_{\rm proj}$ measures that separate question.

Definition. Path-emulation regret [ftip-0089]

Fix the evaluation interface of Convention [ftip-005D]. Let $P_{t+s}^{\rm act}$ be the protocol that returns the artifact reached after $s\geq 1$ additional realized training steps from checkpoint $t$. Let $\mathcal K_{\leq t}$ be the saved checkpoint history available through $t$, and let $\widehat P_{t+s}(\mathcal K_{\leq t})$ instead return the artifact forecast from that history. Their signed path-emulation regret and absolute path-emulation regret are \[ \begin {aligned} \operatorname {Reg}^{\rm path}_{\rm ev}(t,s) &=J_{\rm ev}(P_{t+s}^{\rm act}) -J_{\rm ev}(\widehat P_{t+s}(\mathcal K_{\leq t})),\\ \operatorname {AReg}^{\rm path}_{\rm ev}(t,s) &=\left |\operatorname {Reg}^{\rm path}_{\rm ev}(t,s)\right |. \end {aligned} \]

The sign distinguishes an optimistic forecast from a pessimistic one; the absolute value measures discrepancy without cancellation across checkpoints. Both use the same task law, inference budget, and evaluator. Parameter error alone is not substituted for evaluation error. The NExt extrapolation and recovery schedule described in Sections 4.1--4.3 and 5.1 motivates this comparison Low-rank optimization trajectories modeling for LLM RLVR acceleration[chen2026lowrank], but the paper does not state this regret definition.

Example. Counterexample: early low-rank agreement does not determine continuation [ftip-008A]

Let $e_1=(1,0)^{\mathsf T}$ and $e_2=(0,1)^{\mathsf T}$ be the standard basis vectors of $\mathbb R^2$. In a two-dimensional matrix chart, set $P=e_1e_1^{\mathsf T}$ and $Q=e_2e_2^{\mathsf T}$. Two paths share the entire observed history \[ W_0=0,\qquad W_1=P,\qquad W_2=2P. \] Every observed local difference is the same rank-one matrix $P$. The paths then fork: path A takes $W_3^A=3P$, whereas path B takes $W_3^B=2P+Q$. Each next local difference is again rank one.

$\text{[math]}$

A deterministic history-only forecaster must return the same matrix $\widehat W_3$ in both worlds. Since $\|W_3^A-W_3^B\|_F=\|P-Q\|_F=\sqrt {2}$, the triangle inequality forces its Frobenius error to be at least $1/\sqrt {2}$ on one continuation. Low-rank early motion therefore does not identify the next subspace or the next checkpoint.

Remark. Compression of a charted path is not capability acquisition [ftip-008B]

NExt extracts global, local, and target differences from LoRA checkpoint matrices, keeps leading singular factors, learns a nonlinear predictor, jumps in that parameter chart, and resumes RLVR [chen2026lowrank, Sections 3.2, 4.1--4.3, and 5.1]. Its reported results are empirical; the paper states no theorem that a leading subspace remains stable or that a forecast preserves a policy.

The quantities in Definition [ftip-0086]--Definition [ftip-0089] separate four tests. Squared singular-energy share concerns one matrix, spectral gap controls whether its leading subspace is identifiable, projector drift compares those subspaces over time, and path-emulation regret returns to fixed independent evaluation. None, by itself, is evidence that new feedback was acquired or that a new capability was learned. Such a claim must use a declared post-training gain such as Definition [ftip-005F] and must charge forecast training, the parameter jump, and recovery updates.

Alignment gain and preference information [ftip-008E]

Two finite models ask different questions about post-training. The first assumes an exact KL-regularized optimizer and characterizes the reward gain that its exponential tilt can produce. The second asks what ordinal comparisons reveal when post-training may reroute a fixed set of response circuits.

Neither source model is a general account of neural post-training. The first assumes exact optimization of a declared scalar reward. The second is a stylized routing model with a fixed circuit set. Extending the KL identity to an approximate optimizer requires further analysis. Extending the ordinal routing conclusion to a changing circuit set requires additional assumptions on that change.

Finite KL-regularized alignment [ftip-008F]

This subsection studies an exact, finite optimization model. A reference law is exponentially tilted by a declared scalar reward. In this setting, reward gain admits two exact descriptions: a Jeffreys-divergence identity and a covariance under the reference law.

The statements do not model approximate optimization, neural training cost, reward validity, or independent capability evaluation.

Remark. Exact optimization on a finite response set [ftip-008G]

Theorems 1 and 2 of the v1 preprint [paes2026theoretical] concern an exactly optimal KL-regularized policy and a fixed query. On a finite response set, their expectations and normalizers are ordinary finite sums.

These exact identities do not establish a comparison among best-of-$N$, PPO, and GRPO, or a general guarantee for proxy rewards and reward ensembles. Those questions require assumptions beyond exact optimization of one reward.

Notation. Translating the alignment source notation [ftip-008H]

The source writes $\lambda >0$ for the KL penalty. Here we write $\beta >0$, matching the DPO notation in Notation [ftip-003M]--Definition [ftip-003O]. This avoids collision with the generalized-advantage parameter in Definition [ftip-003E] and the protocol law in Definition [ftip-007Q].

The source's gain $\Delta (r,r')$ is written $G_p(r;s)$: $r$ is the reward used to tilt the reference law $p$, while $s$ is the reward used to evaluate the tilted law. This avoids collision with both the simplex notation $\Delta (X)$ of Notation [ftip-000A] and the paired DPO margin of Notation [ftip-003M]. For a finite set $\mathcal Y$, a law $p\in \Delta (\mathcal Y)$, and functions $f,g:\mathcal Y\to \mathbb R$, set

\[ \operatorname {Cov}_p(f,g) =\mathbb E_{y\sim p}[f(y)g(y)] -\mathbb E_{y\sim p}[f(y)]\, \mathbb E_{y\sim p}[g(y)]. \]

Definition. Finite KL-alignment instance [ftip-008I]

A finite KL-alignment instance consists of a nonempty finite response set $\mathcal Y$, a reference law $p\in \Delta (\mathcal Y)$, a penalty $\beta >0$, and finite real functions $r,s:\mathcal Y\to \mathbb R$. The reference support is

\[ S_p=\{y\in \mathcal Y:p(y)>0\}. \]

A prompt is fixed and suppressed. The function $r$ determines the alignment update; $s$ measures its result. They may coincide, but they need not. Finiteness makes every exponential weight and expectation below finite.

These objects specialize the fixed-query setting in [paes2026theoretical, Section 2, equations (2.1)--(2.2)]; they are not a definition of alignment in general.

Definition. Exponential weight and partition function [ftip-008J]

For a finite KL-alignment instance, define the exponential weight and partition function by

\[ a_r(y)=\exp \left (\frac {r(y)}{\beta }\right ), \qquad Z_r=\mathbb E_{y\sim p}[a_r(y)]. \]

Every $a_r(y)$ is finite and strictly positive. Since $p$ is a probability law on a nonempty finite set, $0<Z_r<\infty $.

This weight and normalizer are the fixed-query form of [paes2026theoretical, Equation (2.2)].

Definition. Aligned law in finite-source notation [ftip-008K]

The aligned law induced by $r$ is $q_r\in \Delta (\mathcal Y)$ given by

\[ q_r(y)=\frac {p(y)a_r(y)}{Z_r}. \]

Normalization follows from the definition of $Z_r$. Moreover, $q_r(y)>0$ exactly when $p(y)>0$. This is the finite notation for the same exponential-tilt optimizer stated in Definition [ftip-003O]; support preservation was proved in Theorem [ftip-007C].

This formula is [paes2026theoretical, Equation (2.2)]. It assumes the exact optimizer, not an iterate returned by a particular training algorithm.

Definition. Cross-reward gain under an alignment reward [ftip-008L]

The cross-reward gain from aligning with $r$ and evaluating with $s$ is

\[ G_p(r;s) =\mathbb E_{y\sim q_r}[s(y)] -\mathbb E_{y\sim p}[s(y)]. \]

The semicolon records two roles. Its left argument changes the response law; its right argument scores both laws. Thus $G_p(r;s)$ is not an independent evaluation unless $s$ has separately been declared to serve that role.

This is the finite notation for $\Delta (r,r')$ in Theorem 2, equation (3.4), of Theoretical limits of language model alignment[paes2026theoretical].

Definition. Jeffreys divergence [ftip-008M]

For probability laws $p$ and $q$ for which both terms are finite, their Jeffreys divergence is

\[ J(p,q) =D_{\mathrm {KL}}(p\Vert q) +D_{\mathrm {KL}}(q\Vert p). \]

The KL divergence and its argument order are defined in Definition [ftip-006X]. Unlike either directed term, $J(p,q)=J(q,p)$. In the finite tilt of Definition [ftip-008K], $p$ and $q_r$ have the same support, so both terms are finite.

See [paes2026theoretical, Theorem 1, equation (3.2)].

Lemma. Log-density ratio of the exact tilt [ftip-008N]

For every $y\in S_p$, the aligned law satisfies

\[ \log \frac {q_r(y)}{p(y)} =\frac {r(y)}{\beta }-\log Z_r. \]

Proof.

On $S_p$, both $p(y)$ and $q_r(y)$ are positive. Dividing the formula of Definition [ftip-008K] by $p(y)$ and taking logarithms gives the identity.

This is equation (B.1) followed by the logarithmic step in [paes2026theoretical, Appendix B.1, equations (B.1)--(B.2)].

Theorem. Exact reward gain equals scaled Jeffreys divergence [ftip-008O]

For a finite KL-alignment instance,

\[ G_p(r;r)=\beta J(p,q_r). \]

Proof.

Rearranging Lemma [ftip-008N] gives $r(y)=\beta \log (q_r(y)/p(y))+\beta \log Z_r$ on the common support. Taking expectation first under $q_r$ and then under $p$ yields

\[ \begin {aligned} \mathbb E_{q_r}[r] &=\beta D_{\mathrm {KL}}(q_r\Vert p)+\beta \log Z_r,\\ \mathbb E_p[r] &=-\beta D_{\mathrm {KL}}(p\Vert q_r)+\beta \log Z_r. \end {aligned} \]

Subtracting cancels the common normalizer and gives the result.

This is the finite form of [paes2026theoretical, Theorem 1, equation (3.2), with proof in Appendix B.1].

Remark. What the Jeffreys identity does and does not identify [ftip-008P]

The identity Theorem [ftip-008O] is an equality inside one declared model. It says that exact gain in the optimizing reward equals a symmetric divergence from the reference law. It does not say that a larger divergence improves a different utility, nor that a training algorithm reaches the exact tilt.

The identity accounts for neither rollout and update work nor the cost of obtaining $r$. Consequently it is not, by itself, a bound on the costed post-training potential of Definition [ftip-005M].

Theorem. Cross-reward gain is a base-law covariance [ftip-008Q]

For a finite KL-alignment instance,

\[ G_p(r;s) =\operatorname {Cov}_p\left (s,\frac {a_r}{Z_r}\right ). \]

Proof.

The aligned expectation can be written under the reference law as $\mathbb E_{q_r}[s]=\mathbb E_p[s a_r/Z_r]$. Also $\mathbb E_p[a_r/Z_r]=1$. Substituting these two identities into Definition [ftip-008L] gives the covariance defined in Notation [ftip-008H].

This is the finite form of Theorem 2, equation (3.4), in Theoretical limits of language model alignment[paes2026theoretical]. Its proof is in Appendix B.1, equations (B.5)--(B.7).

Remark. The covariance is predictive only for a declared reward [ftip-008R]

The identity Theorem [ftip-008Q] expresses exact tilted-law gain using expectations under $p$. This makes the population quantity accessible from the reference law in principle. A finite-sample estimator still needs its own sampling law, moment assumptions, and error analysis.

The formula also remains indexed by both $r$ and $s$. It cannot turn a proxy reward into an independent utility. If $s=r$, it measures improvement in the same quantity that defined the exact optimizer; if $s\ne r$, its sign is determined by the displayed covariance.

Example. A two-response alignment identity [ftip-008S]

Take $\mathcal Y=\{a,b\}$, $p=(1/2,1/2)$, $\beta =1$, and $r(a)=0$, $r(b)=\log 3$. Exponential weighting changes the reference law as follows.

$\text{[math]}$

The reward gain is

\[ G_p(r;r) =\left (\frac 34-\frac 12\right )\log 3 =\frac 14\log 3. \]

Direct calculation gives

\[ \begin {aligned} D_{\mathrm {KL}}(q_r\Vert p) &=\frac 14\log \frac 12+\frac 34\log \frac 32,\\ D_{\mathrm {KL}}(p\Vert q_r) &=\frac 12\log 2+\frac 12\log \frac 23, \end {aligned} \]

whose sum is $\frac 14\log 3$. Thus the example checks Theorem [ftip-008O] exactly; it is not an empirical alignment result.

Corollary. Averaging the fixed-query identity over prompts [ftip-008T]

Let $\mathcal X$ be finite with prompt law $\mu $. For each $x\in \mathcal X$, let $p_x$, $r_x$, and $q_{r_x}$ be a finite KL-alignment instance with the same penalty $\beta >0$. Then

\[ \sum _{x\in \mathcal X}\mu (x)G_{p_x}(r_x;r_x) =\beta \sum _{x\in \mathcal X}\mu (x)J(p_x,q_{r_x}). \]

Proof.

Apply Theorem [ftip-008O] at each prompt and take the finite $\mu $-weighted sum.

The source theorem is stated for each fixed query. This corollary performs only finite averaging; it does not introduce a shared neural parameterization across prompts.

Lemma. Additive reward shifts leave the aligned law unchanged [ftip-008U]

For constants $c,d\in \mathbb R$,

\[ q_{r+c}=q_r, \qquad G_p(r+c;s+d)=G_p(r;s). \]

Proof.

The shifted weight is $a_{r+c}=e^{c/\beta }a_r$, while its partition function is $Z_{r+c}=e^{c/\beta }Z_r$; the common factor cancels in the aligned law. Adding $d$ to the evaluation reward adds $d$ to both expectations in the gain, so it also cancels.

This finite lemma records the same prompt-dependent additive non-identifiability that appears in the DPO reparameterization of Definition [ftip-003P].

Remark. A penalty coefficient is not a hard KL budget [ftip-008V]

For fixed $\beta $, the exponential tilt solves the penalized problem

\[ \max _{q\in \Delta (\mathcal Y)} \left \{\mathbb E_q[r] -\beta D_{\mathrm {KL}}(q\Vert p)\right \} \]

under the conventions of Definition [ftip-008I]. This is not the same specification as choosing a number $\kappa $ and solving

\[ \max _q\mathbb E_q[r] \quad \text {subject to}\quad D_{\mathrm {KL}}(q\Vert p)\leq \kappa . \]

A Lagrange multiplier can relate the two problems when the relevant duality and activity conditions hold. The coefficient $\beta $ alone does not declare a hard budget, and the identities Theorem [ftip-008O]--Theorem [ftip-008Q] do not supply those conditions.

Remark. Fixed-query identities and their assumptions [ftip-008W]

The source setup writes rewards as $r(\mathbf x,\mathbf y)$, while the display of Theorem 1 reverses the two arguments in places. This subsection uses the setup order and then suppresses the fixed prompt. The source also moves between a dataset-level penalized objective and fixed-query identities. The finite-averaging result Corollary [ftip-008T] makes that step explicit.

The identities specialize Theoretical limits of language model alignment[paes2026theoretical] to a finite response set. They require no extension to the full sequence space.

Finally, exact exponential tilting is a distributional optimizer. It does not account for rollout, gradient, optimizer, or systems cost, and it does not show that PPO, GRPO, DPO, or any frontier training run attains the displayed law. Applying the identities to a training run therefore requires a separate argument that its output law is the exact optimizer.

Preference-feedback distortion [ftip-008X]

A finite routing model supports preference-distortion lower bounds in Sections 2--3 and Appendix A of The limits of preference data for post-training[zhao2025limits]. Cardinal queries permit a different guarantee under the hypotheses of Peeking behind the ordinal curtain: Improving distortion via cardinal queries[amanatidis2021peeking].

Both comparisons depend on the queries, circuits, routing maps, utility, feedback profile, algorithm, and comparator class. The noiseless and Bradley--Terry bounds retain those model restrictions.

Remark. The routing-model assumption behind preference limits [ftip-008Y]

The routing model and preference limits below are based on the v1 preprint The limits of preference data for post-training[zhao2025limits]. It models a pretrained system as a finite collection of response circuits plus learned maps that route queries to those circuits. Post-training changes the routing maps while retaining the circuit collection.

The noiseless lower bound is proved here under explicit pointwise response separation and a deterministic preference-only learner, whose returned router may be stochastic. Its finite partition estimate and cardinal-utility construction give a restricted reconstruction of the source rate; the proof does not establish a randomized-learner extension.

The routing model is a source assumption, not an architectural theorem about language models. In particular, its lower bounds do not prove that real post-training creates no circuits, that a neural network decomposes into the displayed objects, or that every preference-learning algorithm obeys the same bound outside this model.

Notation. Notation for the finite preference model [ftip-008Z]

The finite preference model uses the following symbols, corresponding to the notation of The limits of preference data for post-training[zhao2025limits]:

\[ \mathcal Q\mapsto Q,\qquad \mathcal R\mapsto Y,\qquad \mathcal S_0\mapsto C_0,\qquad \mathcal Z\mapsto H, \] \[ \Phi \mapsto \mathfrak E,\qquad \mathcal D\mapsto \mu _Q,\qquad M\mapsto \mathfrak m. \]

Thus $H$ denotes the source's finite internal-representation set, not a public interaction history. The family $\mathfrak E$ contains admissible query encoders and is unrelated to the costed potential $\Phi $ of Definition [ftip-005M].

Definition. Queries, responses, and retained circuits [ftip-0090]

Following the formal model in Section 2 of The limits of preference data for post-training[zhao2025limits], let $Q$ be a finite nonempty query set and $Y$ a response set. A response circuit is a function $c:Q\to Y$. The finite nonempty set $C_0$ contains the circuits retained from the pretrained model.

The word ``circuit'' is source terminology for this abstract function. No claim is made here that an element of $C_0$ is localized in a neural network or corresponds to one human-named capability.

Definition. Query representation and circuit routing [ftip-0091]

Let $H$ be a finite nonempty representation set and $\mathfrak E\subseteq H^Q$ a finite nonempty family of admissible encoders. A query encoder is $e\in \mathfrak E$. A circuit router is a map

\[ h:H\longrightarrow \Delta (C_0). \]

For a query $\xi \in Q$, the composition $h(e(\xi ))$ is a law on the retained circuits. Sampling $c\sim h(e(\xi ))$ and returning $c(\xi )$ gives the response law. These are the two routing components in the source model [zhao2025limits, Section 2, ``Formal model''].

Definition. Pretrained and post-trained routing models [ftip-0092]

A routing model is a triple $\mathfrak m=(e,h,C_0)$ with the types fixed in Definition [ftip-0090]--Definition [ftip-0091]. The pretrained model is $\mathfrak m_0=(e_0,h_0,C_0)$. In the source intervention, a post-trained model may replace $e_0$ and $h_0$, but it retains exactly $C_0$.

Write $\mathfrak m(\xi )$ for the random response obtained by sampling $c\sim h(e(\xi ))$ and returning $c(\xi )$. The source sometimes writes this as though the router selected a circuit rather than a distribution; the stochastic reading follows its declared codomain $\Delta (C_0)$.

Definition. Utility and the uniform query law [ftip-0093]

The source fixes the uniform probability law $\mu _Q$ on $Q$ and a utility function

\[ u:Q\times Y\longrightarrow \mathbb R. \]

For a routing model $\mathfrak m$, its expected utility is

\[ U_u(\mathfrak m) =\mathbb E_{\xi \sim \mu _Q} \mathbb E\left [u\left (\xi ,\mathfrak m(\xi )\right )\mid \xi \right ]. \]

This is the outcome objective in [zhao2025limits, Section 2, ``Outcome-based optimization with preference data'']. It is an assumed cardinal utility, not something ordinal comparisons directly reveal.

Definition. Complete noiseless ordinal preference profile [ftip-0094]

Assume that distinct retained circuits never tie at a query. For each $\xi \in Q$, utility then induces a strict comparison by

\[ c_i\succ _{\xi ,u}c_j \quad \Longleftrightarrow \quad u\left (\xi ,c_i(\xi )\right ) >u\left (\xi ,c_j(\xi )\right ). \]

The complete noiseless ordinal profile is $\succ _u=(\succ _{\xi ,u})_{\xi \in Q}$. The source gives the learner unlimited, unbiased access to every such comparison [zhao2025limits, Section 2, final paragraph]. Utilities with ties require a separately declared deterministic tie-break; the lower bounds below use the source's tie-free constructions.

Remark. Full preference data and an online oracle expose the same source information [ftip-0095]

Within this finite model, a table containing every comparison in $\succ _u$ and a noiseless online oracle that answers every possible query expose the same ordinal information. The source deliberately grants this ideal access so that its obstruction is not caused by finite sampling or an offline split [zhao2025limits, Section 2].

This equivalence does not price the number of queries, labeler time, or adaptivity. It therefore cannot be transferred to a finite-feedback protocol without a separate query-complexity statement.

