The shaded strip is only the visible part of the layer: the same stickers cross the front, right, back, top, and bottom faces; viewed from above, the top-face part is the corresponding rightmost strip. A whole-cube rotation would change the viewing frame, whereas this face move keeps the six centres fixed and changes one layer. The layered-cube family makes this distinction for every \(N\); the complete three-layer cube spells it out for the six outer generators.

A player sees colours; a proof must also track each sticker's cubie and transported local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-frame coordinate. A colour quotient forgets labels, while a larger cube adds positions, so the state space changes with the question.

Take a solved cube and twist one corner in place. Every sticker still has a cubie-sized home, and the three colours at that corner look locally consistent. But the corner orientations now have sum \(1\) modulo \(3\), whereas every face turn preserves sum \(0\). No sequence of legal turns can produce this state.

The complete three-layer cube derives the matching edge and permutation restrictions from the six generators. The calculation is useful at the table: it tells a solver when a scrambled-looking object was assembled illegally, and it tells a proof which state space is worth searching.

A human algorithm has to preserve these legal states while it borrows pieces from completed areas.

A formula often disturbs pieces that were already placed. The disturbance is acceptable only when the setup is undone and the completed pieces return. A working slot, a trigger, and an inverse setup make that promise visible. The complete layer-first method repeats this pattern through the cross, the first layer, the middle layer, and the last layer. Its completeness depends on the preservation and termination checks, not on the formula's familiarity.

The worked replay and projected lower bound fixes the viewing convention, records the ten tokens, and applies their inverse. Sticker replay returns to solved. A corner-only projected search finds no solution of depth at most two, so the evidence is the bounded statement

The inverse word is an upper-bound witness; the projected search is a lower-bound witness. The computer account explains why these checks do not prove that ten is shortest. A state, a word, and a replay log are enough to test the notation before asking for a general solver.

Once the move, its frame, its local repair, and its replay are explicit, three variations separate the effects of enlarging the cube, changing the shape, and forgetting labels.

The layered-cube family develops the first variation; the incidence and facet construction in incidence, symmetry and moves develops the second; and which moves survive a quotient? gives the descent condition for the third. Framed orientations, label transport and topology, and scrambling, spectra and search then study the extra information each variation introduces or forgets. Their hypotheses stay attached to the concrete move that prompted the question.

An abstract \(n\)-polytope, for \(n\geq 0\), is a partially ordered set \(P\) of faces with a least face \(F_{-1}\) and greatest face \(F_n\). Every chain extends to a maximal chain, called a flag, with exactly \(n+2\) faces. Faces in each flag have ranks \(-1,0,\ldots ,n\), consistently across flags. Faces other than the least and greatest are proper; vertices, edges and facets have ranks \(0,1,n-1\).

The diamond condition requires exactly two faces of rank \(i\) between faces \(F<G\) of ranks \(i-1\) and \(i+1\), for \(0\leq i<n\). Two flags are adjacent if they differ in exactly one proper face. Strong flag connectivity requires any two flags to be joined by adjacent flags, each containing their intersection.

Thus, for each \(0\leq i<n\), each flag has a unique adjacent flag differing in rank \(i\). Strong flag connectivity restricts paths to these adjacent flags, each containing the endpoint flags' intersection; arbitrary paths through the least face would connect every pair without imposing an axiom. These conditions prescribe incidence without choosing lengths or angles.

A rank-zero polytope is a two-element chain. A rank-one polytope has two vertices between its least and greatest faces. These low-rank cases occur as sections of higher-rank polytopes.

Let \(P\) be a abstract polytope. For faces \(F\leq G\) of ranks \(i,j\), its section \(G/F\) is the interval \(\{H:F\leq H\leq G\}\), with ranks shifted so that \(F\) has rank \(-1\). For \(F<G\), it is an abstract polytope of rank \(j-i-1\geq 0\). For \(F=G\), the singleton section is called the rank-\(-1\) polytope. These low ranks allow all intervals, including vertex figures of polygons and intervals between consecutive faces.