Definition. Preference-only post-training algorithm and attainable route class [ftip-0096]

A preference-only post-training algorithm $\mathcal A$ maps the pretrained routing model and the complete profile to

\[ \mathfrak m_{\mathcal A}(u) =\mathcal A(\mathfrak m_0,\succ _u). \]

The deterministic comparator class displayed in the source theorems is

\[ \mathcal M_{\rm det}(C_0) =\left \{(e,h,C_0):e\in \mathfrak E, h:H\to C_0\right \}. \]

The source's formal model initially permits stochastic routers $h:H\to \Delta (C_0)$, but its theorem comparator uses deterministic $h:H\to C_0$. The comparator class is therefore the displayed deterministic class.

Definition. Multiplicative post-training distortion [ftip-0097]

Let $\mathcal B$ be any post-training algorithm with output $\mathfrak m_{\mathcal B}(u)$ under the feedback law induced by $u$. When the denominator is positive, define its multiplicative post-training distortion by

\[ \operatorname {Dist}_u(\mathcal B;\mathfrak m_0) =\frac { \max _{\mathfrak m^*\in \mathcal M_{\rm det}(C_0)}U_u(\mathfrak m^*) }{ U_u(\mathfrak m_{\mathcal B}(u)) }. \]

If the numerator is positive and the denominator is zero, set the ratio to $+\infty $. The source lower bounds construct bounded nonnegative utilities for which the comparison is well defined [zhao2025limits, Equation (2) and Appendix A]. Distortion measures loss relative to the declared comparator class; it is not an absolute capability score.

Definition. Borda count [ftip-0098]

Let $m=|C_0|$, and suppose each $\succ _{\xi ,u}$ is a strict total order on $C_0$. If $\operatorname {rank}_{\succ _{\xi ,u}}(c)$ places the most preferred circuit at rank one, its Borda score is

\[ B_{\xi }(c)=m-\operatorname {rank}_{\succ _{\xi ,u}}(c). \]

The Borda-count choice is any circuit in

\[ \operatorname *{arg\,max}_{c\in C_0} \sum _{\xi \in Q}B_{\xi }(c). \]

This is [zhao2025limits, Definition 3.1]. The source cites a prior equivalence between a standard RLHF model and Borda count; the definition here does not assert that every RLHF implementation is a Borda rule.

Example. A compromise circuit can maximize utility without winning Borda count [ftip-0099]

Example 3.2 of The limits of preference data for post-training[zhao2025limits] prints only $a\geq 0$ and $2a<b$, but those conditions do not force its stated rankings or Borda totals. Sufficient conditions are $0<a<1/2$ and $2a<b<1$. Take three queries, three circuits, and a singleton representation set, so every query must use one common circuit. Set

\[ \begin {array}{c|ccc} &c_A&c_B&c_C\\ \hline \xi _1&1&0&1-a\\ \xi _2&1&0&1-a\\ \xi _3&0&1&b \end {array} \]

These inequalities give the rankings $c_A\succ c_C\succ c_B$ for $\xi _1,\xi _2$ and $c_B\succ c_C\succ c_A$ for $\xi _3$. Hence the Borda totals are $(4,2,3)$, so the ordinal rule selects $c_A$. The utility totals are $(2,1,2-2a+b)$. Since $2a<b$, circuit $c_C$ has strictly greater total utility than $c_A$.

$\text{[math]}$

This finite example isolates information lost by ranking. The compromise circuit is never first for an individual query, yet its aggregate utility is largest. It does not show that Borda is always suboptimal or that a neural model must collapse all queries to one representation.

Theorem. Noiseless preference distortion for deterministic learners [ftip-009A]

Let $Q,H$ be nonempty finite sets, let $\varnothing \neq \mathfrak E\subseteq H^Q$ be finite, and let $C_0=\{c_1,\ldots ,c_m\}$ be a nonempty finite set of deterministic maps $Q\to Y$. Write $N=|Q|$, $r=|H|$, $p=|\mathfrak E|$ and $m=|C_0|$. The query law is uniform on $Q$. Call $C_0$ pointwise response-separating when

\[ c_i\neq c_j\quad \Longrightarrow \quad c_i(q)\neq c_j(q) \qquad (q\in Q). \]

Assume this separation. Let $\mathcal A$ be a deterministic preference-only learner: from a pretrained model $\mathfrak m_0$ and the complete strict ordinal profile, it returns $(e,h,C_0)$ with $e\in \mathfrak E$ and a possibly stochastic router $h:H\to \Delta (C_0)$. The comparator class is exactly $\mathcal M_{\rm det}(C_0)$ of Definition [ftip-0096], including every map $H\to C_0$ for each $e\in \mathfrak E$.

For every such pretrained model and learner there exists $u:Q\times Y\to [0,1]$, inducing a strict profile, such that

\[ \sum _{j=1}^m u(q,c_j(q))=1\quad (q\in Q), \]\[ R_{Q,\mathfrak E,H}=\frac {N}{\sqrt {N(r+\log p)}+r}, \]\[ \operatorname {Dist}_u(\mathcal A;\mathfrak m_0) \geq \frac 1{80}\min \{\sqrt m,R_{Q,\mathfrak E,H}\}. \]

Logarithms are natural. Utility averages over the uniform query and the learner's router draw as in Definition [ftip-0093]. The comparator maximum exists because its class is finite and nonempty. It is positive for the normalized utilities above: some constant-circuit route has mean at least $1/m$. A zero learner utility therefore gives distortion $+\infty $, with no $0/0$ case.

Proof.

First, for every normalized nonnegative utility, the comparator value is at least the learner value. Keep the learner's representation map and, on each fiber, choose a circuit maximizing its total utility there. That deterministic choice dominates the stochastic mixture and belongs to the comparator class. Thus distortion is at least one.

Put

\[ \begin {aligned} D&=\sqrt {Nr}+\sqrt {\frac N2\log (4p)},& B&=\sqrt {N(r+\log p)},\\ T&=\min \{\sqrt m,R_{Q,\mathfrak E,H}\},& x&=\min \{\sqrt m,N/(2D)\}. \end {aligned} \]

Since $(\log 4)/2<1\leq r$ and $\log p\geq 0$, we have $D\leq 2B$. Consequently $N/(2D)\geq N/(4B)\geq R_{Q,\mathfrak E,H}/4$ and $x\geq T/4$. Every denominator is positive.

If $m=1$, assign utility one to the sole circuit response at every query. Distortion is one and $T\leq 1$. If $m\geq 2$ but $x<2$, assign the same strict normalized values $2(m-j)/(m(m-1))$ to the circuits in index order at every query. Distortion is at least one and $T<8$, which proves the claimed bound. Extend both assignments by zero to all other responses.

In the remaining case set $k=\lfloor x\rfloor $. Then

\[ 2\leq k\leq \sqrt m,\qquad kD\leq N/2, \qquad k\geq x/2\geq T/8. \]

Write $[k]=\{1,\ldots ,k\}$ and fix a map $f:Q\to [k]$ supplied by Lemma [ftip-009C]. At a query with label $i=f(q)$, fix the strict order putting $c_i$ first and the remaining circuits in increasing index order. Let $\operatorname {rank}_i(j)\in \{1,\ldots ,m\}$ be the position of $c_j$, with the first circuit at rank one, and define

\[v_i(j)=\frac {2(m-\operatorname {rank}_i(j))}{m(m-1)}.\]

These values decrease strictly in the fixed order, sum to one and lie in $[0,2/m]$. The ordinal profile is now fixed independently of the learner's output. Run the deterministic learner on this profile and $\mathfrak m_0$, obtaining $(e,h,C_0)$. Write $h_z(j)=h(z)(c_j)$ for the probability of circuit $c_j$ at representation $z$. For each $z\in H$, choose $i_z\in [k]$ minimizing $h_z(j)$ over $j\in [k]$, breaking ties by index. Since $\sum _{j=1}^k h_z(j)\leq 1$, we have $h_z(i_z)\leq1/k$.

Let $A=\{q:f(q)=i_{e(q)}\}$ and $S=|A|$. The lower partition bound gives

\[ S=\sum _zX_{z,i_z}\geq \sum _z\min _{j\in [k]}X_{z,j} \geq N/k-D\geq N/(2k). \]

Fix $\epsilon =1/4$. For $i=f(q)$, assign

\[ u(q,c_j(q))= \begin {cases} (1-\epsilon )\mathbf 1_{j=i}+\epsilon v_i(j),&q\in A,\\ (1-\epsilon )/m+\epsilon v_i(j),&q\notin A. \end {cases} \]

Each row is nonnegative and sums to one. On selected queries the added peak favors the already highest-ranked circuit; on the other queries the same constant is added to every circuit. Hence every row induces exactly the fixed strict order. Pointwise response separation makes these assignments well-defined on $Q\times Y$; assign zero to responses outside the attained circuit responses. Because the final utility has the unchanged ordinal profile, the deterministic learner returns the same $(e,h,C_0)$.

Write $W=NU$ for total utility. A selected query gives the learner at most $(1-\epsilon )/k+2\epsilon/m$; any other query gives at most $(1-\epsilon )/m+2\epsilon/m$. Averaging over the stochastic router yields

\[ W_{\mathcal A} \leq (1-\epsilon )S/k+(1-\epsilon )(N-S)/m+2\epsilon N/m \leq S/k+2N/m. \]

Choose the deterministic comparator $e^*=e$ and $h^*(z)=c_{i_z}$. It belongs to the declared class, attains utility at least $1-\epsilon \geq1/2$ on each selected query and nonnegative utility elsewhere. Thus its total utility is at least $S/2$. If learner utility is zero the bound follows. Otherwise, using $S\geq N/(2k)$ and $k^2\leq m$,

\[ \begin {aligned} \operatorname {Dist}_u &\geq \frac {S/2}{S/k+2N/m} =\frac {k}{2(1+2kN/(mS))}\\ &\geq \frac {k}{2(1+4k^2/m)} \geq \frac {k}{10}\geq \frac {T}{80}. \end {aligned} \]

Appendix Theorem A.1 of The limits of preference data for post-training[zhao2025limits] states the corresponding rate for its routing model. Its displayed balancing parameter retains $\log N$ factors that are absent from its final rate; the second-moment partition bound in Lemma [ftip-009C] supplies a separate argument for the stated restricted model. Its index-based perturbation on queries outside the selected group need not preserve their prescribed favorite; the query-dependent rank weights used here preserve it. When circuits collide at a query, a circuitwise prescription need not define a response utility; pointwise separation excludes that issue.

The utility is a worst-case existential choice for each fixed learner. The proof permits stochastic inference, but it does not cover internal learner randomness: a utility chosen after a realized random output need not be one utility valid before the learner's random draw. No claim about response-colliding circuits or the separate noisy-preference bound follows.

Corollary. A square-root corollary under explicit query growth [ftip-009B]

Under all hypotheses of Theorem [ftip-009A], including a deterministic preference-only learner and pointwise response separation, assume

\[R_{Q,\mathfrak E,H}\geq \sqrt {|C_0|}.\]

For every pretrained model and every such learner, the utility supplied by that theorem satisfies

\[ \operatorname {Dist}_u(\mathcal A;\mathfrak m_0) \geq \frac 1{80}\sqrt {|C_0|}. \]

Proof.

The displayed growth condition makes the minimum in Theorem [ftip-009A] equal to $\sqrt {|C_0|}$.

Theorem 3.3 of The limits of preference data for post-training[zhao2025limits] gives a square-root rate in its large-query regime. The explicit condition above gives a finite regime for the reconstructed theorem with its stated learner and response assumptions.

Lemma. A simultaneous finite partition bound [ftip-009C]

\[ \begin {aligned} \sum _{z\in H}\max _{j\in [k]}X_{z,j}&\leq \frac Nk+D,\\ \sum _{z\in H}\min _{j\in [k]}X_{z,j}&\geq \frac Nk-D, \end {aligned} \]

where $[k]=\{1,\ldots ,k\}$, logarithms are natural, and

\[ X_{z,j}=|\{q\in Q:e(q)=z,\ f(q)=j\}|, \qquad D=\sqrt {Nr}+\sqrt {\frac N2\log (4p)}. \]

Proof.

Choose the labels $f(q)$ independently and uniformly from $[k]$. Fix $e,z$, write $n_z=|e^{-1}(z)|$, and put $a_j=X_{z,j}-n_z/k$. Each occupancy has variance $n_z(1/k)(1-1/k)$, so

\[\mathbb E\sum _{j=1}^k a_j^2=n_z(1-1/k).\]

The pointwise maximum and minimum satisfy

\[ \begin {aligned} \max _jX_{z,j}&\leq n_z/k+\|a\|_2,\\ \min _jX_{z,j}&\geq n_z/k-\|a\|_2. \end {aligned} \]

Jensen's inequality gives $\mathbb E\|a\|_2\leq \sqrt {n_z(1-1/k)}$. Summing over $z$ and applying Cauchy--Schwarz yields

\[ \begin {aligned} \mathbb E\sum _z\max _jX_{z,j}&\leq N/k+\sqrt {Nr(1-1/k)},\\ \mathbb E\sum _z\min _jX_{z,j}&\geq N/k-\sqrt {Nr(1-1/k)}. \end {aligned} \]

Changing one label changes only one representation group: one occupancy decreases by one and another increases by one. Its maximum and its minimum each change by at most one. Thus both sums have bounded differences with all $N$ constants equal to one. McDiarmid's inequality, also used in Appendix A.1 of The limits of preference data for post-training[zhao2025limits], bounds each relevant one-sided deviation of size $t$ by $\exp (-2t^2/N)$.

Choose $t=\sqrt {(N/2)\log (4p)}$. A union bound over both tails and all $p$ representations gives total failure probability at most

\[2p\exp (-2t^2/N)=\frac 12<1.\]

At least one deterministic labeling satisfies both claimed bounds, since $\sqrt {Nr(1-1/k)}\leq \sqrt {Nr}$. Empty representation fibers have all occupancies zero and require no exception.

The bound uses a second-moment estimate in place of the occupancy estimate in source Lemma A.2. The chosen labeling depends only on $Q,H,\mathfrak E,k$; its existence uses proof randomness and does not assume randomness in the learning algorithm.

Definition. Bradley--Terry preference from a score map [ftip-009D]

Fix a query $\xi $ and nonnegative circuit scores $a_{\xi ,c}\geq 0$ such that $a_{\xi ,c_i}+a_{\xi ,c_j}>0$ for every compared pair. The Bradley--Terry comparison law is

\[ \Pr (c_i\succ _{\xi }c_j) =\frac {a_{\xi ,c_i}} {a_{\xi ,c_i}+a_{\xi ,c_j}}. \]

Only score ratios are identified: multiplying every score for a fixed query by the same positive constant leaves all comparison probabilities unchanged. The linear and exponential score links discussed in [zhao2025limits, Section 3.2] therefore impose different assumptions on the relation between rewards and comparisons.

Definition. Exponential-score Bradley--Terry feedback [ftip-009E]

The exponential-score link sets

\[ a_{\xi ,c}=\exp \left (u(\xi ,c(\xi ))\right ). \]

For this link, exact comparison probabilities reveal pairwise utility differences through

\[ \log \frac {\Pr (c_i\succ _{\xi }c_j)} {\Pr (c_j\succ _{\xi }c_i)} =u(\xi ,c_i(\xi ))-u(\xi ,c_j(\xi )). \]

This algebra explains why a known link and infinite noiseless frequency information can expose more than an ordinal order. It does not apply when the link is unknown or misspecified.

Definition. Linear-score Bradley--Terry feedback [ftip-009F]

The linear-score link sets

\[ a_{\xi ,c}=u(\xi ,c(\xi )), \qquad \Pr (c_i\succ _{\xi }c_j) =\frac {u(\xi ,c_i(\xi ))} {u(\xi ,c_i(\xi ))+u(\xi ,c_j(\xi ))}. \]

Write $p^u_{\xi ,ij}$ for the displayed probability and define the complete pairwise-probability profile

\[ P_u^{\rm lin} =\left (p^u_{\xi ,ij}\right )_{\xi \in Q,\,c_i\neq c_j\in C_0}. \]

This requires nonnegative utilities and a positive denominator for each queried pair. A linear-score algorithm $\mathcal A_{\rm lin}$ takes $(\mathfrak m_0,P_u^{\rm lin})$ as input and returns $\mathfrak m_{\mathcal A_{\rm lin}}(u)$. The following lower bound concerns this specific link; it is not a lower bound for every stochastic preference model.

Theorem. General linear-score noisy preference lower bound [ftip-009G]

Assume pointwise response separation as defined in Theorem [ftip-009A]. For every pretrained routing model and linear-score algorithm $\mathcal A_{\rm lin}$, there exists a utility

\[ u:Q\times Y\longrightarrow [0,1]. \]

Post-training from even the complete comparison-probability profile of the linear-score link then satisfies

\[ \operatorname {Dist}_u(\mathcal A_{\rm lin};\mathfrak m_0) \geq \widetilde \Omega \left ( \min \left \{|C_0|,R_{Q,\mathfrak E,H}\right \} \right ), \]

where $R_{Q,\mathfrak E,H}$ is defined in Theorem [ftip-009A]. The theorem is again existential in $u$ and uses the source's deterministic comparator. The same circuitwise-utility issue recorded in Theorem [ftip-009A] prevents us from asserting the source's unrestricted pretrained-model quantifier here. Its noise is informative because probabilities depend on cardinal scores; the obstruction survives under the particular linear link.

This lower bound is the form of Appendix Theorem A.5 in The limits of preference data for post-training[zhao2025limits] with the additional pointwise response-separation assumption stated above.

Remark. The noisy main theorem and appendix have different asymptotic notation [ftip-009H]

The main-text Theorem 3.4 of The limits of preference data for post-training[zhao2025limits] displays $\Omega (|C_0|)$ in its stated large-query regime. Appendix Theorem A.5 displays the general bound with $\widetilde \Omega $ and the minimum in Theorem [ftip-009G]. The tilde can hide logarithmic factors, so these statements are not textually identical.

The general bound in Theorem [ftip-009G] has the appendix's rate with a possible logarithmic loss. Eliminating that loss to obtain the main-text rate requires an additional argument.

Definition. Cardinal value query [ftip-009I]

In the social-choice source, agents rank alternatives and also have nonnegative cardinal values. A value query takes an agent $i$ and an alternative $j$ and returns $v_{ij}$ [amanatidis2021peeking, Definition 1].

Under the routing analogy, a query $\xi \in Q$ plays the role of an agent and a circuit $c\in C_0$ plays the role of an alternative. A cardinal query therefore reveals one value $u(\xi ,c(\xi ))$. Query counts inherited from the social-choice theorem are per query/agent, not totals across $Q$.

Theorem. Acceptable Range Voting information--distortion tradeoff [ftip-009J]

Let $m$ be the number of alternatives and let $k\in \{1,\ldots ,m\}$. Theorem 4 of Peeking behind the ordinal curtain: Improving distortion via cardinal queries[amanatidis2021peeking] states that $k$-Acceptable Range Voting uses

\[ O(k\log m) \]

value queries per agent and has distortion

\[ O\left (m^{1/(k+1)}\right ). \]

This is a social-choice mechanism theorem. It transfers directly to the singleton-representation specialization $|H|=1$, where every query must use one common circuit. Under that restriction, queries are agents and circuits are alternatives as in Definition [ftip-009I]. The theorem does not by itself cover a routing comparator that may choose different circuits for different representations, and it is not an implementation of neural post-training.

This statement is Theorem 4 of Peeking behind the ordinal curtain: Improving distortion via cardinal queries[amanatidis2021peeking].

Corollary. Constant distortion with logarithmically many value queries per query [ftip-009K]

Taking $k$ proportional to $\log m$ in Theorem [ftip-009J] yields constant distortion with

\[ O(\log ^2 m) \]

value queries per agent. This is Corollary 2 of Peeking behind the ordinal curtain: Improving distortion via cardinal queries[amanatidis2021peeking], inherited as Theorem 3.5 by The limits of preference data for post-training[zhao2025limits]. Under the routing analogy, the count is per query, so a full table over $|Q|$ queries can require $O(|Q|\log ^2m)$ values. The statement is an upper bound for the named mechanism in the common-circuit specialization of Theorem [ftip-009J]. It is not a result for the full routing comparator, nor a lower bound saying that this many values are necessary.

Remark. Information and optimization assumptions in the two finite models [ftip-009L]

The two source families assume different information models; the diagram compares them without imposing an information order.

$\text{[math]}$

The KL identities characterize the exact optimizer for a declared scalar reward. The preference lower bounds are worst-case statements inside a fixed- circuit routing model. Bradley--Terry conclusions depend on the chosen score link. In the common-circuit specialization, cardinal queries change the information modality and admit a positive mechanism result. Outside that specialization, the preference and cardinal-query results are not directly comparable without further assumptions.

None of these statements derives the finite transcript theorem Theorem [ftip-007S], and none proves that benchmark gain is capability acquisition. Applying either result to a training system requires that system to satisfy the corresponding information, optimizer, and comparator assumptions.