The section \(F_n/v\) at a vertex \(v\) is its vertex figure. Its vertices are the edges through \(v\); its faces record the incidences above \(v\). A facet \(H\) is viewed as the section \(H/F_{-1}\).

The face-poset construction of [schulte2017regular, Section 2] gives the following elementary examples of flags. A triangle has the empty face, three vertices, three edges and the whole triangle, of ranks \(-1,0,1,2\). The flag \(\varnothing <1<12<123\) has rank-zero neighbor \(\varnothing <2<12<123\) and rank-one neighbor \(\varnothing <1<13<123\). There are \(3\cdot 2=6\) flags.

Each vertex in this flag graph records the vertex and edge of one flag; the common empty and greatest faces are suppressed. Edge labels give the rank changed by adjacency.

A tetrahedron has all subsets of \(\{1,2,3,4\}\) as faces, ordered by inclusion; a nonempty face with \(k\) vertices has rank \(k-1\). Its flag \(\varnothing <1<12<123<1234\) exhibits all five ranks. The vertex figure at \(1\) is a triangle: its vertices correspond to the three incident edges. A cube also has a triangular vertex figure, although its facets are squares. Local and global shapes carry different information.

An automorphism of a abstract polytope \(P\) is a bijection \(a:P\to P\) satisfying \(F\leq G\iff a(F)\leq a(G)\). Automorphisms form a group \(\Gamma (P)\) under composition; in \(ab\), the map \(b\) acts first. They preserve ranks and act on flags face by face.

The polytope is regular if this action is transitive on flags. The tetrahedron is regular: a permutation of its four vertices sends any flag to any other.

The underlying order automorphism is represented by OrderIso with equal source and target. This declaration does not impose the polytope axioms or assert regularity.

An automorphism of an abstract polytope fixing every face of one flag is the identity. Consequently the automorphism group of a regular polytope acts freely and transitively on its flags.

The cited paragraph preceding Lemma 2.1 states simple transitivity for regular polytopes. The elementary argument below also proves freeness without assuming regularity.

Choose a base flag \(\Phi \) in a regular abstract polytope \(P\) of rank \(n\). By flag freeness, there is a unique automorphism \(\rho _i\) taking \(\Phi \) to its rank-\(i\) neighbor, for \(0\leq i<n\). The diamond condition and uniqueness give \(\rho _i^2=1\). For \(|i-j|>1\), the two rank changes are independent; \(\rho _i\rho _j\) and \(\rho _j\rho _i\) agree on \(\Phi \), hence are equal. Connectivity shows that the \(\rho _i\) reach every flag, so they generate \(\Gamma (P)\). These are the distinguished generators of [schulte2017regular, Section 5, pp. 13--14].

For \(I\subseteq \{0,\ldots ,n-1\}\), put \(G_I=\langle \rho _i:i\in I\rangle \). They also satisfy

This intersection property is part of the string C-group description [schulte2017regular, Section 5, equation (18)]. Involution and nonadjacent commutation relations alone do not reconstruct a polytope; the group-theoretic characterization includes this additional condition [schulte2017regular, Section 5, p. 14].

Let \(H\) be a facet of a regular abstract polytope \(P\) of rank \(n\geq 2\). Restriction gives an isomorphism

In particular the facet is regular, and each of its automorphisms has a unique extension fixing \(H\). This is a consequence of the flag action in [schulte2017regular, Section 2 and Lemma 2.1]; the argument follows.

Let \(Q\) be a regular abstract polytope of rank \(m\geq 2\). Choose a base flag and its distinguished generators (involutions) \(\rho _0,\ldots ,\rho _{m-1}\). Its rotation subgroup is

The index starts at \(1\): for a polygon, the single adjacent pair \(\rho _0\rho _1\) generates its rotations. For triangles and squares these are the usual cyclic rotation groups.