Controlled coverage, feedback resolution, and prompt breadth [ftip-009M]

This section studies finite measurements and controlled comparisons that separate starting-policy coverage from the information supplied by a reward. It also records how a training prompt law limits the scope of an observed post-training effect.

The empirical route is a controlled language-model post-training study. Its reported outcomes remain experiment-specific observations. Theorems in this section follow from the stated finite-probability assumptions; the empirical results alone do not imply those conclusions.

Starting-policy interventions [ftip-009N]

A comparison among starting policies is meaningful only after the task, training grid, feedback rule, and evaluation procedure have been declared. Observed differences are conditional on those experimental coordinates.

Remark. The source experiment as an intervention matrix [ftip-009O]

[clay2026demystifying, Sections 4.1--4.3 and Figure 1] organize the reported experiments along three declared axes: a starting model distribution, a training-prompt distribution, and a reward function. Each reported measurement therefore belongs to a cell of a matrix rather than to an unqualified ``RL post-training'' condition. The source holds its broad training recipe fixed while varying selected axes; it does not study variation among RL algorithms.

Each experimental comparison is conditional on its model, task, prompt law, executable reward rule, training budget, and evaluation procedure. The reward rule is especially consequential here because the main text and Appendix C do not state identical sparse-reward semantics. Section 4.2 describes target containment with a length penalty. Appendix C's prose says an exact target receives unit reward, but its displayed equation requires the response to be strictly longer than the target for any positive reward. These two descriptions do not identify a single sparse-reward implementation.

Definition. Target-response event [ftip-009P]

Fix an evaluation instance $x$, a response space $\mathcal Y_x$, and a declared target predicate $h_x^\star :\mathcal Y_x\to \{0,1\}$. For a sampled response $Y$, the target-response event is

\[ E_x^\star =\{h_x^\star (Y)=1\}. \]

The predicate is part of the evaluation specification. It may test exact string equality after a declared normalization, containment of a target substring, equality of a parsed answer, or another explicitly typed condition. These predicates are not interchangeable. The movie-quote and AIME experiments in [clay2026demystifying, Sections 4.1--4.2] use different response objects and therefore require different target predicates.

Remark. Exact-string success and task utility are different predicates [ftip-009Q]

Let $y_x^\star $ be a designated string and let $N_x$ be a declared normalizer. Exact-string success is the specialization $h_x^\star (y)=\mathbf 1\{N_x(y)=N_x(y_x^\star )\}$ of Definition [ftip-009P]. A task utility may instead accept semantically equivalent answers, assign partial credit, invoke a verifier, or depend on evaluator randomness as in Convention [ftip-005D].

Consequently, a change in target-string match rate is not automatically a change in task utility. [clay2026demystifying, Sections 4.1--4.2] deliberately use narrow target behaviours to study controlled coverage. Transferring their measurements to broader capability requires a separately justified evaluator.

Definition. Base, SFT-positive, and SFT-negative starting-policy triple [ftip-009R]

For one declared model family and target predicate, a starting-policy triple is

\[ \left (\pi ^{\mathrm {base}},\pi ^+,\pi ^-\right ). \]

Here $\pi ^{\mathrm {base}}$ is the unmodified starting policy, $\pi ^+$ is obtained by a declared supervised update intended to increase target-response frequency, and $\pi ^-$ is obtained by a declared update intended to decrease it. The superscripts name producing interventions, not mathematical order relations between the resulting policies.

[clay2026demystifying, Section 4.1] instantiates the triple with the source's Base, SFT-positive, and SFT-negative checkpoints. For the movie quote, its SFT mixture places twenty percent weight on the target quote and eighty percent on other quotes from the Cornell Movie-Dialogs corpus; the negative intervention maximizes cross-entropy loss on the target.

Remark. SFT-positive and SFT-negative are composite weight interventions [ftip-009S]

The policies $\pi ^+$ and $\pi ^-$ of Definition [ftip-009R] are produced by optimization. Their weights, output probabilities, representations, and off-target behaviours can all change together. Calling one intervention ``positive'' and the other ``negative'' describes its intended effect on the declared target response; it does not assert that only one probability mass was edited.

The comparisons in [clay2026demystifying, Sections 4.1 and 5.1] therefore test dependence on three constructed starting checkpoints. They do not identify a causal effect of initial target probability alone without an additional intervention model.

Definition. Controlled post-training experiment cell [ftip-009T]

A controlled post-training experiment cell is a record

\[ \mathfrak c= (M,\pi ^{\mathrm {init}},\mathsf T,\mathcal D_{\mathrm {tr}}, r^{\mathrm {id}},\mathcal A,b_{\mathrm {tr}},\mathsf E), \]

where $M$ identifies the model architecture and checkpoint lineage, $\pi ^{\mathrm {init}}$ is its starting policy, $\mathsf T$ is the task, $\mathcal D_{\mathrm {tr}}$ is the training-prompt law, $r^{\mathrm {id}}$ identifies an executable reward rule and version, $\mathcal A$ is the update procedure, $b_{\mathrm {tr}}$ is the training budget, and $\mathsf E$ is the evaluation interface. Random seeds and hyperparameters belong to the relevant record fields even when suppressed from the notation.

[clay2026demystifying, Figure 1 and Sections 4.1--4.3] motivates the coordinates. The source varies starting distribution, prompt distribution, and reward while using a common broad RL setup. Because its main-text and Appendix-C sparse rewards differ, the reward field is an identifier for an executable rule, not merely the word ``sparse.''

Definition. Matched cells and held-fixed coordinates [ftip-009U]

Let $J$ index the coordinates of an experiment cell $\mathfrak c$ from Definition [ftip-009T]. For $S\subseteq J$, two cells $\mathfrak c$ and $\mathfrak c'$ are matched outside $S$ when

\[ \mathfrak c_j=\mathfrak c'_j\qquad \text {for every }j\in J\setminus S. \]

The coordinates in $J\setminus S$ are the held-fixed coordinates; those in $S$ are the declared intervention coordinates. Equality means equality at the resolution recorded by the experiment, including evaluation sample size when that size affects the reported statistic.

A matched comparison licenses a contrast between the recorded cells. It does not show that a named coordinate is internally atomic. In particular, matching outside the starting-policy coordinate leaves the composite-checkpoint qualification of Remark [ftip-009S] in force.

Example. Three starting laws under one declared training grid [ftip-009V]

A source comparison may place the three starting policies of Definition [ftip-009R] into cells matched outside the starting-policy coordinate. The shared boxes below mean equality of the declared grid fields, not that the three checkpoints differ in only one scalar probability.

$\text{[math]}$

This is the comparison pattern used in [clay2026demystifying, Sections 4.1 and 5.1]. An actual cell still needs its model, task, reward implementation, and numerical record; the diagram is not a claim that all experiments in the paper instantiate one identical grid.

Remark. What the starting-policy comparison can and cannot identify [ftip-009W]

Within a declared model, task, training grid, and evaluation procedure, the three-cell comparison can establish that the measured post-training outcome differs across the source's constructed starting checkpoints. The movie-quote and AIME entries in [clay2026demystifying, Section 5.1, Tables 1--2] are empirical records of that form.

The comparison does not isolate initial target probability from every other SFT-induced weight change, prove that an observed zero has zero support, or establish acquisition of a broader capability. The reported model sets differ across the paper: [clay2026demystifying, Section 4.1] names OLMo 3, Qwen2-7B, Qwen3-1.7B, and Qwen2.5-7B-Instruct, while Table 1 and Appendix G also report Qwen3-8B and Qwen2-1.5B. Each measurement is specific to the model and configuration in its table row.

Finite initial-coverage measurement [ftip-009X]

A finite evaluation records hits, not mathematical support. This subsection introduces the hit count and its sampling law, derives the exact zero-hit bound under independent sampling, and then returns to the limits of the source measurement.

Definition. Initial evaluation hit count [ftip-009Y]

Fix a starting policy, an evaluation sampling law, and the target predicate of Definition [ftip-009P]. Before post-training, draw evaluation records $(X_j,Y_j)_{j=1}^m$. Define the hit indicators and the initial evaluation hit count by

\[ I_j^\star =h_{X_j}^\star (Y_j), \qquad K_m=\sum _{j=1}^{m}I_j^\star . \]

The sampling law determines whether the instances $X_j$ are fixed, resampled, or stratified. The adjective ``initial'' locates the measurement before the post-training intervention; it does not assert that the samples are independent or that the starting policy is a foundation checkpoint.

Definition. Empirical initial success rate [ftip-009Z]

For a nonempty initial evaluation sample of size $m$, the empirical initial success rate is

\[ \widehat p_m=\frac {K_m}{m} =\widehat {\mathbb E}_{j\in \{1,\ldots ,m\}}[I_j^\star ], \]

using the empirical-average convention of Convention [ftip-006Z]. This is a statistic of the declared sample. It is not, without a sampling model and an uncertainty statement, the exact successful-support probability $p_P(x)$ of Definition [ftip-005U].

Remark. Zero observed hits are not zero probability [ftip-00A0]

The equality $K_m=0$ says that no target response occurred in one finite sample. Every success probability $0\leq p<1$ assigns positive probability $(1-p)^m$ to this observation under independent Bernoulli sampling. Thus $\widehat p_m=0$ does not imply $p=0$, absence from mathematical support, or impossibility of discovery under a larger budget.

The entries printed as 0.00 percent in [clay2026demystifying, Section 5.1, Tables 1--2] are empirical rates at the stated evaluation sizes. They must not be rewritten as exact support claims.

Theorem. Zero-hit likelihood under iid evaluation [ftip-00A1]

Let $m\in \mathbb N_{\geq 1}$. Assume the hit indicators from Definition [ftip-009Y] are independent Bernoulli variables with a common success probability $p$. Then $K_m$ has the binomial law and

\[ \Pr _p(K_m=0)=(1-p)^m. \]

Proof.

The event $K_m=0$ is $\bigcap _{j=1}^m\{I_j^\star =0\}$. Independence and $\Pr _p(I_j^\star =0)=1-p$ give the displayed product.

This finite statement follows from the displayed hypotheses. Applying it to a source table requires the additional iid and stationary-success assumptions stated above.

Corollary. Exact one-sided bound after zero hits [ftip-00A2]

Assume Theorem [ftip-00A1] with $m\in \mathbb N_{\geq 1}$ and fix $0<\delta <1$. If $K_m=0$, inversion of the exact zero-count likelihood gives the one-sided upper endpoint

\[ u_{m,\delta }=1-\delta ^{1/m}. \]

Indeed, $\Pr _{u_{m,\delta }}(K_m=0)=\delta $, and for every $p>u_{m,\delta }$ one has $\Pr _p(K_m=0)<\delta $. Thus the observed zero count excludes probabilities above $u_{m,\delta }$ at exact level $\delta $ under the declared iid model.

Proof.

By Theorem [ftip-00A1], the zero-count likelihood is $(1-p)^m$, which is strictly decreasing in $p$. Solving $(1-p)^m=\delta $ gives the endpoint and the stated strict inequality.

This is a likelihood inversion derived from the displayed finite setup, not a source theorem and not a support certificate.

Example. The 128-sample zero-hit bound [ftip-00A3]

Take $m=128$ and $\delta =0.05$ in Corollary [ftip-00A2]. After zero hits, the exact one-sided upper endpoint is

\[ u_{128,0.05} =1-0.05^{1/128} \approx 0.02313. \]

Under the iid Bernoulli model, probabilities above approximately 2.313 percent make a zero count less than five percent likely. The calculation does not turn a recorded $0/128$ into proof that the true probability is zero, and it does not apply if the 128 evaluations have a different joint sampling law.

Definition. First positive sparse-reward time [ftip-00A4]

Fix an executable sparse-reward rule and let $r_j^{\mathrm {sp}}$ be the reward returned on rollout $j$. The first positive sparse-reward time is the extended-valued index

\[ J_+=\inf \{j\geq 1:r_j^{\mathrm {sp}}>0\}, \qquad \inf \varnothing :=\infty . \]

This definition concerns the reward implementation named by an experiment cell. The event $r_j^{\mathrm {sp}}>0$ coincides with a target-response event only when that equality is part of the declared reward semantics. In particular, a length penalty or partial-credit rule can separate positive reward from exact target match.

Corollary. Sparse-reward discovery within a rollout budget [ftip-00A5]

Suppose the events $\{r_j^{\mathrm {sp}}>0\}$ are independent and have a common probability $q$. For every positive rollout budget $B$,

\[ \Pr (J_+\leq B)=1-(1-q)^B. \]

Proof.

Apply the independent-attempt theorem Theorem [ftip-0078] with $E_j=\{r_j^{\mathrm {sp}}>0\}$. Its discovery event is exactly $\{J_+\leq B\}$.

This is a specialization proved from the displayed hypotheses. It does not supply independence, stationarity, or the value of $q$ for a training run, and it does not equate positive proxy reward with task utility.

Example. The AIME 128-sample measurement record [ftip-00A6]

[clay2026demystifying, Section 5.1, Table 2] reports the following empirical target-answer rates for AIME Problem 4 with Qwen2.5-7B-Instruct and evaluation size $m=128$. The task asks for the number of integer pairs $(x,y)\in [-100,100]^2$ satisfying $12x^2-xy-6y^2=0$; the target answer is $117$.

\[ \begin {array}{c|ccc} &\mathrm {SFT{+}}&\mathrm {Base}&\mathrm {SFT{-}}\\ \hline \text {No Reward}&26.6&3.92&0.00\\ \text {Sparse Reward}&85.9&10.2&0.00\\ \text {Dense Reward}&86.7&92.2&0.00 \end {array} \]

Every entry in the displayed array is a percentage.

$\text{[math]}$

These are source-reported empirical percentages, not exact policy probabilities. ``No Reward'' is retained as the source's row label; the table alone does not define it as a post-training protocol. The three zero entries for SFT-negative are finite observations and remain subject to Remark [ftip-00A0]. Appendix Figure 5's caption prints a different set of Qwen2.5-7B-Instruct values that duplicates the neighbouring movie-quote caption and is incompatible with Table 2. The displayed percentages are those of Table 2; the caption does not provide a consistent alternative measurement.

Remark. An observed plateau is not a universal probability threshold [ftip-00A7]

[clay2026demystifying, Section 5.2 and Figure 2] report that the Qwen3-1.7B Base movie-quote cell, whose initial empirical match rate is about 0.5 percent, does not optimize under the source's sparse-reward run, whereas its dense-reward run rises toward fifty percent. [clay2026demystifying, Appendix Figure 8] gives final match rates 10.0 percent for sparse reward and 48.8 percent for dense reward. Thus ``failed'' here means failure of the reported sparse run to optimize as intended, not zero final matches.

A plateau is indexed by the model, target, reward implementation, optimizer, sampling process, and budget in its experiment cell. One observed transition therefore does not identify a universal initial-probability threshold for RL learning, support, or capability acquisition.

Sparse, dense, and process feedback [ftip-00A8]

Sparse, dense, and process rewards expose different observations about a response. This subsection treats that difference as a change in the feedback channel, rather than as a free change in the smoothness of one fixed objective.

Definition. Sparse exact reward [ftip-00A9]

Let $\mathcal Y$ be a finite response set and fix a target response $\tau \in \mathcal Y$. The sparse exact reward is the map $r_{\rm exact}:\mathcal Y\to \{0,1\}$ defined by

\[ r_{\rm exact}(y)=\mathbf 1\{y=\tau \}. \]

This idealized reward exposes one bit: whether the whole response equals the target. It does not expose a prefix, edit location, or partial milestone. The movie-quote source uses related but textually inconsistent substring and length-penalized rewards; the distinct formulas are compared in Remark [ftip-00AA].

Remark. Main-text and Appendix-C sparse-reward discrepancy [ftip-00AA]

Section 4.2 of Demystifying Reinforcement Learning Post-Training of Language Models[clay2026demystifying] displays a substring indicator $\mathbf 1\{\tau \subseteq y\}$ and says that a length penalty is applied. In Appendix C, $n$ is the maximum generation length and $s$ is the number of tokens beyond the target. The appendix first defines the excess-length penalty

\[ p(s)= \begin {cases} 0,&s=0,\\ s/n,&s>0, \end {cases} \]

and then gives Equation (3):

\[ r_{\rm C}(y,\tau )= \begin {cases} \max \{0.5,1-p(s)\},&\tau \text { occurs in }y \text { and }|y|>|\tau |,\\ 0,&\text {otherwise}. \end {cases} \]

As printed, Equation (3) assigns zero when $y=\tau $, because its first branch requires strict excess length. This conflicts with the immediately preceding Appendix-C prose, which assigns base reward one when the target is generated, and it is not the exact-match reward of Definition [ftip-00A9]. The two printed descriptions therefore specify different reward rules.

Definition. Edit-distance reward [ftip-00AB]

Let $y=y_1\cdots y_m$ and a nonempty target $\tau =\tau _1\cdots \tau _n$ be finite strings. Their Levenshtein distance is determined by $D(0,j)=j$, $D(i,0)=i$, and, for $i,j>0$,

\[ D(i,j)=\min \left \{ \begin {aligned} &D(i-1,j)+1,\\ &D(i,j-1)+1,\\ &D(i-1,j-1)+\mathbf 1\{y_i\neq \tau _j\} \end {aligned} \right \}. \]

Set $L(y,\tau )=\max \{|y|,|\tau |\}$. The edit-distance reward is

\[ r_{\rm edit}(y,\tau ) =\max \left \{0,1-\frac {D(|y|,|\tau |)}{L(y,\tau )}\right \}. \]

The recurrence is Equation (1) in the main text and Equation (4) in Appendix C of Demystifying Reinforcement Learning Post-Training of Language Models[clay2026demystifying]; the normalized reward is its Equation (5). The nonempty-target assumption makes the displayed denominator positive. This reward does not establish that edit similarity is the correct utility for a different task.

Remark. Dense reward imports target structure [ftip-00AC]

The edit reward of Definition [ftip-00AB] is not obtained from the binary value in Definition [ftip-00A9] by a numerical smoothing operation. It also receives the target string, a character-level edit model, and the ordering of symbols. Two non-target responses that both receive sparse reward zero can therefore receive different edit rewards.

That extra resolution can improve credit assignment, but it changes the feedback channel and its assumptions. The source calls edit distance a dense proxy and reports controlled movie-quote experiments [clay2026demystifying, Sections 4.2 and 5.2]. Neither the formula nor those observations establish that edit proximity is an independent measure of general response quality.

Definition. Milestone process reward [ftip-00AD]

For the source's fixed AIME problem, let

\[ M(y)=(m_1(y),\ldots ,m_5(y))\in \{0,1\}^5 \]

record whether a completed response contains five declared milestones: a valid algebraic setup, correct root relations, correct integer bounds, a correct count for at least one branch, and correct subtraction of the overlap at the origin. With

\[ (w_1,\ldots ,w_5)=(0.05,0.05,0.10,0.15,0.25), \]

the milestone process reward is

\[ r_{\rm proc}(y)=\sum _{i=1}^{5}w_i m_i(y). \]

Appendix D.1 of Demystifying Reinforcement Learning Post-Training of Language Models[clay2026demystifying] gives Equation (6) and the milestone list above. In the experiment, a 32-billion-parameter instruction-tuned model classifies the completed trajectory. The displayed map is therefore richer than a deterministic final-answer verifier, even though it returns one scalar after the rollout.

Remark. Process feedback assumptions and milestone-weight discrepancy [ftip-00AE]

Appendix D.2 of Demystifying Reinforcement Learning Post-Training of Language Models[clay2026demystifying] supplements $r_{\rm proc}$ with an extracted-answer override and two penalties. Translating its notation, let $\tau =117$, $\tau _{\rm near}=118$, and define

\[ r_{\rm base}(y)= \begin {cases} 1.0,&\operatorname {extract}(y)=\tau ,\\ 0.6,&\operatorname {extract}(y)=\tau _{\rm near},\\ r_{\rm proc}(y),&\text {otherwise}. \end {cases} \]

Let $p_{\rm loop}(y)=-0.3$ when the judge flags a loop without the exact answer, and zero otherwise. Let $p_{\rm format}(y)=-0.5$ when the response lacks the required boxed delimiter, and zero otherwise. Equations (7)--(8) then give

\[ r_{\rm PRM}(y) =\max \{0,r_{\rm base}(y)+p_{\rm loop}(y)+p_{\rm format}(y)\}. \]

These clauses assume a reliable milestone judge, parser, near-miss choice, loop flag, and formatting rule. They are part of the feedback definition, not consequences of reinforcement learning. The main text describes ``exponentially increasing'' dense rewards, and Appendix D calls the weights ``exponential scaling.'' The printed vector $(0.05,0.05,0.10,0.15,0.25)$ is neither strictly increasing at every milestone nor a geometric progression. We preserve the exact weights and do not infer an exponential law from that wording.

Definition. Reward-induced observational equivalence [ftip-00AF]

Let $\mathcal Y$ be a finite response set and $r:\mathcal Y\to \mathcal R$ any reward map. Two responses are observationally equivalent under $r$, written $y\sim _r y'$, when

\[ y\sim _r y'\quad \Longleftrightarrow \quad r(y)=r(y'). \]

Equality makes $\sim _r$ an equivalence relation. Its quotient $\mathcal Y/{\sim _r}$ is the finite set of response classes distinguished by the reward alone. This proposed equivalence relation ignores any information in the response that is not returned by $r$; equal rewards need not imply equal latent utility.