Equivalently, \(\Gamma ^+(Q)\) consists of the elements represented by even-length words in the distinguished involutions. Indeed every pair \(\rho _i\rho _j\) is a product of adjacent pairs or their inverses, by cancellation along the index interval. Conjugating an even word by a distinguished involution again gives an even word, so this subgroup is normal. A change of base flag conjugates the distinguished generators and therefore leaves the subgroup unchanged. We do not require it to have index two for every abstract polytope.

Let \(P\) be a finite regular abstract polytope of rank \(n\geq 3\), with least and greatest faces \(F_{-1},F_n\). Write \(P^\circ =P\setminus \{F_{-1},F_n\}\). The incidence positions are

Choose a facet \(H\) and an element \(\alpha \) of the rotation subgroup \(\Gamma ^+(H/F_{-1})\). The facet extension lemma gives its unique extension \(\widehat \alpha \in \Gamma (P)\) fixing \(H\). The corresponding facet move is

The first entry decides where the shape automorphism is applied; the second entry is transported with it. The facet is part of the move's name: one shape automorphism may stabilize several facets and give different local moves. Each move is a permutation of \(\mathcal R(P)\).

Let \(P\) be a finite regular abstract polytope of rank \(n\geq 3\), \(H\) a facet and \(\alpha \in \Gamma ^+(H/F_{-1})\) a facet rotation. The facet move \(\mu _{H,\alpha }\) is a permutation of \(\mathcal R(P)\), with

This follows directly from the construction in [marchetti2025rubik, Definitions 5--6].

The general construction combining permutations of complementary subsets is Equiv.Perm.subtypeCongr; its inverse formula is Equiv.Perm.subtypeCongr.symm. Here the predicate is \(F\leq H\), the moving permutation is the restricted shape automorphism and the other permutation is the identity. These declarations supply that general combination; the polytope-specific preservation argument is the proof above.

Represent the faces of a tetrahedron by subsets of \(\{1,2,3,4\}\). We calculate facet moves from [marchetti2025rubik, Definitions 5--6]. For the facet \(H=123\), take an element of the rotation subgroup with global extension \(\widehat \alpha =(123)\), fixing \(4\). The vertex-edge records follow the cycle

Every arrow is the local move \(\mu _{H,\alpha }\). The record \((4,14)\) stays fixed because \(4\not \leq H\), although the global shape symmetry sends it to \((4,24)\). The diagram records this permutation of incidence positions, rather than the tetrahedron's face graph.

For records whose second entry is a facet, the same move gives

The last record is fixed even though the global symmetry changes its second entry. Now let \(K=124\) and take an element of the rotation subgroup with global extension \(\widehat \beta =(124)\). Applying the \(H\)-move and then the \(K\)-move sends

Reversing their order gives \((1,124)\mapsto (2,124)\mapsto (3,234)\), so the two moves do not commute. Each has order three: its cube is the identity on both branches, and its displayed action is nontrivial.

For a finite regular abstract polytope \(P\) of rank \(n\geq 3\), the Rubik group is the subgroup generated by all facet moves:

This is Marchetti's generated group of permutations [marchetti2025rubik, Section 3, p. 6, following Definition 6]. The inverse lemma ensures that the generators are permutations.

Give each incidence position its own initial label. A labelled configuration can be represented by the permutation carrying each initial label to its current position. The identity is the solved configuration; the reachable labelled configurations are the elements of \(\operatorname {Rub}(P)\). A new move acts on the left: applying \(\mu \) to configuration \(g\) gives \(\mu g\), with \(g\) acting first. A visible coloring may identify several labelled configurations, so a solved visible coloring need not determine the labelled permutation.

Realize the cube as \([-1,1]^3\) and let \(H\) be the facet \(z=1\). The facet move construction of [marchetti2025rubik, Definitions 5--6] gives the following explicit outer-face action. The shape rotation

is a clockwise quarter-turn when viewed from positive \(z\), with \(x\) rightward and \(y\) upward. Its restriction to \(H\) supplies our move. The corner \((1,1,1)\) goes to \((1,-1,1)\), and its color on \(y=1\) goes to the color position on \(x=1\). The corresponding corner at \(z=-1\) is fixed by the puzzle move, although the global map \(q\) moves it.