Lemma. Refining feedback separates at least as many responses [ftip-00AG]

Let $\mathcal Y$ be finite and let $r_1:\mathcal Y\to \mathcal R_1$ and $r_2:\mathcal Y\to \mathcal R_2$. Say that $r_2$ refines $r_1$ when

\[ r_2(y)=r_2(y')\quad \Longrightarrow \quad r_1(y)=r_1(y') \qquad (y,y'\in \mathcal Y). \]

If $r_2$ refines $r_1$, then

\[ \left |\mathcal Y/{\sim _{r_2}}\right | \geq \left |\mathcal Y/{\sim _{r_1}}\right |. \]

Proof.

Send the $r_2$-class of $y$ to the $r_1$-class of $y$. The refinement condition makes this map well defined. It is surjective because every $r_1$-class contains some $y$, whose $r_2$-class maps to it. A surjection between finite sets has a domain at least as large as its codomain.

This finite lemma compares observational partitions only. It does not say that the refined reward is cheaper, more accurate, or better aligned with utility.

Example. Counterexample: equal sparse reward, unequal dense reward [ftip-00AH]

Take the target string $\tau =\texttt {abc}$ and two responses $y=\texttt {abx}$ and $y'=\texttt {xyz}$. Both fail exact matching, while their edit distances and normalized edit rewards are

\[ \begin {array}{c|c|c|c} \text {response}&r_{\rm exact}&D(\,cdot\,,\tau )&r_{\rm edit}\\ \hline \texttt {abx}&0&1&2/3\\ \texttt {xyz}&0&3&0 \end {array} \]

Thus $y\sim _{r_{\rm exact}}y'$ but $y\not \sim _{r_{\rm edit}}y'$. The example witnesses a strict separation inside one sparse-reward class. It does not claim that every dense proxy refines every sparse verifier.

Example. Three feedback resolutions on one response set [ftip-00AI]

One finite response set can be observed through three different maps. The arrows below share a domain; they do not assert that the three codomains form a refinement chain.

$\text{[math]}$

The exact reward forgets every difference among failures. Edit reward can retain character-level proximity, while the source process reward retains a declared milestone vector only after judge, override, and penalty choices. The partition lemma Lemma [ftip-00AG] applies to a pair only after its refinement hypothesis has been checked.

Remark. Reward density is not cost-free smoothing [ftip-00AJ]

A denser reward can distinguish more responses and supply more frequent update signal. It can also require information absent from a sparse verifier. In the controlled source, edit feedback assumes the full target and computes a string metric; process feedback uses a 32-billion-parameter judge, five problem-specific milestones, answer extraction, an enumerated near miss, and loop and format penalties [clay2026demystifying, Appendices C--D].

The richer feedback channel incurs specification, computation, and validation costs. Replacing sparse feedback by dense feedback can alter both the information available to training and the objective being optimized. The resulting comparison is an intervention on feedback resolution, not evidence that one fixed reward was smoothed at zero cost.

Prompt breadth and effect locality [ftip-00AK]

Training on one prompt law can change behavior unevenly across evaluation slices. Localized and distributed degradation are distinguished relative to that training law and the specified evaluation slices.

Definition. Training prompt law and evaluated prompt slice [ftip-00AL]

Let $\mathcal X$ be a task-instance set. Write $\mu _{\rm tr}=\mathcal D_{\rm tr}$ for the training-prompt coordinate of an experiment cell in Definition [ftip-009T]. A training prompt law is this probability law on $\mathcal X$, used to draw instances during a post-training protocol. Let $\mu _{\rm ev}$ be an independently declared evaluation law on the same set.

An evaluated prompt slice is a measurable set $C\subseteq \mathcal X$ with $\mu _{\rm ev}(C)>0$. Its conditional evaluation law is

\[ \mu _{\rm ev}(A\mid C) =\frac {\mu _{\rm ev}(A\cap C)}{\mu _{\rm ev}(C)}. \]

The training and evaluation laws need not agree. A slice records where an effect is measured; it does not assert that prompts inside the slice are equally difficult or represented equally in pretraining. We specialize the task-law convention of Definition [ftip-001Q].

Definition. Source-specific narrow and broad configurations [ftip-00AM]

A prompt configuration is the record

\[ \mathsf c=(M_0,\mu _{\rm tr},m_{\rm tr},r,\mathsf A,b), \]

containing the starting artifact, training prompt law, finite prompt-pool size, reward, update algorithm, and training budget. We call two particular records $\mathsf c_{\rm nar}$ and $\mathsf c_{\rm brd}$ only when a cited experiment names them narrow and broad.

Section 4.3 and Appendix F of Demystifying Reinforcement Learning Post-Training of Language Models[clay2026demystifying] instantiate these labels in two model-family experiments. The Qwen comparison uses 100 DeepScaleR prompts versus 10,000 WildChat prompts. The OLMo comparison uses 100 math-only prompts versus 10,000 prompts split evenly among mathematics, instruction following, and code. These labels identify the two reported configurations; they do not define a general measure or ordering of prompt breadth.

Remark. Prompt breadth is not a total order [ftip-00AN]

The records in Definition [ftip-00AM] change several coordinates together. The prompt-pool size, domain mixture, and repetition frequency differ. Across the paper's Qwen and OLMo studies, the starting artifact and evaluated tasks also differ.

We therefore use ``narrow'' and ``broad'' as names for the source cells. They do not define a total order on prompt laws, and their comparison does not identify a causal effect of breadth alone. A breadth theorem would need a declared statistic and matched configurations that differ only in that statistic.

Definition. Slice-conditioned success change [ftip-00AO]

Fix an evaluated slice $C$ from Definition [ftip-00AL], and let $P_0$ and $P_1$ denote the pre- and post-training protocols whose successful-support probabilities are defined in Definition [ftip-005U]. The slice-conditioned success change is

\[ \Delta _{\rm suc}(C;P_1,P_0) =\mathbb E_{X\sim \mu _{\rm ev}(\cdot \mid C)} \left [p_{P_1}(X)-p_{P_0}(X)\right ]. \]

The evaluation interface, inference budget, and success predicate are held fixed across the two terms. The quantity is an average change on one declared slice. It neither locates the change inside the model nor identifies which training coordinate caused it.

Definition. Localized and distributed degradation [ftip-00AP]

Fix pairwise disjoint evaluated slices $C_1,\ldots ,C_J$ and a declared degradation threshold $\kappa >0$. Define the set of materially degraded slices

\[ \mathcal H_\kappa (P_1,P_0) =\left \{j:\Delta _{\rm suc}(C_j;P_1,P_0)\leq -\kappa \right \}. \]

Degradation is localized relative to this slice family and threshold when $|\mathcal H_\kappa |=1$. It is distributed when $|\mathcal H_\kappa |\geq 2$.

These terms depend on the chosen partition, threshold, and evaluation budget. They do not turn a finite benchmark vector into a statement about all capabilities.

Example. Narrow and broad random-reward paths [ftip-00AQ]

Section 5.3 and Appendix F of Demystifying Reinforcement Learning Post-Training of Language Models[clay2026demystifying] supply two OLMo paths; Figure 4 reports their outcomes. Both replace the usual verifiable reward with a random scalar drawn from $\operatorname {Unif}[0,1]$, but their prompt configurations differ.

$\text{[math]}$

In the narrow SFT-start path, GSM8K accuracy falls from 86 to about 32 by step 400, while the reported MMLU and IFEval changes are much smaller. In the broad path, the source reports an entropy spike near step 400 together with collapse on GSM8K, MMLU, and IFEval. The diagram records that interpretation; it does not isolate prompt breadth from pool size, mixture, or repetition.

Remark. What the source observes about spurious reward [ftip-00AR]

Section 5.3 and Figures 3--4 of Demystifying Reinforcement Learning Post-Training of Language Models[clay2026demystifying] report that the Qwen narrow configuration retains higher MATH and AMC Acc@1 than its broad configuration under random reward. The OLMo experiment reports the different evaluation patterns summarized in Example [ftip-00AQ].

These are finite empirical observations. The random scalar reward removes task alignment from one feedback channel, but it does not hold all other training coordinates fixed. The reported entropy is a named token statistic, not a direct measure of capability or acquisition.

Example. Counterexample: equal marginal reward, unequal prompt-local damage [ftip-00AS]

Let two evaluation slices $C_1,C_2$ have equal mass. A feedback channel returns an independent Bernoulli reward with mean $1/2$ under either of two post-training protocols. Suppose their fixed-interface success changes are

\[ \left (\Delta _{\rm suc}(C_1),\Delta _{\rm suc}(C_2)\right ) =(-1,0) \quad \hbox {or}\quad \left (-\tfrac 12,-\tfrac 12\right ). \] $\text{[math]}$

The reward law and its mean are identical, but the slice-level evaluation vectors differ. Hence marginal training reward does not determine whether damage is localized or distributed. This is a finite counterexample, not a claim about the mechanism in the source experiment.

Remark. Coverage, feedback resolution, and prompt-conditioned effects [ftip-00AT]

The source cells motivate tests of starting-policy coverage, feedback resolution, and prompt-conditioned damage. They do not establish that dense reward creates mathematical support, that zero sampled hits imply zero probability, or that random reward has one model-independent effect.

Separating the effects of coverage, feedback resolution, and prompt breadth requires stronger controls. One comparison fixes prompt laws and varies only the feedback sigma-algebra. Another fixes feedback and evaluation while varying one starting-policy coordinate. Confidence regions for the complete vector of slice-conditioned changes would quantify effects that an aggregate score can hide.

The reported observations remain conditional on their experimental configurations. They do not establish general results about capability acquisition, elicitation, or the optimal allocation of post-training compute.

Finite consequences and counterexamples [ftip-00JE]

Finite sampling, reward information, and deterministic execution give conditional conclusions about post-training. Their counterexamples show why the task law, feasible set, and available information cannot be omitted.

This synthesis brings together the finite results used in the evaluation analysis. The conceptual-discovery chapter asks whether such arguments can constrain a complete learning lineage, including changes to representations, curricula and research procedures.

Probability, information, and execution [ftip-00I1]

Discovery probabilities, support, proxy error, feedback resolution, and replay each constrain a different part of a comparison. Capability remains conditional behavior under a declared intervention and evaluation law, rather than a scalar latent property.

Finite probability and execution results [ftip-00I2]

The conclusions depend on their declared probability laws, finite index sets, positivity conditions, and execution inputs. Changing one of these assumptions can change the result even when the observed score is unchanged.

Convention. Domains of the finite results [ftip-00I5]

Each result concerns its declared carriers $X_1,\ldots ,X_n$, with fixed equality and order conventions. Finiteness, probability laws, and positivity conditions are hypotheses of the result; they cannot be inferred from a finite observation alone.

Theorem. Independent discovery probability [ftip-00I7]

For $B\in \mathbb N$ independent discovery events with the probabilities declared in Convention [ftip-0077], the identity in Theorem [ftip-0078] gives $\Pr (D_B)=1-\prod _{i=1}^{B}(1-p_i)$.

Theorem. Discovery budget for a failure threshold [ftip-00I8]

Under $0<p<1$, $0<\delta <1$, and an integer budget $B\geq 1$, the failure constraint is equivalent to the following bound.

\[B\geq \left \lceil \frac {\log \delta }{\log (1-p)}\right \rceil .\]

The result is proved in Corollary [ftip-0079].

Theorem. Support under finite exponential tilting [ftip-00I9]

Under the finite-carrier and positive-temperature hypotheses of Theorem [ftip-007C], normalized exponential tilting preserves the base law's positive support.

Theorem. Uniform proxy error and objective regret [ftip-00IA]

For a finite common feasible set and a common regularizer, the uniform proxy error hypothesis in Definition [ftip-007G] yields the two-epsilon objective regret bound proved in Theorem [ftip-007H]; the conclusion concerns the regularized objective named there.

Theorem. Utility under a change of evaluation law [ftip-00IB]

If the evaluation laws and finite response space satisfy the total-variation hypotheses of Theorem [ftip-00GM], then the expectation difference is bounded by the stated sup-norm times total variation.

Theorem. Response classes under feedback refinement [ftip-00IC]

For finite response set $\mathcal Y$ and deterministic rewards, a reward channel that refines the equality partition of another channel separates at least as many response classes, as proved in Lemma [ftip-00AG].

Theorem. Equal execution inputs give equal replay traces [ftip-00IJ]

If the execution map is deterministic in all coordinates fixed by Definition [ftip-00HF], then equal input, revision, environment, and seed records produce equal finite traces, as proved in Theorem [ftip-00HG].

Theorem. Observational equivalence under a common update kernel [ftip-00IK]

For the finite transcript and world kernels declared in Definition [ftip-008C]--Definition [ftip-008D], observationally equivalent worlds induce the same output law for any common randomized post-training kernel, as proved in Theorem [ftip-007S].

Theorem. Conditions for an admissible commit decision [ftip-00IL]

Under the typed audit record of Definition [ftip-00HL], a commit is admissible exactly when all required checks pass, the digest matches, and the recorded decision is $commit$, as proved in Theorem [ftip-00HM].

Theorem. Admission of a finite audit plan [ftip-00IM]

A finite audit plan with nonnegative stage costs is admissible exactly when its declared additive cost is within budget; adding a positive stage beyond slack is inadmissible, by Theorem [ftip-00HX].

Counterexamples and restricted assumptions [ftip-00ID]

Finite constructions can separate average success from coverage, proxy accuracy from evaluation quality, and observed gradients from the hypotheses needed to bound them. Each separation identifies an assumption that a stronger conclusion would require.

Remark. What a finite counterexample establishes [ftip-00IE]

A finite assignment satisfying a claim's hypotheses and violating its conclusion disproves that universal claim. It does not estimate how often the failure occurs under another distribution or establish a general failure rate.

Remark. Changing the model or evaluation changes the claim [ftip-00IF]

A conclusion about fixed carriers, policies, feedback, budgets, and evaluation laws need not survive a change to those objects. Finite zero hits do not imply zero support, empirical benchmark movement does not imply acquisition, and an exact-optimizer result need not hold for an approximate optimizer without an additional error bound.

Example. Equal averages and zero hits do not identify coverage [ftip-00IG]

The two-task examples in Example [ftip-007E] and Example [ftip-00A3] show that equal one-shot averages or zero observed hits can coexist with different coverage or nonzero latent rates. These conclusions use their declared iid model and do not estimate rates outside it.

Example. Small training-law error can reverse evaluation selection [ftip-00IH]

In the two-point construction of Example [ftip-007J], training-law average error is small while evaluation selection reverses. The missing hypothesis is a uniform bound on the evaluated feasible set; an informal promise of similar distributions does not supply it.

Remark. Untied heads, shared gradients, and checkpoint forecasts [ftip-00II]

For DGG, the untied head identity and its finite-batch Jensen bound in Theorem [ftip-007Z] and Theorem [ftip-0081] differ from an intermediate-occurrence bound. A shared intermediate weight requires the sum over all occurrences in Theorem [ftip-007Z]; the source's local hypotheses do not establish its claimed total-gradient bound in [miao2026when, Section 4.2.1, Lemma 1, Theorem 1, and Appendix A.3]. The cancellation example Example [ftip-0084] also separates the active clipping interval from the raw-surrogate extension.

NExt's empirical checkpoint forecasts in [chen2026lowrank, Sections 3.2, 4.1--4.3, and 5.1] do not establish a universal subspace-collapse, capability, or speed theorem. The spectral and evaluation quantities in Definition [ftip-0086] through Remark [ftip-008B] measure different trajectory properties.

Mathematical notation [ftip-00IN]

Prompt and response spaces, probability laws, policies, rewards, evaluation scores, and execution traces are distinct mathematical objects.

Convention. Prompt spaces, response spaces, and laws [ftip-00IO]

Use $\mathcal X$ for prompts, $\mathcal Y$ for finite responses, $\Omega $ for sample space, $\mathsf P$ for a declared protocol, and $\mu $ for a prompt law. A probability law is written $\Pr $ only after its sample space is named.

Convention. Policies and reference policies [ftip-00IP]

Use $\pi $ for a policy, $\pi _{\rm base}$ for a reference policy, and $\pi _\theta $ for a parameterized policy. A policy is always typed as a kernel from the declared prompt carrier to the declared response carrier.

Convention. Rewards and evaluation utilities [ftip-00IQ]

Use $r:\mathcal X\times \mathcal Y\to \mathbb R$ for a deterministic reward and $u$ for an evaluation utility. Proxy, verifier, and process rewards retain their local subscripts; no bare symbol silently changes meaning.

Convention. Model artifacts and evaluation scores [ftip-00IR]

Use $M$ for a model artifact, $\mathsf E$ for an evaluation protocol, and $J_{\mathsf E}(M)$ for its declared score. A comparison must state the invariant evaluation law before subtracting two scores.

Convention. Traces, states, lineages, and costs [ftip-00IS]

Use $z_{0:n}$ for a finite trace, $s_t$ for a harness state, $\ell $ for a lineage, and $c$ for an execution cost. Event logs and summaries are distinct carriers even when one is computed from the other.

Backlinks

Architecture-conditioned potential and computation [ftip-00JC]

This chapter refines the evaluation framework for claims that depend on architecture, optimization geometry or a systems implementation. It begins with the decoder-only Transformer, then introduces architecture-indexed frontiers and matched-compute comparisons, and studies how an optimizer can depend on the representation of its parameters.

The reference architecture specifies how token probabilities are computed; it does not replace the shared task and evaluation interfaces. Readers whose question does not depend on an architecture or optimizer can continue directly to conceptual discovery. Changes to retained context and rollout computation are treated with agent state.

Decoder-only Transformer architecture [ftip-000K]

The continuation law in Definition [ftip-000I] specifies what probabilities a language model exposes. A decoder-only Transformer specifies how a sequence of token identifiers is transformed into those probabilities. This section introduces one component at a time, with dimensions fixed before the component formulas are used.

Notation. Tensor dimensions, positions, heads, and layers [ftip-000L]

Let $T$ be the maximum token positions processed together, $d$ the residual-stream width, $d_{\mathrm {ff}}$ a feed-forward hidden width, $H$ the number of attention heads, $d_h$ the query-key width of one head, and $L$ the number of decoder layers. Positions are indexed by $t,s\in \{1,\ldots ,T\}$, heads by $h\in \{1,\ldots ,H\}$, and layers by $\ell \in \{1,\ldots ,L\}$.

A batch of $B$ hidden-state sequences is a tensor in $\mathbb R^{B\times T\times d}$. When the batch coordinate is irrelevant, write $X\in \mathbb R^{T\times d}$ and let $X_t\in \mathbb R^d$ denote its row at position $t$. Matrix products act on the final coordinate.

Definition. Token embedding [vaswani2017attention, sec. 3.4, Embeddings and Softmax] [ftip-000M]

Let $\mathcal V$ be the vocabulary of Definition [ftip-000D] and $d$ the residual width of Notation [ftip-000L]. A token embedding matrix is a learned matrix $E\in \mathbb R^{|\mathcal V|\times d}$. The hidden vector assigned to token $a\in \mathcal V$ is the row $E_a\in \mathbb R^d$.

For a sequence $x_{1:n}$, embedding lookup produces the matrix $X^{\mathrm {tok}}\in \mathbb R^{n\times d}$ with row $X^{\mathrm {tok}}_t=E_{x_t}$. This operation maps token identifiers to vectors; it does not yet encode their positions.

Definition. Embedding-stage positional transformation [ftip-000N]

Using the dimensions of Notation [ftip-000L], an embedding-stage positional transformation is a specified family of position-dependent transformations $r_t:\mathbb R^d\to \mathbb R^d$, one for each $t\in \{1,\ldots ,T\}$. Applied after the token embedding of Definition [ftip-000M], it produces the initial hidden vector \[ X^{(0)}_t=r_t(E_{x_t}). \] The family $(r_t)_{t=1}^T$ is part of the architecture, even when it has no learned parameters.

Remark. Where positional information enters [ftip-000O]

Definition [ftip-000N] covers only mechanisms acting on the hidden vector at the embedding stage. The original Transformer adds a fixed or learned vector $p_t$ to the token embedding, so $r_t(z)=z+p_t$; see [vaswani2017attention, §3.5].

Relative-bias methods instead alter an attention score as a function of the displacement $t-s$. ALiBi gives one such construction in [press2021train, §3].

Rotary position embedding applies position-indexed rotations to queries and keys rather than adding a vector to the residual stream; see RoFormer, Section 3.2, equations (14)--(16) [⧉]. These mechanisms are not interchangeable choices of one tensor: they act at different points in the computation, so a model specification must name the mechanism and placement.

Definition. Query, key, and value projections [vaswani2017attention, sec. 3.2.1] [ftip-000P]

Let $X\in \mathbb R^{T\times d}$ use the notation of Notation [ftip-000L]. For attention head $h$, choose learned matrices \[ W_h^Q,W_h^K\in \mathbb R^{d\times d_h}, \qquad W_h^V\in \mathbb R^{d\times d_v}, \] where $d_v$ is the value width of one head. The query, key, and value matrices are \[ Q_h=XW_h^Q,\qquad K_h=XW_h^K,\qquad V_h=XW_h^V. \] Their rows associate a query, key, and value vector with each token position.

Definition. Scaled dot-product attention [vaswani2017attention, sec. 3.2.1, equation (1)] [ftip-000Q]

Using the head widths in Definition [ftip-000P], let $T_q,T_k\geq 1$ be query and key-value sequence lengths. For $Q\in \mathbb R^{T_q\times d_h}$, $K\in \mathbb R^{T_k\times d_h}$, and $V\in \mathbb R^{T_k\times d_v}$, scaled dot-product attention is \[ \operatorname {Att}(Q,K,V) =\operatorname {softmax}\left (\frac {QK^{\mathsf T}}{\sqrt {d_h}}\right )V. \] The softmax is applied across each row of the $T_q\times T_k$ score matrix, so every output row is a convex combination of the value rows.

Example. A two-token attention calculation [ftip-000R]

A scalar attention head on two positions makes the masked softmax and its resulting value average explicit.

$\text{[math]}$

Take head width $d_h=1$, so the scale $1/\sqrt {d_h}$ is one. At position two both keys are causally available. Therefore \[ \operatorname {softmax}(0,\log 3) =\frac {(1,3)}{1+3} =\left (\tfrac 14,\tfrac 34\right ), \qquad z_2=\tfrac 14\cdot 2+\tfrac 34\cdot 6=5. \]

Substitution into the scaled dot-product attention of Definition [ftip-000Q] produces the displayed value. Projection matrices, multiple heads, normalization, and residual connections remain outside this scalar calculation; it represents only one attention operation.

Definition. Causal attention mask [vaswani2017attention, sec. 3.2.3] [ftip-000S]

For a length-$T$ sequence, the causal attention mask is the matrix $M\in (\mathbb R\cup \{-\infty \})^{T\times T}$ with \[ M_{ts}= \begin {cases} 0,&s\leq t,\\ -\infty ,&s>t. \end {cases} \] Masked self-attention replaces the score matrix in Definition [ftip-000Q] by $QK^{\mathsf T}/\sqrt {d_h}+M$. Consequently, the output at position $t$ has zero attention weight on every later position $s>t$.

Example. A length-four causal mask [ftip-000T]

For four positions, a causal mask admits the current and earlier keys and excludes every future key.

$\text{[math]}$

Rows index querying positions and columns index key positions. In row three, the first three entries remain available and the fourth receives $-\infty $; after softmax the fourth token therefore contributes exactly zero.

The construction is the autoregressive mask of [vaswani2017attention, Section 3.2.3]. It enforces a dependency restriction. Effective use of the available earlier positions is a separate model property.

Definition. Multi-head self-attention [vaswani2017attention, sec. 3.2.2] [ftip-000U]

For each head $h\in \{1,\ldots ,H\}$, form $Q_h,K_h,V_h$ as in Definition [ftip-000P] from the same input $X$, and apply the masked attention of Definition [ftip-000S]: \[ Z_h=\operatorname {softmax}\left ( \frac {Q_hK_h^{\mathsf T}}{\sqrt {d_h}}+M \right )V_h. \] With $W^O\in \mathbb R^{H d_v\times d}$, multi-head self-attention is \[ \operatorname {MHA}(X) =\operatorname {Concat}(Z_1,\ldots ,Z_H)W^O. \] It returns one width-$d$ vector at every input position.

Remark. Multi-head, grouped-query, and compressed-attention variants [ftip-000V]

The multi-head construction in Definition [ftip-000U] stores a key and value projection per query head. Grouped-query attention instead partitions query heads into groups that share key and value heads; see Ainslie et al., Section 2 [⧉]. This changes parameter and cache shapes while preserving the query-key-value semantics.

Multi-head latent attention first compresses key-value information through a lower-dimensional latent representation and reconstructs head-specific quantities; see DeepSeek-V2, Section 2.1 [⧉]. DeepSeek-V3, Section 2.1, records a later use of that architecture [⧉]. These architectural differences do not establish that their post-training effects equal those of ordinary multi-head attention.

Definition. Positionwise feed-forward sublayer [vaswani2017attention, sec. 3.3, equation (2)] [ftip-000W]

With widths $d$ and $d_{\mathrm {ff}}$ from Notation [ftip-000L], a positionwise feed-forward sublayer is a map $F:\mathbb R^d\to \mathbb R^d$ of the form \[ F(z)=\phi (zW_1+b_1)W_2+b_2, \] where $W_1\in \mathbb R^{d\times d_{\mathrm {ff}}}$, $W_2\in \mathbb R^{d_{\mathrm {ff}}\times d}$, the vectors $b_1,b_2$ have the corresponding widths, and $\phi $ is applied coordinatewise. The same map is applied independently at each token position.

Definition. Sparse mixture-of-experts routing [ftip-000X]

Let $J\geq 1$ and $1\leq k\leq J$. Let $F_1,\ldots ,F_J:\mathbb R^d\to \mathbb R^d$ be feed-forward experts and let $g:\mathbb R^d\to \mathbb R^J$ be a router. For a hidden vector $z$, let $S_k(z)$ be the indices of the $k$ largest coordinates of $g(z)$, breaking equal scores by a fixed total order on expert indices. A sparse mixture-of-experts layer returns \[ F_{\mathrm {MoE}}(z) =\sum _{j\in S_k(z)}\alpha _j(z)F_j(z), \qquad \alpha _j(z)= \frac {\exp g_j(z)}{\sum _{i\in S_k(z)}\exp g_i(z)}. \] Only the selected experts are evaluated for that token.

Remark. Dense and routed feed-forward computation [ftip-000Y]

The dense sublayer in Definition [ftip-000W] evaluates one learned map for every token. The sparsely gated construction of Shazeer et al., Section 2, selects among multiple experts and introduces an additional routing and load-balancing problem [⧉]. The top-$k$ selection rule is described in Definition [ftip-000X].

Parameter count, active parameter count, and executed operations therefore need not agree. A post-training comparison involving routed models must state which quantity is held fixed and whether the router itself is updated.

Example. Top-k expert routing and its DeepSeek counterpart [ftip-000Z]

A routed mixture-of-experts layer uses router scores to activate only a declared subset of routed experts, then combines their outputs with any shared expert path.

$\text{[math]}$

The revision-pinned configuration [⧉] declares $256$ routed experts, $6$ experts selected per token, and one shared expert. This instantiates the selection interface while leaving the artifact's router scores, load-balancing mechanism, and expert outputs unspecified.

Definition. Residual connection [vaswani2017attention, sec. 3.1] [ftip-0010]

Using the sequence and residual dimensions of Notation [ftip-000L], let $F:\mathbb R^{T\times d}\to \mathbb R^{T\times d}$ be a sublayer whose input and output have the same shape. A residual connection forms \[ R_F(X)=X+F(X). \] The addition is coordinatewise. Shape equality is part of the construction; the residual path is not a concatenation of features.

Definition. Layer normalization [ba2016layer, sec. 3] [ftip-0011]

For the residual width $d$ of Notation [ftip-000L], a declared numerical stabilizer $\varepsilon _{\mathrm {LN}}>0$, and $z=(z_1,\ldots ,z_d)\in \mathbb R^d$, define $\mu (z)=d^{-1}\sum _i z_i$ and $\sigma ^2(z)=d^{-1}\sum _i(z_i-\mu (z))^2$. With learned vectors $\gamma ,\beta \in \mathbb R^d$, layer normalization is \[ \operatorname {LN}(z) =\gamma \odot \frac {z-\mu (z)\mathbf 1} {\sqrt {\sigma ^2(z)+\varepsilon _{\mathrm {LN}}}}+\beta . \] It is applied independently to the hidden vector at each token position. The declared stabilizer makes the numerical operator total on $\mathbb R^d$; the unstabilized expression is recovered where $\sigma ^2(z)>0$ by setting $\varepsilon _{\mathrm {LN}}=0$.

Definition. Normalization placement [ftip-006E]

For a residual stream $X$, a sublayer $F$, and a normalization operator $N$, post-normalization applies $N(X+F(X))$. In pre-normalization, the sublayer receives a normalized input and the residual update is $X+F(N(X))$.

Placement is an architecture coordinate, separate from the formula for $N$. Every decoder block must state which residual branches use which placement.

Definition. Root mean square layer normalization [zhang2019root, Section 4, equation (4)] [ftip-006F]

For $z\in \mathbb R^d$, learned scale $\gamma \in \mathbb R^d$, and a declared numerical stabilizer $\varepsilon _{\mathrm {RMS}}>0$, root mean square layer normalization is

\[ \operatorname {RMSNorm}(z) =\gamma \odot \frac {z}{\sqrt {d^{-1}\sum _{i=1}^{d}z_i^2+ \varepsilon _{\mathrm {RMS}}}}. \]

Unlike layer normalization in Definition [ftip-0011], this operator does not subtract the coordinate mean. The declared stabilizer makes it total at the zero vector; setting the stabilizer to zero recovers the source expression on nonzero inputs.

Remark. Normalization operator and placement are separate choices [ftip-006G]

Ba et al. define layer normalization in [ba2016layer, Section 3]. The original Transformer applies it after residual addition in [vaswani2017attention, Section 3.1]. Xiong et al. distinguish post-normalized and pre-normalized Transformers in [xiong2020layer, Section 3.3]. Either the normalization operator or its placement can change while the other is held fixed.

A later stability statement must therefore name both choices. A result about the normalization formula alone does not determine the residual path.

Definition. Decoder-only Transformer language model [phuong2022formal, sec. 6, Decoder-only transformers] [ftip-0012]

A decoder-only Transformer language model consists of:

the token interface in Definition [ftip-000E];
an initial representation from Definition [ftip-000M] and Definition [ftip-000N];
a stack of $L$ causal decoder layers built from the attention, feed-forward, residual, and normalization constructions in Definition [ftip-000S]--Definition [ftip-0011]; and
an output map from the final hidden vector to token logits.

If the final hidden vector at position $t$ is $h_t\in \mathbb R^d$, an output matrix $W_U\in \mathbb R^{|\mathcal V|\times d}$ and bias $b_U\in \mathbb R^{|\mathcal V|}$ give logits $z_t=W_Uh_t+b_U$. The softmax \[ \pi _\theta (a\mid x_{1:t}) =\frac {\exp (z_{t,a})} {\sum _{v\in \mathcal V}\exp (z_{t,v})} \] realizes the conditional next-token law of Definition [ftip-000H]. The parameter tuple $\theta $ contains every learned tensor named by this architecture.

Example. Decoder-only dataflow and a DeepSeek V4-Flash provenance overlay [ftip-0013]

A decoder pipeline maps token identifiers through hidden states to logits; the released model instance and family report provide different evidence about that pipeline.

$\text{[math]}$

The pinned configuration [⧉] declares 43 layers, hidden width 4096, 64 attention heads, and one key--value head. Section 2 of the family report [⧉] describes the preview architecture. Neither source identifies a complete 0731-specific architecture or training delta. The diagram records the interface, not a small copy of the released model.

Architecture-indexed ceilings and matched compute [ftip-00JF]

This optional section refines the architecture-neutral capability framework only when the question names an architecture, optimizer, or systems implementation. The general model/interface formulation remains available when these fields are unnecessary.

Architecture comparisons below are conditional on a declared task family, evaluation law, intervention class, and scalar cost. Toy bounds are finite results under their displayed assumptions and do not transfer to concrete model instances without verified interface, task, and cost assumptions.

Architecture, optimizer, and systems variables [ftip-00JG]

Architecture, optimizer, and systems fields refine the model/interface description. A comparison depends only on the fields named in its assumptions and conclusion.

Definition. Optional architecture refinement [ftip-00JH]

Let $M$ denote an existing model intervention. An architecture refinement is a record $A=(\mathcal X,\mathcal Y,\operatorname {Map}_A)$ whose map realizes the same declared input and output interface. Forgetting $A$ returns $M$; no architecture claim is made when the field is omitted.

Definition. Optimizer and systems records [ftip-00JI]

When needed, extend the record by $O$ for optimizer and state-update rules, and $S$ for kernels, precision, memory, cache, and scheduling. The triplet $(A,O,S)$ is descriptive; it is not a scalar intelligence score.

Definition. Architecture-indexed evaluation functional [ftip-00JJ]

Fix an architecture $A$, its base artifact, a declared intervention set $\mathfrak I_A$, and the common evaluation interface of Convention [ftip-005D]. Each $\eta \in \mathfrak I_A$ specifies a protocol $P_{A,\eta }=\operatorname {PostTrain}(A,\eta )$ in that measurable and absolutely integrable evaluation domain, with real performance $J_{\rm ev}(P_{A,\eta })$. Let $c_A(\eta )\in [0,+\infty ]$ be its scalar cost under the study's declared accounting rule: a fixed measured cost or an expected nonnegative cost, as specified. For a finite budget $C\in \mathbb R_{\geq 0}$, define \[ V_A(C)=\sup \left \{ J_{\rm ev}(P_{A,\eta }): \eta \in \mathfrak I_A,\ c_A(\eta )\leq C \right \}\in \overline {\mathbb R}. \] Infinite-cost interventions are infeasible at every such budget.

As in Definition [ftip-005M], the empty feasible set has value $-\infty $; an unbounded-above feasible score set has value $+\infty $. The value is real exactly when that set is nonempty and bounded above. Increasing $C$ enlarges the feasible set, so $V_A$ is nondecreasing. A finite supremum need not be attained by any intervention.

Frontier differences and derivatives are ordinary real operations only where the relevant values are finite, with differentiability additionally required for a derivative. Cost ratios use the domain in Definition [ftip-00JP]. The allowed intervention set, evaluation law, and accounting rule are part of the definition, so $V_A$ is not a universal intelligence function.

Remark. Architecture-neutral projection and gradual complexity [ftip-00JK]

A statement that depends only on the model/interface map is invariant under changes to architecture $A$, optimizer $O$, or systems $S$ that leave that map fixed. A conclusion about one of those fields instead requires assumptions that distinguish its possible values.

Definition. Three ceiling layers [ftip-00JL]

For fixed task and evaluation laws, distinguish the representational ceiling of $A$, the optimizer-reachable ceiling of $(A,O)$, and the systems-feasible frontier of $(A,O,S,C)$. Each layer is conditional on the objects named; none implies that the next layer attains it.

Remark. Architecture effects are comparative estimands [ftip-00JM]

A difference between two architectures is meaningful only after the task, data, post-training, inference, evaluation, and cost records identify which coordinates are held fixed and which are allowed to vary.

Depth reuse and latent recurrence [ftip-00KG]

Repeated depth, latent feedback, and recirculation change the computation performed at inference. Relative to the fixed reference autoregressive Transformer, their costs include any extra passes, state, and prefill work.

Definition. Computation-axis record [ftip-00KH]

For an architecture record $A$, distinguish token steps, feed-forward depth passes, recurrent passes, and latent-state updates. Let $d(A,r)$ be the declared number of depth passes in run $r$; it is a computation coordinate, not a synonym for parameter count or reasoning quality.

Remark. Looped Transformers as a source result [ftip-00KI]

Giannou et al. show that a fixed shallow Transformer placed in a loop can execute programmed iterative computations, including conditional branches and in-context algorithms, in Looped Transformers as Programmable Computers[giannou2023looped]. This is a source result under its program, state, and precision assumptions; it is not a universal claim about language-model capability.

Remark. Parameter sharing and width tradeoffs [ftip-00KJ]

Xue et al. report that depth sharing can reduce trainable parameters while limiting modeling capacity, and study width and mixture-of-experts remedies in Go Wider Instead of Deeper[xue2021wider]. The comparison is empirical and task-specific; it does not identify a depth-independent ceiling.

Remark. Virtual logical depth and scaling [ftip-00KK]

Zhu et al. vary virtual logical depth by reusing weights and report reasoning gains at nearly fixed parameter count in Beyond Parameters: Exploring Virtual Logic Depth for Scaling Laws[zhu2025virtualdepth]. Their results motivate a depth coordinate in a scaling study, while leaving knowledge capacity, optimization, and transfer dependent on the declared training and evaluation protocol.

Remark. Recirculation as an inference intervention [ftip-00KL]

Mozer et al. introduce inference-time recirculation that feeds latent states back through an off-the-shelf model and report task improvements in Recirculation[mozer2026recirculation]. The added serial prefill and adaptive tuning belong in the systems and inference records; the observation is not a proof of a new representational ceiling.

Remark. Latent feedback with a preserved Transformer interface [ftip-00KM]

Wang et al. widen the feedback channel between decoding steps with latent feedback while retaining a Transformer and language-modeling interface in Full-bandwidth transformer[wang2026fullbandwidth]. Any reported token or accuracy savings are conditional on the scheduled multi-pass training and measured decoding cost.

Remark. Latent prediction as a training intervention [ftip-00KN]

Teoh et al. add next-latent prediction to next-token training and report compact predictive states without changing the Transformer interface in Next-Latent Prediction Transformers Learn Compact World Models[teoh2025nextlatent]. This belongs to the training intervention record, not to an architecture-only comparison.

Example. Equal parameters do not fix effective depth [ftip-00KO]

Two runs can share parameter count while using different numbers of recurrent or latent-feedback passes. A matched study must therefore report the computation-axis record and cannot infer an architecture ceiling from parameters alone.

Definition. Depth-reuse model-instance record [ftip-00KP]

For a depth-reuse comparison, record shared weights, pass count, stopping rule, recurrent state, training objective, token and pass FLOPs, prefill and decode latency, memory, and evaluation seeds. A pass-count change is an intervention even when the parameter tensor is unchanged.

Example. Fixed-recipe depth comparison [ftip-00KQ]

In a fixed-recipe arm, hold data, optimizer, feedback, stopping rule, and evaluation law fixed while varying only the declared depth-reuse intervention. The resulting difference is an estimand for that recipe, not a best-achievable comparison.

Remark. Scaling surfaces rather than a single law [ftip-00KR]

A depth-reuse study may fit a surface over parameters, tokens, passes, and cost. The fit is an empirical summary over its measured range; it does not establish an asymptotic law or a universal saturation point.

Remark. Architecture and training are separable records [ftip-00KS]

Weight sharing, latent objectives, and recirculation can alter optimization without changing the declared external interface. The comparison must retain architecture, training intervention, and systems coordinates separately.

Remark. Depth reuse across passes, parameters, and tasks [ftip-00KT]

The cited looped, virtual-depth, latent-feedback, and recirculation results do not show that more passes always improve capability, that parameter sharing dominates added parameters, or that inference-time gains transfer to training or to another task family. Each transfer requires a matched model-instance record.

Ceilings, frontiers, and saturation [ftip-00JN]

Definition. Fixed-recipe and tuned frontiers [ftip-00JO]

The fixed-recipe frontier evaluates one common $\eta $. The equal-tuning frontier permits architecture-specific choices from a predeclared trial set with the same tuning data and budget. The restricted envelope permits a predeclared class of interventions under $C$. These are distinct estimands.

Definition. Iso-quality cost ratio [ftip-00JP]

For a finite real target $q\in \mathbb R$ and the frontier of Definition [ftip-00JJ], define the inverse cost \[ C_A(q)=\inf \{C\in \mathbb R_{\geq 0}:V_A(C)\geq q\} \in [0,+\infty ], \qquad \inf \varnothing =+\infty . \] The iso-quality ratio is defined only on the domain \[ \rho _{A/B}(q)=\frac {C_A(q)}{C_B(q)}, \qquad 0\leq C_A(q)<+\infty , \quad 0<C_B(q)<+\infty . \] The architectures must use the same target, evaluation interface, cost units, and comparison arm. The ratio compares efficiency under these choices.

An inverse cost is a threshold infimum, not an executable minimum. The infimum over budgets may be unattained; even if a budget satisfies $V_A(C)\geq q$, its performance supremum may be unattained at $q$. An actual target-achieving intervention requires a separate witness.

Positive individual costs do not guarantee a positive inverse cost. For interventions $\eta _n$, $n\geq 1$, with score $1$ and cost $1/n$, the target $q=1$ has $C_A(1)=0$, although every intervention costs more than zero. If both architectures have this family, the putative ratio is $0/0$ and is excluded by the displayed domain.

Definition. Operational cost vector and scalarization [ftip-00JQ]

Record training FLOPs, inference FLOPs, wall time, memory, energy, and hardware separately as $\mathbf c$. A scalar cost $C=w\cdot \mathbf c$ is a declared study choice with nonnegative units $w$; changing $w$ changes the frontier and must not be hidden as architecture quality.

Theorem. Finite-window representational obstruction [ftip-00JR]

Consider a causal system whose state after each prefix has at most $K$ distinct values, and a task with $K+1$ prefixes requiring pairwise distinct continuation labels. By the pigeonhole principle two prefixes share a state, so at least one continuation label is wrong. This is a finite toy obstruction only; it does not bound a concrete Transformer or KDA instance without a proved reduction to this state model.

Theorem. A conditional saturation certificate [ftip-00JS]

Let $V_A$ be the nondecreasing frontier of Definition [ftip-00JJ], bounded above by a finite $U\in \mathbb R$. If an intervention reaches a real target $q<U$ at a finite cost $C_q\geq 0$, then at every finite $C\geq C_q$, $q\leq V_A(C)\leq U$. In particular $V_A(C)$ is real and $0\leq U-V_A(C)\leq U-q$. This elementary certificate is conditional on the bound $U$; it does not assert that real intelligence saturates.