The signs give the \(x,y\) coordinates of the four corners on \(z=1\). The arrows show their clockwise motion; the central incidence record \((H,H)\) stays fixed. No orientation of a marked center is recorded.

Restricting the second entry to facets gives \(8\cdot 3+12\cdot 2+6=54\) incidence positions: three per corner, two per edge and one per face center. The turn moves twelve corner stickers and eight edge stickers, fixing the other thirty-four positions, including the unmarked center of the turned face. This is the usual outer-face action on visible stickers of a \(3\times 3\times 3\) cube in a fixed frame.

The full incidence set has \(8\cdot 7+12\cdot 3+6=98\) elements: above each vertex are that vertex, three edges and three facets; above each edge are itself and two facets. The facet-colored subset is invariant under every move. Its restricted action supplies the comparison with ordinary stickers; a faithful restriction for all abstract polytopes is not needed. Marked center orientations and inner-layer turns are additional data, absent from this comparison.

The intermediate coordinates are \((\pi (i),a_i x)\). Applying \(h\) evaluates its local map at that intermediate position, giving \((\sigma \pi (i),b_{\pi (i)}a_i x)\). Thus a local factor must be multiplied at its transported position index.

For \(h=(\sigma ,b)\), the composition law and commutativity of the target give

The middle equality reindexes using the bijection \(\pi \). In the new frames the local map is \(a_i'=t_{\pi (i)}^{-1}a_i t_i\). Its abelianized product contains each \([t_i]\) and its inverse exactly once, so these cancel and leave \(\chi (g)\).

Apply the coordinate action \(r\) times. Each local map is applied after the preceding one, giving the stated product with the first map on the right. In the abelianization its factors may be reordered; multiplying over disjoint cycles uses each factor exactly once. A fixed position is a cycle of length one and must also be checked.

Let \(H\leq G\) have finite index. Choose representatives \(r_i\) for the left cosets \(r_iH\), indexed bijectively by a finite set \(I\). For \(g\in G\), left multiplication permutes the cosets, so uniquely write

Write \(H_{\mathrm {ab}}=H/[H,H]\) and \([h]\) for the class of \(h\in H\). The abelianization is the target of the transfer

is a homomorphism independent of the chosen representatives. This is the left-coset construction of [mayer2015artin, Definition 3.1, Proposition 3.1 and Theorem 3.1, pp. 4--6]. Neither finiteness of \(G\) nor normality of \(H\) is required.

The linked transfer definitions specialize the homomorphism \(\phi :H\to A\) to the quotient \(H\to H_{\mathrm {ab}}\). The indexed formula takes \(f(i)=r_i\) and \(\pi =\pi _g\); its factor \(r_{\pi _g(i)}^{-1}g r_i\) is precisely \(h_i(g)\). Its finite-index instance follows from the finite bijective family of cosets.

Let \(G\) act transitively on a finite nonempty position set \(I\), and on a tagged union of fibers \(X_i\), carrying \(X_i\) bijectively to \(X_{g i}\). Fix \(i_0\in I\) and put \(H=G_{i_0}\). Let \(K\) act faithfully on a nonempty set \(\Omega \); choose a bijective base frame \(c_0:\Omega \to X_{i_0}\) and a homomorphism \(\rho :H\to K\) satisfying \(h c_0=c_0\rho (h)\) for every \(h\in H\). Choose \(r_i\in G\) with \(r_i i_0=i\), and set \(c_i=r_i c_0\). Then the total-orientation character of this framed transport is

where \(T_{G,H}\) is transfer and \(\rho _{\mathrm {ab}}([h])=[\rho (h)]\). This identifies the coset construction in [mayer2015artin, Section 3.3, p. 6] with the local-frame coordinates of [marchetti2025rubik, Section 3.1, pp. 6--7].