Example. Equal scalar cost does not identify architecture [ftip-00JT]

Two systems can have the same scalar $C$ while differing in memory, latency, and training allocation. A cost-weight change can reverse their ordering without changing either system. Thus a one-factor comparison needs a fixed scalarization and a reported cost vector; no architecture effect follows from equal $C$ alone.

Matched-compute comparison protocol [ftip-00JU]

Definition. Common-recipe comparison arm [ftip-00JV]

A common-recipe arm fixes training data, optimizer family, schedule, post-training procedure and feedback, training and post-training budgets, inference budget, and evaluation protocol and draws. Interface-compatible parameter choices are declared in advance as part of each permitted architecture package. The comparison varies that package under the common recipe; any further deviation is recorded as a separate comparison coordinate. Its score difference measures performance under this recipe, not the best attainable result for either architecture.

Definition. Equal-tuning-budget comparison arm [ftip-00JW]

An equal-tuning-budget arm permits architecture-specific tuning under a predeclared protocol that fixes the tuning data and its volume, trial count, selection rule, stopping rule, and tuning cost for both architectures. The cost uses the same declared accounting rule. The resulting comparison measures practical performance under this equal optimization effort; it does not determine best possible training or a representation-only limit.

Definition. Restricted-envelope comparison arm [ftip-00JX]

A restricted-envelope arm declares an allowed intervention class $\mathcal E_A\subseteq \mathfrak I_A$ for each architecture, specifying the permitted training, post-training, and inference procedures. The classes expose exclusions, search budgets, seeds, and stopping rules. Restrict the evaluation functional of Definition [ftip-00JJ] to $\mathcal E_A$ and compare the resulting suprema under the same finite cost cap $C\geq 0$, common evaluation law, and common declared cost accounting.

The measurable, integrable evaluation domain and extended-real conventions of Definition [ftip-00JJ] apply: an empty feasible class has value $-\infty $, an unbounded-above feasible score set has value $+\infty $, and a finite supremum need not be attained. The allowed class is part of the comparison, so its envelope is a restriction-dependent quantity, not an unrestricted architectural ceiling.

Definition. Model-instance comparison evidence record [ftip-00JY]

A model-instance record contains architecture/checkpoint identity, parameter count, data and post-training recipe, optimizer, precision, hardware, cache policy, training and inference cost vectors, task suite, and independent evaluation seed. A missing field makes an architecture attribution unknown.

Remark. Systems optimizations are measured interventions [ftip-00JZ]

FlashAttention, KV-cache layouts, quantization, batching, and kernels can change feasible cost. If they approximate, truncate, or alter precision of the mathematical computation, the record must mark that as a systems intervention rather than as a pure architecture comparison.

Theorem. Matched-protocol difference is an estimand [ftip-00K0]

Given $n\geq 1$ evaluation draws, let $\widehat V_A(C)$ be the sample mean of finite real evaluation scores for architecture $A$ at a finite cost cap $C\geq 0$. Under a fixed evaluation law and common-recipe arm, the finite difference $\widehat V_A(C)-\widehat V_B(C)$ is a well-defined empirical estimand.

It does not identify a causal architecture effect when data, tuning, or systems coordinates differ. The conclusion follows directly from the declared record.

Remark. Conditions for applying a toy model [ftip-00K1]

A toy theorem applies to a concrete model only if an explicit map from its state and interface to the model preserves the theorem's assumptions, including its cost accounting. Without such a map, the conclusion concerns only the abstract model; its validity for the concrete system is unknown.

Kimi Delta Attention versus Transformer [ftip-00K2]

Definition. KDA model-instance record [ftip-00K3]

For a Kimi Delta Attention (KDA) versus full-attention baseline study, record the exact Kimi Linear checkpoint, layer mix, context length, hardware, kernel, precision, batch, and decoding workload. The paper describes KDA as a fine-grained gated delta-rule module in a hybrid architecture; those are source observations from Kimi Linear: An Expressive, Efficient Attention Architecture[kimi2025linear], not universal theorems.

If the baseline is called a Transformer, record whether it is the paper's MLA baseline or another full-attention implementation; the label alone does not identify a common architecture or cost.

Example. Matched-cost KDA and Transformer comparison [ftip-00K4]

A matched comparison uses the same task prompts, post-training data and feedback, optimizer family, evaluation protocol, and quality target. The corresponding measurements include parameter count, training FLOPs, inference FLOPs, wall time, memory, KV-cache bytes, and context length for each architecture. A reported speed or quality difference is conditional on these experimental conditions and measurements.

Remark. What the Kimi Linear paper establishes [ftip-00K5]

Kimi Linear reports a hybrid KDA/MLA model and fair-comparison experiments in Sections 3--5, with setup in Section 5.4 and efficiency comparisons in Sections 5.5--5.6 of Kimi Linear: An Expressive, Efficient Attention Architecture[kimi2025linear]. These include long-context efficiency.

The reported throughput and quality are empirical under its training recipe and hardware; they are lower bounds on demonstrated performance, not an architecture-independent ceiling.

Example. Parameter matching is not cost matching [ftip-00K6]

Two models with equal parameter count can have different attention FLOPs, KV-cache memory, kernel utilization, and attainable context. Conversely, equal wall time can hide different hardware and precision. Therefore a parameter-matched result cannot by itself identify a compute frontier.

Remark. Cache, throughput, and quality in KDA comparisons [ftip-00K7]

The KDA example does not prove that every linear-attention model beats every Transformer, that cache savings imply equal quality, or that a measured throughput gain is a capability gain. Any such statement requires a new matched study with the evidence record of Definition [ftip-00JY].

Definition. Model-instance comparison decision rule [ftip-00K8]

Call architecture A more cost-efficient than B at quality $q$ only when the compared threshold costs use the same declared protocol, the confidence procedure and stopping rule are fixed, and all cost-vector coordinates needed by the claim are present. A ratio additionally requires finite costs and a strictly positive denominator as in Definition [ftip-00JP]. State whether the costs are measured target-achieving witnesses or justified inverse-frontier values $C_A(q)$ and $C_B(q)$; observing one successful run does not identify those infima. Otherwise the comparison is descriptive or unknown.

Interface boundaries for alternative architectures [ftip-00K9]

Definition. JEPA comparability interface [ftip-00KA]

A JEPA-like architecture is comparable to an autoregressive model only after fixing the predictor, target representation, decoder or downstream interface, training objective, inference procedure, and evaluation law. A parameter-count match alone is not a typed comparison.

Remark. Representation and task interface are separate [ftip-00KB]

A latent predictor can be excellent under one decoder and unusable under another. The evaluation record must therefore include the interface that maps the representation to the declared task output.

Theorem. Interface-preserving architecture reduction [ftip-00KC]

If two architecture records induce the same conditional output law on a fixed task and evaluation interface, every evaluation functional $J$ gives the same value. This is an immediate pushforward identity under the stated output-law and interface assumptions. It does not imply that distinct architectures are generally equivalent.

Example. Equal benchmark score can hide different ceilings [ftip-00KD]

Two architectures may tie on a finite benchmark while differing on an unmeasured task slice or longer context. Thus a single score cannot establish equal representational ceilings; the task family and evaluation coverage must be declared.

Remark. Architecture-independent model interfaces [ftip-00KE]

A result stated solely in terms of a model/interface map applies to every architecture realizing that map and satisfying its hypotheses. An architecture-specific conclusion can require further assumptions, as in Remark [ftip-00JK].

Remark. Open model-instance questions [ftip-00KF]

The relative capability and cost of KDA, full attention, and JEPA candidates remain empirical questions. A reproducible comparison requires identified checkpoints or reproducible training, fixed evaluation seeds, and measured cost vectors. Architecture-specific ceilings remain estimands and hypotheses rather than established constants.

Published hybrid-attention comparison [ftip-00KU]

Kimi Linear combines recurrent KDA layers with periodic full MLA attention. Its published comparison shows task-dependent score changes and lower long-context decoding times. The selected scores in Example [ftip-00LC], the memory estimate in Example [ftip-00KY], and the discovery calculation in Example [ftip-00L3] connect these observations to a post-training question: when can a cheaper attempt compensate for a possible lower per-attempt success probability?

Example. Published Kimi Linear operating point [ftip-00KV]

The Kimi Linear report compares experimental models pretrained on 1.4 trillion tokens, each with 48 billion total and 3 billion active parameters. Kimi Linear uses three Kimi Delta Attention layers for each full Multi-head Latent Attention layer, with no positional encoding (NoPE) in its MLA layers; the reference uses full MLA attention [kimi2025linear, Sections 4 and 5.4]. These are properties of the reported hybrid architecture.

These experimental models are distinct from the released Kimi Linear model, pretrained on 5.7 trillion tokens and supporting up to one million tokens of context. Appendix D compares that release with Moonlight, which has 16 billion total and 3 billion active parameters [kimi2025linear, Section 5.4.1 and Appendix D].

Example. Published efficiency observations [ftip-00KW]

For batch size one at one million tokens, Figure 7 labels a $2.9\times $ prefill speedup and a $2.2\times $ decoding speedup for Kimi Linear against MLA. Section 6.3 gives $2.3\times $ for the latter point, so the graphic and prose differ slightly [kimi2025linear, Figure 7 and Sections 5.6 and 6.3].

The larger-batch comparison in Figure 1(b) gives decoding time per output token of 1.84 milliseconds for Kimi Linear and 11.48 milliseconds for MLA, with a reported $6.3\times $ speedup. Section 6.3 describes this as a theoretical speedup from reallocating saved KV-cache memory to larger batches. It illustrates a different use of the memory saving from the batch-one latency result [kimi2025linear, Figure 1(b) and Section 6.3].

Figure 1(a) reports RULER scores of 84.3 for Kimi Linear and 81.3 for MLA at 128k context, with $3.98\times $ decoding acceleration [kimi2025linear, Figure 1(a)]. The report also gives a reduction of up to 75 percent in KV-cache use [kimi2025linear, Abstract]. Together these observations motivate studying context length and batch size as separate cost coordinates. The unequal RULER scores and different timing regimes are retained when interpreting those points.

Definition. Quality target and cost measurement [ftip-00KX]

For a reported benchmark score q, use the same task, prompt law, scoring rule, and post-training condition for both systems. The comparison records the smallest measured cost at which each system reaches q, when such a measurement exists. A throughput or cache ratio is not substituted for this iso-quality cost.

Example. A cache-memory estimate for hybrid attention [ftip-00KY]

The cited FLA KDA implementation at commit 6b6f546f stores one active recurrent state per sequence, with dimensions determined by the value heads and key/value dimensions rather than accumulated context length. Its newly allocated state uses float32 [⧉]. The optional short-convolution state has one fixed-width kernel buffer per channel and sequence [⧉]. This supplies a concrete implementation model for the recurrent part of the 3:1 mixture in Example [ftip-00KV].

Consider a stylized comparison with $4\ell $ layers, batch size $b$, and context length $T$, where $\ell ,b,T$ are positive integers. Assume each MLA layer stores $\kappa T$ bytes per sequence, with the same $\kappa >0$ in both designs, and each KDA layer stores one terminal recurrent-plus-convolution state of $\sigma >0$ bytes. The active attention-state memories are

\[ M_{\rm MLA}=4b\ell \kappa T,\qquad M_{\rm hybrid}=b\ell (\kappa T+3\sigma ),\qquad \frac {M_{\rm hybrid}}{M_{\rm MLA}} =\frac {1}{4}+\frac {3\sigma }{4\kappa T}. \]

The hybrid uses less active state exactly when $T>\frac {\sigma }{\kappa }$, and the ratio tends to $1/4$ as context grows. This explains why replacing three quarters of the growing caches can approach a 75-percent saving, while fixed recurrent-state overhead can erase the advantage at short contexts. The retained MLA layers keep the hybrid's state memory linear in $T$; the whole model is not a fixed-memory recurrent system.

Weights, training activations, workspaces, allocator padding and saved prefix states are outside this active-state calculation. At a fixed device memory budget, the estimate predicts more room for simultaneous or longer rollouts when attention state dominates. It does not predict a fourfold speedup. Training and decoding also use different execution paths: FLA selects chunk kernels for gradient-enabled computation and fused recurrence for short inference calls [⧉]. The fixed-state estimate is therefore a useful serving hypothesis, not a training-memory or training-speed formula.

Example. Common-recipe reading of the Kimi result [ftip-00KZ]

The report holds its stated pretraining and SFT recipes common across the experimental models [kimi2025linear, Sections 5.4--5.5 and Table 4]. The selected observations in Example [ftip-00LC] compare the resulting trained designs, including the hybrid's 3:1 KDA/MLA pattern and NoPE choice. The mixed score directions suggest a task-dependent tradeoff rather than one ordering of the two architectures.

The theoretical mechanism is a change in the cost of maintaining and using the prefix: recurrent layers compress it into state, while the retained MLA layers preserve growing attention caches. The model in Example [ftip-00KY] predicts the largest memory advantage at long contexts. If this advantage reduces the cost of generating useful candidates, Example [ftip-00L3] predicts when more attempts can offset a reduction in per-attempt quality. This is a conditional explanation to test through rollout costs and verified success rates; it does not require attributing the whole result to KDA alone.

Example. Measured iso-quality cost ratio [ftip-00L0]

Let $c_K(q)$ and $c_T(q)$ be measured inference costs for Kimi Linear and the Transformer reference at the same finite real score $q$ and context length, under one accounting rule. If $0\leq c_K(q)<+\infty $ and $0<c_T(q)<+\infty $, the measured ratio $\widehat \rho _{K/T}(q)=c_K(q)/c_T(q)$ is reported together with the full cost vectors. These measured costs need not be the inverse frontier infima of Definition [ftip-00JP]. If either threshold is unmeasured or the denominator is zero, the ratio is left undefined rather than inferred from the paper's speedup plot.

Remark. Hybrid attention is not pure linear attention [ftip-00L1]

Kimi Linear combines KDA with periodic full-attention layers. A measured advantage therefore identifies the reported hybrid intervention, not a claim that a purely linear recurrent architecture has the same quality or ceiling. The layer pattern belongs in the architecture record.

Remark. Demonstrated performance is a lower bound [ftip-00L2]

The published benchmark scores demonstrate attainable points for the reported checkpoints and protocol. They do not establish a representational ceiling, a universal scaling law, or a claim that additional cost cannot improve either architecture.

Example. When cheaper attempts compensate for lower success [ftip-00L3]

Smaller caches can support more concurrent requests or avoid preemption and recomputation. These are actual serving mechanisms in vLLM's memory and scheduling guidance [⧉]. For post-training, the useful prediction is therefore about completed candidate attempts under a budget, including verification and optimization costs, rather than decoding speed alone.

Use the following cost model on the same hardware. One baseline attempt costs $c>0$; generation accounts for a fraction $f\in [0,1]$. Suppose generation becomes $s_{\rm gen}>0$ times as fast while all other cost per attempt stays fixed. The new cost and total attempt-throughput factor are

\[ c_K=c\left (1-f+\frac {f}{s_{\rm gen}}\right ),\qquad S=\frac {c}{c_K} =\frac {1}{1-f+f/s_{\rm gen}}. \]

For a fixed budget $B\geq 0$, this model permits $N_M=\lfloor B/c\rfloor $ baseline attempts and $N_K=\lfloor B/c_K\rfloor $ hybrid attempts. Assume $N_M,N_K\geq 1$ and verified-success events independent within each set of attempts, with respective probabilities $p_M,p_K\in [0,1]$. Then Theorem [ftip-0078] gives discovery probabilities $1-(1-p_M)^{N_M}$ and $1-(1-p_K)^{N_K}$. The hybrid matches or exceeds baseline discovery exactly when

\[ p_K\geq 1-(1-p_M)^{N_M/N_K}. \]

This follows by comparing the two failure probabilities and taking the nonnegative $N_K$-th root.

For an illustrative estimate, suppose generation consumes 80 percent of the baseline cost and use $s_{\rm gen}=2.2$, motivated by the batch-one decode figure in Example [ftip-00KW]. Applying that decode factor to the whole generation stage is a working approximation, not a measured rollout result. Then $S\approx 1.774$; a budget of $B=10c$ buys 10 baseline attempts or 17 hybrid attempts. If $p_M=0.10$ and $p_K=0.08$, the discovery probabilities are approximately $0.651$ and $0.758$. The break-even hybrid probability is about $0.0601$. Thus this model predicts that more attempts can outweigh a moderate per-attempt quality loss. The assumed probabilities are illustrative verified-success rates, not conversions of the benchmark scores in Example [ftip-00LC].

The prediction is strongest for long-context, generation-heavy work whose saved memory becomes usable rollout capacity. As verification or optimization dominates, $f$ shrinks and $S$ approaches one. Correlated attempts require the joint-law analysis of Convention [ftip-0077] instead of the independent product. Measure completed attempts, verified-success rates and total cost under a fixed prompt, decoding and verifier protocol to test the tradeoff. In the RLVR round of Example [ftip-004M], additional successful candidates can improve the available feedback; whether they produce useful accepted updates is a further optimizer-and-evaluation question, as Example [ftip-0085] demonstrates.

Paired measurements for model comparison [ftip-00L4]

A paired measurement specifies both model instances and their task, intervention, evaluation, and cost coordinates. These conditions determine the meaning of each point in the measured frontier.

Definition. Comparison pair identity [ftip-00L5]

A paired comparison identifies the two systems, exact checkpoint or training commit, architecture variant, parameter counts, tokenizer, and context limit. A label such as ``Transformer'' is insufficient when attention, cache, or layer patterns differ.

Definition. Common task and evaluation conditions [ftip-00L6]

Both systems use the same task family, prompt and data law, scoring rule, sampling seeds, stopping rule, and evaluator version. Any exception is a separate comparison arm, not an unrecorded architecture effect.

Definition. Training and post-training coordinates [ftip-00L7]

Record data volume, token order policy, optimizer and schedule, supervised tuning, feedback or reinforcement procedure, update count, and selection rule. The common-recipe arm fixes these coordinates; the equal-tuning arm allows predeclared alternatives with equal search resources.

Definition. Inference and systems coordinates [ftip-00L8]

Record hardware, software and kernel versions, precision, batch, cache policy, sequence lengths, warm-up procedure, concurrency, and decoding settings. A kernel or cache change that alters numerical computation is marked as an intervention rather than hidden as an implementation detail.

Definition. Per-system cost vector [ftip-00L9]

For each system, the cost vector $\mathbf c$ contains training FLOPs, inference FLOPs, wall time, memory, energy when available, and context length. The primary scalar $C$ and its weights are declared beside the vector; omitted coordinates are marked unknown.

Definition. Repeated measurement rule [ftip-00LA]

For each fixed workload and system, record warm-up runs, repetition count, random seeds, and the aggregation rule. Report mean and dispersion for time, memory, and score; a single fastest run cannot define a cost frontier.

Definition. Paired uncertainty estimates [ftip-00LB]

When the same evaluation items are used, report paired score differences and an uncertainty procedure fixed before inspecting the result. Confidence intervals quantify sampling variation; they do not repair unmatched training, systems, or task coordinates.

Example. Published Kimi Linear comparison [ftip-00LC]

Table 4 of the Kimi Linear report compares the experimental models pretrained on 1.4 trillion tokens in Example [ftip-00KV], after the same supervised fine-tuning (SFT) recipe. Both have 48 billion total and 3 billion active parameters. Three selected benchmark scores are [kimi2025linear, Sections 4 and 5.4 and Table 4]:

Arm	MMLU-Pro	LiveBench	EvalPlus
Kimi Linear	67.4	45.2	61.0
Full MLA	65.7	45.7	62.6

The entries use the paper's score units, with larger values better on each selected benchmark; LiveBench is reported as Pass@1. Kimi Linear is higher on MMLU-Pro and lower on LiveBench and EvalPlus in these observations. These are reported point estimates on different tasks, not a single capability score; the table supplies no uncertainty estimate.

The common recipe uses the K2 pretraining corpus, MuonClip and a shared training schedule. SFT proceeds from broad instruction data to targeted reasoning tasks; evaluation uses temperature 1.0 and an internal framework derived from LM-Harness [kimi2025linear, Section 5.4]. Thus the scores provide a concrete example of how the trained hybrid and full-MLA designs respond to the same recipe. The cost mechanism and its possible post-training consequences are developed in Example [ftip-00KZ].

Remark. Measured and assumed costs [ftip-00LD]

A useful cost model can combine measurements with assumptions justified by the architecture and its implementation. In Example [ftip-00L3], the generation share $f$ and speed factor $s_{\rm gen}$ determine a predicted attempt-throughput factor. Their assumed values can be varied without pretending they were measured in the original experiment.

For fixed $f\in [0,1]$, write $S(s)=1/(1-f+f/s)$ for $s>0$. This function is nondecreasing. An interval $s_{\rm gen}\in [s_-,s_+]$, with $0<s_-\leq s_+$, therefore gives the conditional range $S\in [S(s_-),S(s_+)]$. This is sensitivity to a chosen assumption range; it is a statistical confidence interval only if a sampling argument supplies that interpretation.

Definition. Measured contrasts and conditional predictions [ftip-00LE]

A measured paired contrast evaluates the same declared quantity for two observed model instances under a specified task, protocol and cost accounting. The common-recipe scores in Example [ftip-00LC] are one example. A confirmatory comparison fixes its matching and selection rules before examining the scores.

A conditional paired prediction instead uses an explicit model to supply one or more cost or response quantities, then derives the comparison under those assumptions. The discovery probabilities in Example [ftip-00L3] are an example. Such a prediction can be motivated by existing observations and guide a later experiment; its unmeasured inputs remain assumptions, and agreement with new matched measurements tests the model.

Example. How generation share changes the predicted gain [ftip-00LF]

Keep the illustrative generation factor $s_{\rm gen}=2.2$ from Example [ftip-00L3] and vary the baseline generation share $f$. Its cost model gives the following total attempt-throughput factors:

Generation share	Predicted throughput factor
20 percent	1.122
50 percent	1.375
80 percent	1.774
100 percent	2.200

These calculated values show why the same decoding improvement can matter much more for generation-heavy search than for a verifier- or optimizer-dominated workload. They suggest measuring the time spent in generation, verification and updates before selecting where to invest systems effort. A change in the bottleneck changes the useful intervention, even when the attention architecture stays fixed.

Remark. Testing the predicted architecture tradeoff [ftip-00LG]

The memory and discovery models predict the strongest benefit when long prefixes make attention state expensive, saved capacity improves useful rollout throughput, and per-attempt success remains above the break-even threshold. Short contexts, expensive verification, or strongly correlated attempts can weaken that benefit. These are distinct mechanisms to test, rather than a universal ranking of KDA and full attention.

A comparison across implementations can retain this reasoning while re-estimating state sizes, generation share and success probabilities for the new system. Report both the prediction and the matched measurements: their agreement or disagreement identifies which mechanism or assumption needs revision.

Conditional frontier and saturation analysis [ftip-00LH]

A finite matched study measures capability at selected values of one declared cost. Its frontier and saturation target are conditional on that protocol; they do not establish a universal intelligence law.

Definition. Finite cost grid [ftip-00LI]

Choose a finite ordered grid $0<C_1<\cdots <C_m$ and evaluate each architecture at every declared grid point under the same task and protocol. The resulting values $\widehat V_A(C_i)$ are observations of the restricted frontier, not its values between grid points.

Definition. Observed upper envelope [ftip-00LJ]

For an observed score $s_A(C_i)$ define the discrete upper envelope $U_A(C_i)=\max _{j\leq i}s_A(C_j)$. It is a descriptive monotone summary of the measured points; it is not evidence that unmeasured costs attain the envelope or that extra cost cannot reduce score.

Definition. Uncertainty bands on the frontier [ftip-00LK]

Attach a predeclared uncertainty interval $I_A(C_i)$ to each score and carry those intervals through score differences and threshold crossings. A finite band describes sampling and measurement variation; it does not cover unmeasured training procedures or architectures.

Definition. Declared saturation target [ftip-00LL]

Fix a quality target $q\in \mathbb R$, tolerance and cost increment $\varepsilon ,\delta \in (0,+\infty )$, and a budget $C\in \mathbb R_{\geq 0}$. Require both $V_A(C)$ and $V_A(C+\delta )$ to be finite real values. A study calls an architecture $\varepsilon $-saturated at $C$ only relative to its allowed intervention class when the restricted frontier has certified gain $V_A(C+\delta )-V_A(C)\leq \varepsilon $.

Theorem. Conditional saturation certificate [ftip-00LM]

Take finite real $U$, $C\geq 0$, $\delta \geq 0$, and $\varepsilon \geq 0$, with the nondecreasing frontier of Definition [ftip-00JJ]. If a justified upper bound gives $V_A(C+\delta )\leq U$ and a justified lower bound gives $V_A(C)\geq U-\varepsilon $, then $U-\varepsilon \leq V_A(C)\leq V_A(C+\delta )\leq U$. Both endpoint values are therefore finite real, and $0\leq V_A(C+\delta )-V_A(C)\leq \varepsilon $. This is a conditional bound for the named frontier; it proves no universal saturation of intelligence.

For the finite controller class with hard resource admission, Corollary [ftip-00MC] obtains the required bounds from a feasible controller and a uniform Bellman certificate.

Example. Finite observations need not identify a frontier [ftip-00LN]

Let $S\subset \mathbb R_{\geq 0}$ be a finite set of measured costs, and choose $C_0>0$ larger than every element of $S$. Suppose the exact frontier value observed at each cost in $S$ is zero. For finite budgets $C\geq 0$, suppose the declared class of possible frontiers permits both \[ V_0(C)=0, \qquad V_1(C)= \begin {cases} 0,&0\leq C<C_0,\\ 1,&C\geq C_0. \end {cases} \] These nondecreasing frontiers with scores in $[0,1]$ agree on every observed cost and differ at $C_0$. The observations alone do not distinguish them.

Both possibilities have finite realizations in the framework of Definition [ftip-00JJ]: allow two interventions with costs $0$ and $C_0$. Give the first score zero and the second score $\theta \in \{0,1\}$. For example, on a single deterministic evaluation task with utility equal to the output bit, let the two resulting protocols return $0$ and $\theta $. The two possible choices of $\theta $ give $V_0$ and $V_1$, respectively, under the same intervention and cost specification. Every score is finite, and the feasible score maximum is attained at each budget.

This example does not supply a strictly better agreeing frontier for every possible data set or every admissible class. If a proved global score bound is $1$ and an intervention attains it at a finite cost $C_*\geq 0$, monotonicity forces $V_A(C)=1$ for every $C\geq C_*$; no higher value is admissible. A singleton class of possible frontiers can also identify the frontier without such an alternative. A conditional certificate such as Theorem [ftip-00LM] therefore requires its stated upper-bound evidence; that evidence does not follow merely from the absence of an observed improvement.

Definition. One-factor cost interpretation [ftip-00LO]

The primary cost $C$ is a declared scalarization of the recorded cost vector. A frontier statement is conditional on its units and weights; the same paired measurements may produce a different ordering under a different scalarization.

Example. Crossing architecture frontiers [ftip-00LP]

Two architectures may alternate in the observed ordering across costs: one can score higher at small $C$ while the other catches up at larger $C$. Thus a single comparison point cannot establish a global ordering or a common ceiling.

Example. Frontier report for the KDA study [ftip-00LQ]

A Kimi Delta Attention versus Transformer study should publish the cost grid, paired scores, uncertainty intervals, scalarization weights, and the exact identities and configurations of the compared models. The report may then state which measured points are Pareto or iso-quality comparisons under that protocol.

Remark. Frontiers depend on the intervention and evaluation [ftip-00LR]

The conditional frontier is indexed by architecture, intervention class, task family, evaluation law, and cost scalarization. Changing any of these coordinates creates a new estimand; no ordering transfers automatically to a new checkpoint, optimizer, hardware stack, or task family.

Study design for comparisons across model instances [ftip-00LS]

This subsection specifies how a conditional architecture comparison may be instantiated without changing the architecture-neutral formulation. It is a study design, not a universal ranking of architectures.

Definition. Matched model-instance record [ftip-00LT]

A model-instance record names the architecture, parameterization or checkpoint, training and post-training recipe, optimizer, execution stack, inference budget, task family, evaluation law, and recorded cost vector. Two records are comparable only after the fields held fixed and the fields allowed to vary are declared.

Definition. Pair identity and comparison unit [ftip-00LU]

A comparison unit is a pair of model-instance records evaluated on the same task family and evaluation law, together with a declared cost scalarization. A Transformer reference and a Kimi Delta Attention instance may form such a pair only when the remaining coordinates are documented.

Remark. Paired responses to a common recipe [ftip-00LV]

For a pair of matched model-instance records, the common-recipe arm measures how the permitted architecture packages respond to one recipe. A score advantage in this comparison need not persist after either package is retuned.

Remark. Different recipes under equal tuning effort [ftip-00LW]

Under the equal-tuning-budget arm, the two selected model instances may use different recipes. That difference is compatible with equal predeclared tuning effort; forcing the selected recipes to match would answer a different comparison question.

Remark. How an allowed class changes its envelope [ftip-00LX]

In the restricted-envelope arm, enlarging an allowed intervention class at fixed evaluation law, cost accounting, and budget can raise its envelope and cannot lower it. A finite search over feasible interventions supplies a lower bound from the scores it attains. It certifies the supremum only if it covers every feasible intervention or is accompanied by a matching upper-bound argument.

Remark. Coordinate-change rule [ftip-00LY]

Changing the optimizer, post-training method, hardware or kernel, precision, context policy, or evaluation law changes a comparison coordinate. A score difference after such a change belongs to the new intervention, unless the study explicitly estimates the interaction.

Definition. Architecture-family transfer matrix [ftip-00LZ]

A transfer matrix records which interfaces and budgets permit a comparison among a Transformer reference, Kimi Delta Attention, looped or depth-reused variants, and JEPA-like systems. A blank or incompatible cell is an incomparability finding, not a missing score to be imputed.

Remark. Applying a finite model to an architecture [ftip-00M0]

A toy representation or recurrence result may motivate a hypothesis, but it is not transferred to a model instance until its interface, task, cost accounting, numerical regime, and evaluation protocol are instantiated and checked. If any required assumption is unverified, applicability to the model instance remains unestablished.

Definition. Negative-result and stopping report [ftip-00M1]

A completed comparison reports the tested cost grid, excluded configurations, stopping rule, uncertainty, and negative results. Stopping without a detected difference is evidence about the declared study, not proof that the architectures have equal potential.

Remark. Open comparison questions [ftip-00M2]

The remaining questions are whether a declared interface supports a fair Kimi Delta Attention--Transformer comparison, how looped computation changes the frontier under matched budgets, and which JEPA interface can be fixed without silently changing the task. None is settled by the study design alone.

Optimizer geometry and representation dependence [ftip-00AU]

A scalar objective does not by itself determine a training intervention. This section studies an exact finite-dimensional witness: one represented matrix can have many factor bases, the loss can be constant along that gauge orbit, and an update rule can nevertheless distinguish the bases.

The principal source is the version-one arXiv preprint The loss does not see the basis, but Adam does[singh2026lossbasis]. Its algebraic results assume the stated dimensions, rank conditions, state conventions, and real arithmetic. Experimental observations concern the reported configurations.

Factored objectives and gauge symmetry [ftip-00AV]

We first separate a represented matrix from a choice of factors and then separate its full product-preserving symmetry from the orthogonal subgroup that preserves the Euclidean parameter metric.

Notation. Factored objective and represented matrix [ftip-00AW]

Fix integers $d_1,d_2,k\geq 1$. Let $U\in \mathbb R^{d_1\times k}$ and $V\in \mathbb R^{d_2\times k}$ be factors, let

\[ W(U,V)=UV^{\mathsf T}\in \mathbb R^{d_1\times d_2}, \]

and let $f:\mathbb R^{d_1\times d_2}\to \mathbb R$ be differentiable. The corresponding factored objective is $L(U,V)=f(W(U,V))$. The pair $(U,V)$ is a parameterization; $W(U,V)$ is the represented object on which $f$ depends. This is the rectangular form noted in Section 3 of The loss does not see the basis, but Adam does[singh2026lossbasis].

Definition. General linear gauge action [ftip-00AX]

For $A\in \mathrm {GL}(k)$, define the general linear gauge action by

\[ \gamma _A(U,V)=(UA,VA^{-\mathsf T}). \]

The inverse transpose on the second factor is essential for a general invertible $A$. When $A=Q\in \mathrm O(k)$, one has $Q^{-\mathsf T}=Q$, and the action becomes $(UQ,VQ)$.

Lemma. The gauge action preserves the product and loss [ftip-00AY]

For every $A\in \mathrm {GL}(k)$,

\[ W(UA,VA^{-\mathsf T})=UV^{\mathsf T} \quad \hbox {and}\quad L(UA,VA^{-\mathsf T})=L(U,V). \]

Proof.

Since $(VA^{-\mathsf T})^{\mathsf T}=A^{-1}V^{\mathsf T}$, the first identity is $UAA^{-1}V^{\mathsf T}=UV^{\mathsf T}$. Applying $f$ gives the second.

This is the factored-model calculation in Section 3 of The loss does not see the basis, but Adam does[singh2026lossbasis].

Definition. Orthogonal gauge orbit [ftip-00AZ]

The orthogonal gauge orbit of $(U,V)$ is

\[ \mathcal O(U,V)=\{(UQ,VQ):Q\in \mathrm O(k)\}. \]

Every pair in the orbit represents the same $W$. This orbit is generally smaller than the complete fibre of the map $(U,V)\mapsto UV^{\mathsf T}$. The restriction to $\mathrm O(k)$ is geometric: it preserves the Euclidean metric on factor space.

Lemma. The orthogonal gauge is the maximal isometric subgroup [ftip-00B0]

The action $\gamma _A$ preserves $\lVert U\rVert _F^2+\lVert V\rVert _F^2$ for every pair $(U,V)$ if and only if $A\in \mathrm O(k)$.

Proof.

If the action is an isometry, take $V=0$ and let one row of $U$ be an arbitrary $u\in \mathbb R^{1\times k}$. Then $\lVert uA\rVert _2=\lVert u\rVert _2$ for every $u$, hence $AA^{\mathsf T}=I$. Conversely, right multiplication by an orthogonal matrix preserves both Frobenius norms.

See Lemma 3.2 and its proof in Appendix B.1 of The loss does not see the basis, but Adam does[singh2026lossbasis].

Lemma. Gradients transform covariantly along an orthogonal orbit [ftip-00B1]

For $Q\in \mathrm O(k)$,

\[ \nabla _U L(UQ,VQ)=\nabla _U L(U,V)Q, \qquad \nabla _V L(UQ,VQ)=\nabla _V L(U,V)Q. \]

Proof.

Put $G=\nabla f(W)$. Then $\nabla _U L=GV$ and $\nabla _V L=G^{\mathsf T}U$. The represented matrix, and hence $G$, is unchanged along the orbit. Substitution gives both identities.

See Lemma 4.1 and its proof in Appendix B.2 of The loss does not see the basis, but Adam does[singh2026lossbasis].

Definition. Optimizer state action [ftip-00B2]

Use the optimizer-state convention of Definition [ftip-001H]. For every $Q\in \mathrm O(k)$, an optimizer state action is a map $\sigma _Q:\mathcal S_{\rm opt}\to \mathcal S_{\rm opt}$ compatible with the shapes of the factor states and satisfying $\sigma _Q(s_0)=s_0$ for the declared initial state.

For example, a momentum buffer shaped like $U$ transforms by right multiplication by $Q$; a shared scalar second moment is fixed. An entrywise second-moment array need not admit a state action compatible with every orthogonal $Q$.

Definition. Gauge-equivariant optimizer [ftip-00B3]

Let $\mathcal A$ be a deterministic instance of the parameter-update rule of Definition [ftip-001I], acting on $(U,V,s)$. It is gauge-equivariant when for every $Q\in \mathrm O(k)$ there is a state action $\sigma _Q$ such that gauge-related initial states produce

\[ (\widetilde U_t,\widetilde V_t,\widetilde s_t) =(U_tQ,V_tQ,\sigma _Q(s_t)) \qquad (t\geq 0). \]

This is Definition 3.1 of The loss does not see the basis, but Adam does[singh2026lossbasis]. Any randomness, schedules, stopping rule, and mixed update blocks must also be coupled as required by Remark [ftip-001J]; the definition does not hide them.

Example. A commuting optimizer square [ftip-00B4]

One update of a gauge-equivariant optimizer makes the following square commute. The vertical arrows change factor basis and the horizontal arrows apply the same declared update.

$\text{[math]}$

Commutation is stronger than equality of the scalar losses before the step. It states equality of the represented product trajectory after aligning the parameter bases.

Remark. Product symmetry is larger than the isometric gauge [ftip-00B5]

The full $\mathrm {GL}(k)$ action in Definition [ftip-00AX] preserves $W$, but ordinary Euclidean gradients are covariant under the orthogonal specialization used in Lemma [ftip-00B1]. A theorem for $(UQ,VQ)$ must therefore not be silently promoted to every product-preserving reparameterization.

The source calls the orthogonal subgroup the gauge in its optimizer classification while also noting the larger function-preserving group. These are compatible statements only when their different geometric scopes remain explicit.

Equivariant and basis-sensitive update rules [ftip-00B6]

Gradient covariance supplies a test for update rules. Linear and full-matrix operations can commute with a latent rotation, whereas a fixed nonlinear map applied separately to coordinates selects a preferred basis.

Definition. Memoryless right-equivariant update [ftip-00B7]

A memoryless factor-update map is a function $\Phi :\mathbb R^{n\times k}\to \mathbb R^{n\times k}$ applied to a current gradient $G$ without a persistent optimizer state. It is right-equivariant when

\[ \Phi (GQ)=\Phi (G)Q \qquad (G\in \mathbb R^{n\times k},\ Q\in \mathrm O(k)). \]

This local definition is the input of Theorem 4.5 in The loss does not see the basis, but Adam does[singh2026lossbasis]. It does not classify stateful optimizers such as Adam.

proposition. Gradient descent and momentum preserve the orthogonal gauge [ftip-00B8]

Gradient descent and Polyak or Nesterov momentum are gauge-equivariant when their factor gradients, momentum buffers, schedules, and look-ahead points are transformed consistently.

Proof.

By Lemma [ftip-00B1], $G_t$ becomes $G_tQ$. Gradient descent therefore sends $U_tQ$ to $(U_t-\eta _tG_t)Q$. A momentum buffer initialized at zero and formed by linear combinations of covariant gradients transforms as $M_tQ$; a Nesterov look-ahead point is likewise the reference look-ahead right-multiplied by $Q$. Induction proves the claim.

Proposition 4.2(1) and its proof are in Appendix B.3 of The loss does not see the basis, but Adam does[singh2026lossbasis].

proposition. A shared scalar second moment preserves the orthogonal gauge [ftip-00B9]

Replace Adam's entrywise second-moment denominator by one scalar whose updates depend only on gauge-invariant quantities such as the pooled mean of the squared entries of both factor gradients. With a covariant first-moment buffer, the resulting stateful update is gauge-equivariant.

Proof.

The first moment is a linear exponential average and transforms by right multiplication by $Q$. The pooled squared-entry mean is the joint squared Frobenius norm divided by the entry count, hence is unchanged by $G\mapsto GQ$. Its bias correction and positive scalar denominator are unchanged, so the complete update transforms covariantly.

Proposition 4.2(2) and its proof are in Appendix B.3 of The loss does not see the basis, but Adam does[singh2026lossbasis]. The result concerns this declared shared-scalar variant, not stock Adam.

proposition. Full-matrix Muon and Shampoo preserve the orthogonal gauge [ftip-00BA]

Fix one factor, write $n\in \{d_1,d_2\}$ for its row dimension, and let $G_t,M_t\in \mathbb R^{n\times k}$ be its gradient and Muon momentum buffer. For $0\leq \beta <1$, $M_{-1}=0$, and the source's linear momentum update $M_t=\beta M_{t-1}+G_t$, Muon's direction is $\Delta _t=\operatorname {msign}(M_t)$. Under real arithmetic this stateful update and the source's damped Shampoo update are right-equivariant. The Muon claim includes the stated finite Newton--Schulz approximations; the Shampoo claim uses its complete left and right accumulators.

Proof.

By Lemma [ftip-00B1], the primed gradient is $G'_t=G_tQ$. Starting from $M'_{-1}=M_{-1}Q=0$, induction through the linear momentum update gives $M'_t=M_tQ$. If $M_t=A\Sigma B^{\mathsf T}$ is a compact singular-value decomposition, then $M_tQ=A\Sigma (Q^{\mathsf T}B)^{\mathsf T}$, so $\operatorname {msign}(M_tQ)=\operatorname {msign}(M_t)Q$. For $\epsilon _{\rm NS}>0$, the Newton--Schulz initialization $X_0=M_t/(\lVert M_t\rVert _F+\epsilon _{\rm NS})$ is covariant because its Frobenius normalization is invariant; each polynomial iterate formed from $X_j$ and $X_jX_j^{\mathsf T}$ preserves the same covariance. For Shampoo, the left accumulator is invariant while the right accumulator transforms by $R_t\mapsto Q^{\mathsf T} R_t Q$. Orthogonal functional calculus conjugates the right matrix function, so the transformed update is the original update right-multiplied by $Q$.

Proposition 4.2(3)--(4) and its proof are in Appendix B.3 of The loss does not see the basis, but Adam does[singh2026lossbasis]. Variable splitting, coordinatewise clipping, mixed update rules, finite precision, and different damping conventions lie outside this statement. The Muon part is stateful; it does not replace the momentum buffer by the current gradient or invoke the memoryless classification of Definition [ftip-00B7].

proposition. Coordinatewise equivariance forces linearity [ftip-00BB]

Let $k\geq 2$ and let a memoryless update have the fixed entrywise form $\Phi (G)_{ij}=\phi (G_{ij})$. If $\Phi (GQ)=\Phi (G)Q$ for every $G$ and $Q\in \mathrm O(k)$, then $\phi (x)=cx$ for some $c\in \mathbb R$.

Proof.

It suffices to use one row and rotations in the first two coordinates. Equivariance at zero gives $\phi (0)=0$. Rotating $(x,0)$ through an angle whose cosine is $a\in [-1,1]$ gives $\phi (ax)=a\phi (x)$. For nonzero $x,y$, compare each with a third number of at least their absolute magnitudes; the ratio $\phi (x)/x$ is constant.

Proposition 4.3 and its proof are in Appendix B.4 of The loss does not see the basis, but Adam does[singh2026lossbasis]. No continuity assumption is needed.

Corollary. Fixed nonlinear coordinatewise updates break the gauge [ftip-00BC]

A nonlinear fixed entrywise update applied at a zero optimizer state is not gauge-equivariant when $k\geq 2$. In particular, the source applies proposition [ftip-00BB] to the first-step maps of Adam, RMSProp, signSGD, and Lion; it checks Adafactor separately because its factored statistics are not an entrywise map.

This is a first-step obstruction. It suffices to disprove equivariance of the full run, but it does not classify every later state or say that such an optimizer cannot reach a particular solution.

Definition. First-step Adam map and gauge defect [ftip-00BD]

Fix $\epsilon >0$. For zero-state, bias-corrected Adam define the entrywise first-step direction

\[ D_\epsilon (G)=G\mathbin {\oslash }(|G|+\epsilon ) \]

and, for $Q\in \mathrm O(k)$, define its gauge-aligned defect by

\[ E_Q(G)=D_\epsilon (GQ)Q^{\mathsf T}-D_\epsilon (G). \]

Here $|G|$, addition, and division are entrywise. Bias correction makes the first direction independent of Adam's two decay coefficients under the declared convention [singh2026lossbasis, Proposition 4.4].

proposition. Exact represented-matrix defect after one Adam step [ftip-00BE]

Let $G_U=\nabla _UL(U_0,V_0)$, $G_V=\nabla _VL(U_0,V_0)$, $D_U=D_\epsilon (G_U)$, and $D_V=D_\epsilon (G_V)$. Let $W_1$ be the product after one Adam step from $(U_0,V_0)$ and $\widetilde W_1$ the product after one step from $(U_0Q,V_0Q)$. Then

\[ \begin {aligned} \widetilde W_1-W_1={}&-\eta \bigl (E_Q(G_U)V_0^{\mathsf T} +U_0E_Q(G_V)^{\mathsf T}\bigr )\\ &+\eta ^2\bigl (E_Q(G_U)D_V^{\mathsf T} +D_UE_Q(G_V)^{\mathsf T} +E_Q(G_U)E_Q(G_V)^{\mathsf T}\bigr ). \end {aligned} \]

Proof.

Gauge-align the rotated factors back by $Q^{\mathsf T}$. They are $U_0-\eta (D_U+E_Q(G_U))$ and $V_0-\eta (D_V+E_Q(G_V))$. Multiply, subtract $(U_0-\eta D_U)(V_0-\eta D_V)^{\mathsf T}$, and collect powers of $\eta $.

Proposition 4.4 and its proof are in Appendix B.5 of The loss does not see the basis, but Adam does[singh2026lossbasis].

Example. A two-coordinate Adam defect witness [ftip-00BF]

Take $d_1=d_2=1$, $k=2$, $f(w)=w^2/2$, $U_0=V_0=(1,0)$, and

\[ Q=2^{-1/2}\begin {pmatrix}1&1\\-1&1\end {pmatrix}. \]

The original and rotated products after one zero-state Adam step are

\[ W_1=\left (1-\frac {\eta }{1+\epsilon }\right )^2, \qquad \widetilde W_1=\left (1-\frac {\eta }{2^{-1/2}+\epsilon }\right )^2. \] $\text{[math]}$

They differ when $0<\eta <2^{-1/2}+\epsilon $. This is the explicit witness in Proposition 4.4 of The loss does not see the basis, but Adam does[singh2026lossbasis]; it is a statement about one factored quadratic, not about language-model performance.

Example. A rotated sign update fails to commute [ftip-00BG]

For $G=(1,1)$ and the 45-degree rotation matrix $Q$ of Example [ftip-00BF], one has $GQ=(0,\sqrt 2)$. With $\operatorname {sign}(0)=0$,

\[ \operatorname {sign}(GQ)=(0,1), \qquad \operatorname {sign}(G)Q=(0,\sqrt 2). \]

Thus even this two-coordinate update violates the commuting equation in Definition [ftip-00B7]. The calculation is the explicit Appendix B.4 witness of The loss does not see the basis, but Adam does[singh2026lossbasis].

Theorem. Full-rank memoryless equivariant rules are Gram-determined [ftip-00BH]

Let $k\leq n$, and let $\Phi $ be defined on full-column-rank matrices $G\in \mathbb R^{n\times k}$. Then $\Phi (GQ)=\Phi (G)Q$ for every $Q\in \mathrm O(k)$ if and only if there is a matrix-valued function $H$ of $GG^{\mathsf T}$ such that

\[ \Phi (G)=H(GG^{\mathsf T})G. \]

Proof.

The displayed form is immediately equivariant. Let $G^+$ denote the Moore--Penrose pseudoinverse. Conversely, set $X(G)=\Phi (G)G^+$. Then $X(G)G=\Phi (G)$ and $X(GQ)=X(G)$. Two full-column-rank matrices with the same left Gram matrix have the same range and differ by a right orthogonal factor. Hence $X(G)$ depends only on $GG^{\mathsf T}$; take $H=X$.

See Theorem 4.5 and its proof in Appendix B.6 of The loss does not see the basis, but Adam does[singh2026lossbasis]. It classifies the full-rank stratum only.

Corollary. The square invertible preconditioner representation [ftip-00BI]

In the specialization $k=n$ with $G$ invertible, let $P^{1/2}$ denote the unique positive-definite square root of $P$. The function in Theorem [ftip-00BH] has the unique canonical representative

\[ H(P)=\Phi (P^{1/2})P^{-1/2}, \qquad P=GG^{\mathsf T}>0. \]

Proof.

The polar decomposition writes $G=P^{1/2}Q$. Equivariance gives $\Phi (G)=\Phi (P^{1/2})Q=H(P)G$. Evaluating at $G=P^{1/2}$ proves uniqueness in this square invertible case.

For rectangular $G$, the action of $H(P)$ away from the range needed to multiply $G$ is not determined by $\Phi $. The proof above establishes uniqueness in the square invertible specialization.

Remark. What the structure theorem leaves unresolved at rank deficiency [ftip-00BJ]

In the ambient matrix space, full-column-rank matrices form an open dense subset whose complement has Lebesgue measure zero. A probability-one claim requires the random gradient law to be absolutely continuous and not confined to a lower-rank set; random initialization alone does not supply that premise.

Theorem [ftip-00BH] therefore makes no classification claim on a general rank-deficient stratum. Equivariance still constrains values within each orthogonal orbit, and it forces $\Phi (0)=0$, but it does not determine the rule there by continuity unless continuity is separately assumed. The source's generic-rank discussion therefore does not establish a classification at rank deficiency.

Common-scalar flows, clock changes, and momentum [ftip-00BK]

One narrow class of continuous-time preconditioners follows the gradient-flow path at a different speed. The clock change transfers path properties, but not rates, and it does not encompass a stateful momentum method.

Definition. Common-scalar preconditioned flow [ftip-00BL]

Let $\theta =(U,V)$ and let $a(t)>0$ be measurable with $1/a$ locally integrable. A common-scalar preconditioned flow is a locally absolutely continuous trajectory satisfying

\[ \dot \theta (t)=-\frac {\nabla L(\theta (t))}{a(t)} \]

for almost every $t$. The same scalar multiplies every coordinate of both factor gradients. For gauge-uniform conclusions, the source additionally requires $a(t)$ to be determined by gauge-invariant statistics of the trajectory up to time $t$.

Definition. Effective optimizer clock [ftip-00BM]

For the flow of Definition [ftip-00BL], define the effective optimizer clock

\[ \tau (t)=\int _0^t\frac {du}{a(u)}, \qquad \tau _{\max }=\int _0^\infty \frac {du}{a(u)}, \qquad 0<\tau _{\max }\leq \infty . \]

The clock is strictly increasing on its domain. If $\tau _{\max }=\infty $, it reaches every gradient-flow time; otherwise it traverses only the prefix with effective time below $\tau _{\max }$.

Theorem. Common-scalar flow is a time-reparameterized gradient flow [ftip-00BN]

Let $\theta (t)$ satisfy Definition [ftip-00BL], let $t(\tau )$ be the inverse of the clock in Definition [ftip-00BM], and put $\widetilde \theta (\tau )=\theta (t(\tau ))$. Then, for almost every $\tau $,

\[ \widetilde \theta '(\tau )=-\nabla L(\widetilde \theta (\tau )). \]

Proof.

The chain rule for absolutely continuous changes of variable gives $dt/d\tau =a(t)$ almost everywhere. Multiplying $\dot \theta =-\nabla L/a(t)$ by $dt/d\tau $ yields the displayed gradient flow.

See Theorem 4.6 and its proof in Appendix B.9 of The loss does not see the basis, but Adam does[singh2026lossbasis]. Gauge invariance of $a$ is not needed for this single-trajectory clock identity; it makes the clock common across a gauge orbit.

Corollary. Transfer of path and limit properties [ftip-00BO]

Every property depending only on the portion of the gradient-flow path traversed before $\tau _{\max }$ transfers to the common-scalar flow. If $\tau _{\max }=\infty $ and gradient flow converges, both flows have the same limit point. A sufficient condition for clock divergence is an eventual finite upper bound on $a(t)$.

For shallow factorization, the source cites Implicit regularization in matrix factorization[gunasekar2017implicit]. For deep factorization, it cites Implicit regularization in deep matrix factorization[arora2019implicit]. For greedy low-rank dynamics, it cites Towards resolving the implicit bias of gradient descent for matrix factorization: Greedy low-rank learning[li2021greedy].

Those conclusions transfer only when both this clock theorem and every hypothesis of the original result hold. This corollary supplies no missing matrix-sensing assumption.

Remark. Rates and hitting times do not survive time reparameterization [ftip-00BP]

Two flows can traverse the same curve with arbitrarily different physical clocks. A convergence rate, finite-step budget, or hitting time stated in $t$ therefore does not transfer through Theorem [ftip-00BN] without quantitative bounds on $a$ and its integral. If $\tau _{\max }<\infty $, even the tail and limit of the gradient-flow path are not reached.

Remark. Scalar-Adam is outside the common-scalar flow theorem [ftip-00BQ]

The shared-scalar Adam variant in proposition [ftip-00B9] retains a first-moment exponential moving average. It is stateful and is not the memoryless flow in Definition [ftip-00BL]. Remark 4.7 of The loss does not see the basis, but Adam does[singh2026lossbasis] treats its agreement with gradient flow as empirical rather than as a consequence of Theorem 4.6.

The theorem likewise does not apply to discrete Adam, PPO, RLVR update loops, clipping, weight decay, and mixed optimizers. Gauge equivariance and time reparameterization are separate properties.

Example. Equivariance neither guarantees nor precludes recovery [ftip-00BR]

The source records controls on both sides of the proposed implication. Its ScaledGD-inspired control is equivariant but equalizes the spectral schedule and does not recover the planted low-rank target at the declared budget. A long-anneal sign update is non-equivariant but eventually reaches a low-recovery-error regime.

$\text{[math]}$

The controls are reported in Remark 4.7 and Appendix D.10 of The loss does not see the basis, but Adam does[singh2026lossbasis]. The source reports no tuned reproducibility baseline for the ScaledGD control. The diagram is not a convergence theorem.

Optimizer geometry in training interventions [ftip-00BS]

Equivalent parameterizations need not follow equivalent training trajectories. Their evolution also depends on the update geometry; the scalar loss and represented starting object do not always suffice.

Definition. Objective-equivalent parameterizations [ftip-00BT]

Two factor pairs $(U,V)$ and $(\widetilde U,\widetilde V)$ are objective-equivalent for $L=f\circ W$ when

\[ W(U,V)=W(\widetilde U,\widetilde V). \]

They then have the same scalar loss for every objective depending only on that represented matrix. Orthogonal gauge-related pairs are objective-equivalent by Lemma [ftip-00AY]; objective equivalence need not imply membership in the same orthogonal orbit.

Definition. Parameterization-stable training intervention [ftip-00BU]

Fix a factorized objective, an update algorithm, hyperparameter schedule, randomness coupling, stopping rule, and initial optimizer state. The resulting training intervention is parameterization-stable on an orthogonal orbit when every two gauge-related initial states have identical represented trajectories:

\[ \widetilde U_t\widetilde V_t^{\mathsf T}=U_tV_t^{\mathsf T} \qquad \hbox {at every compared step }t. \]

This is a local stability property of a fully declared intervention. It does not say that arbitrary objective-equivalent pairs outside the orbit must agree.

proposition. Gauge equivariance gives parameterization stability [ftip-00BV]

Every gauge-equivariant optimizer is parameterization-stable on each orthogonal orbit, provided the remaining protocol coordinates are coupled as in Definition [ftip-00BU].

Proof.

Definition [ftip-00B3] gives $(\widetilde U_t,\widetilde V_t)=(U_tQ,V_tQ)$. Hence

\[ \widetilde U_t\widetilde V_t^{\mathsf T} =U_tQQ^{\mathsf T}V_t^{\mathsf T}=U_tV_t^{\mathsf T}. \]

This is a sufficient condition on a declared orbit, not a claim that equivariance is necessary for every possible equality of represented trajectories.

proposition. A first-step defect falsifies a loss-only protocol specification [ftip-00BW]

Suppose two orthogonally gauge-related initializations have equal represented object and scalar objective, but a declared update produces different represented objects after one step. Then the training intervention is not determined by the represented initialization and scalar objective alone.

Proof.

If those two objects determined the intervention, equal inputs would give the same represented next state. The witnessed inequality contradicts that factorization.

Proposition proposition [ftip-00BE] and Example [ftip-00BF] instantiate its premise for one factored quadratic and zero-state Adam. It proves underspecification, not a capability difference.

Example. The orthogonal gauge of an attention head [ftip-00BX]

For the row-vector attention weights of Definition [ftip-000P], let $W^Q,W^K\in \mathbb R^{d\times d_h}$ and $Q_h=XW^Q$, $K_h=XW^K$. Attention logits depend on these weights through

\[ Q_hK_h^{\mathsf T}=XW^Q(W^K)^{\mathsf T}X^{\mathsf T}. \]

For $A\in \mathrm O(d_h)$, the right action $(W^Q,W^K)\mapsto (W^QA,W^KA)$ preserves the middle product.

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Section 6 of The loss does not see the basis, but Adam does[singh2026lossbasis] uses column-vector weights and the invariant $W_Q^{\mathsf T}W_K$. Transposition gives the row-vector invariant $W^Q(W^K)^{\mathsf T}$, with the action on the right as displayed.

Remark. What the attention-twin experiment observes [ftip-00BY]

Section 6 and Table 3 of The loss does not see the basis, but Adam does[singh2026lossbasis] compare two function-equivalent initializations of a two-layer, four-head transformer on modular addition. The reported relative logit distance after one Adam step is $3.6\times 10^{-3}$, versus $2.9\times 10^{-7}$ for a same-basis noise twin; the final per-head invariant distance is reported as 56 percent.

The exact first-step defect has an algebraic explanation, but later drift, scale robustness, and downstream behavior are empirical. The tested small transformers, precision, optimizer allocation, seeds, and task do not establish a frontier-wide law.

Example. One loss quotient and two optimizer trajectories [ftip-00BZ]

A loss-only description collapses a gauge orbit to one represented starting point. A basis-sensitive optimizer can split that quotient back into different represented trajectories.

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The right-hand split is possible only because the centre box omits the factor basis and optimizer geometry. An FTIP admissible protocol in Definition [ftip-005J] must retain those intervention coordinates when they affect the executable result.

Remark. Product symmetry and the scope of optimizer conclusions [ftip-00C0]

The relevant optimizer results appear in version 1 of [⧉], submitted 5 August 2026: Sections 3--4 and 6, Appendices B.1--B.6 and B.9, with limitations in Section 11.

The full product symmetry acts on the factors by

\[ (U,V)\longmapsto (UA,VA^{-\mathsf T}). \]

It does not use $(UA,VA)$. For row-vector attention weights, the corresponding action uses right multiplication, as in Example [ftip-00BX].

The general rectangular preconditioner representation need not have the square-case uniqueness. A probability-one full-rank claim additionally needs a law for the gradient; a generic full-rank observation does not specify one.

The coordinatewise theorem is a zero-state, memoryless obstruction. The clock theorem is continuous-time and excludes scalar-Adam's momentum. Experimental optimizer and attention results remain observations reported by the source. Equivariance is not identified with low-rank recovery.

Remark. Continuous-flow conclusions and discrete optimization [ftip-00C1]

The factored-model results prove that update geometry can be a genuine intervention coordinate even when objective and represented initialization are fixed. They do not prove transformer or RLVR convergence, practical optimizer superiority, a scaling law, or any change in reliable capability. Different represented trajectories are not by themselves an acquisition witness of Definition [ftip-005X].

The memoryless theorem does not describe Adam's full state, and the continuous-flow result does not supply discrete rates or finite-step bounds. The source also notes that fixed-rank LoRA changes the learned-rank mechanism; no LoRA transfer is made here.

A dynamic-compute comparison also depends on active research state, persistent harness state, retained history, evaluation configuration, and descendant and retry costs. One realized chain and an archive-best envelope are different outcomes, even at the same declared budget.

How to read these notes [ftip-00NW]

The seven chapters connect learning mechanisms, computational constraints, agent reliability, evaluation, capability growth and economic conditions. The diagram shows their main logical relationships. Solid arrows indicate concepts used by another analysis. The dashed branch supplies architecture-specific detail for comparisons involving a particular model implementation or optimizer.

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Post-training and agent state are complementary sources of capability change: one changes learned parameters, while the other manages interaction, computation and retained artifacts. Evaluation studies the evidence for those changes and their limits. Architecture analysis connects the comparison to the model's computation and optimization. The lineage analysis extends it to discovery and learning across generations. Economic analysis then asks how productive capacity, knowledge renewal and financing make the required resources available, and how learning changes those conditions.

Foundations and interfaces introduces capability claims, token probabilities, pretraining, tasks and environments. These interfaces allow the later analysis to specify a model without requiring a particular neural architecture.
Post-training, alignment, and feedback develops fine-tuning, preference learning, RLHF, DPO, RLVR and agentic training, with concrete objectives, feedback mechanisms and update procedures.
Agent state, computation, and reliability studies persistent state, retained context, recursive execution, failure modes and reliability. It separates changes to agent behavior from changes to model weights while examining their interaction.
Evaluation, evidence, and finite limits fixes the comparison law and costs, studies transport to other evaluations, and develops finite results about discovery, support, feedback and proxy error. Its closing synthesis collects the consequences and counterexamples.
Architecture-conditioned potential and computation begins with a decoder-only Transformer, then studies architecture, optimization geometry and matched-compute capability comparisons. This branch uses the evaluation framework to make implementation-specific claims.
Conceptual discovery across model generations studies discovery and learning across successive artifacts. It develops the complete lineage, independently developed contributions, representation learning, curricula and mathematical research settings, alongside the conjecture and arguments that could establish or defeat it.
Civilization, economic reproduction and capability conditions learning on feasible economic paths. It develops knowledge renewal and financing bounds, reliable acquisition from contributors, shared preparation costs, and alternative models with different implications for continued learning.

The research question and scope introduces the common distinctions between elicitation, acquisition, computation and evaluation. Questions about fine-tuning, preferences, rewards and interaction lead to the post-training chapter. Questions about memory, tools, recursive execution and reliability lead to the agent chapter. Both use the model, task and environment interfaces in the foundations.

Questions about a particular model implementation or optimizer are developed in the architecture chapter. Its decoder-only reference, optimization geometry and matched-compute comparisons explain how capability claims depend on the computation that implements the model. The shared evaluation framework supplies the task laws and resource comparisons used in that analysis.

Claims about a capability gain or limit are examined in independent evaluation and post-training potential and the following finite results and counterexamples. These sections distinguish training reward from evaluation evidence, examine transport to other task laws, and make the assumptions behind a bound explicit. Training, search and deployment costs have distinct roles in those comparisons.

Questions about reusable concepts and capability growth across generations are developed in conceptual discovery. Its complete lineage, acquisition criterion and conjecture specify the autonomous and contributed processes being compared. Contributor development, learned representations, curricula and mathematical research provide concrete settings for that open question.

Questions about the origin of a learning budget begin with economic state and admissible paths and the uniform resource envelope. The knowledge-renewal model and financing account give distinct constraints; the automation game studies an equilibrium effect under a narrower institutional assumption. The contribution protocol and portfolio comparison specify possible cost advantages. Read these with the alternative mathematical models: they show which maintenance, investment, substitution and simulation assumptions can prevent a proposed separation. Each proved implication is conditional; none alone establishes a universal training ceiling or the open conceptual-discovery lower bound.