The vertical arrows are the transported frames. The equality \(g r_i=r_{\pi _g(i)}h_i(g)\) makes the square commute: after the destination representative is removed, the remaining motion fixes \(i_0\) and acts through \(\rho \). Changing representatives changes these frames within \(K\), so the character is unchanged.

The induced map is Abelianization.map. Naturality of transfer is MonoidHom.transfer_comp, with \(\phi :H\to H_{\mathrm {ab}}\) the quotient map and the target map \(\rho _{\mathrm {ab}}\). That algebraic identity supports the last equality; the identification with the actual fiber action is the transported-frame argument above.

Let \(G\) be the actual permutation group generated by the six outer-face quarter-turns of the ordinary cube, with its frame of reference fixed. Use the corner and edge fibers: cyclic corner labels with local group \(C_3\), and edge labels with local group \(C_2\). This specializes the local-orientation construction of [marchetti2025rubik, Section 3.1]; the orbit argument below identifies the stabilizers needed for transfer.

The action on eight corner positions is transitive. Adjacent corners share a face whose quarter-turn takes one to the other, and the cube's edge graph is connected. Hence a corner stabilizer \(H_c\) has index eight. Its action on the three cyclically ordered labels gives a homomorphism \(\rho _c:H_c\to C_3\). The action on twelve edge positions is also transitive: the four edges of one face form a single orbit under its quarter-turn; neighboring faces share an edge; and the face-adjacency graph is connected. Thus an edge stabilizer \(H_e\) has index twelve, with local action \(\rho _e:H_e\to C_2\).

Choose a frame at the fixed corner or edge and transport it using coset representatives. The transfer comparison gives

We identify the abelianizations of \(C_3\) and \(C_2\) with those groups. The face-turn calculation proves that the twist and flip characters vanish on every legal generator, hence on \(G\). It follows that these two local images of transfer vanish. It does not follow that \(T_{G,H_c}\) or \(T_{G,H_e}\) is trivial in the larger stabilizer abelianizations. No classification of \(G\) was needed.

For \(d\geq 3\), a vertex figure of a \(d\)-simplex or a \(d\)-cube is a \((d-1)\)-simplex. Its automorphism group is \(S_d\), and its rotation subgroup is \(A_d\). These are the local corner groups used in [marchetti2025rubik, Section 4, p. 10, and Section 5, p. 14].

Here \(d\) is abstract dimension. Varying the layer count of an \(N\times N\times N\) cube leaves the ambient dimension equal to three; it changes the piece orbits and allowed layer moves instead.

Let \(A_d\) be the alternating group on \(d\) letters, with composition rightmost first. Then

The corresponding commutator subgroups are stated after Proposition 4.1 in [marchetti2025rubik, Section 4.1, p. 11]. We give the elementary calculation, using \([g,h]=ghg^{-1}h^{-1}\).

The three cases have separate formalized ingredients. alternatingGroup.isCyclic_of_card_le_three gives the cyclicity of \(A_3\). For four letters, alternatingGroup.kleinFour_eq_commutator identifies \(V\) with the commutator subgroup, and TauCeti.card_abelianization_alternatingGroup gives the quotient's cardinality three; the displayed cyclic isomorphism then uses the elementary classification of groups of prime order. For \(d\geq 5\), commutator_alternatingGroup_eq_top is the perfection statement. These links concern alternating groups, not the entire puzzle group.

For a \(d\)-simplex or \(d\)-cube with \(d\geq 3\), the local corner groups have rotation subgroup \(A_d\). Their contribution to the total-orientation character passes through \((A_d)_{\mathrm {ab}}\). The abelianization calculation therefore measures exactly what this detector can see, as in [marchetti2025rubik, Section 4.1, p. 11, and equation (19), p. 14].

In dimension three the target is \(C_3\), the familiar corner twist sum. In dimension four the local group has grown to the nonabelian \(A_4\), but the character still sees only a \(C_3\) quotient: every local change in the Klein four group disappears from it. In dimensions five and above the local group is perfect, so this corner character is identically trivial on the ambient framed transports.

A lost detector does not classify the legal moves. Other piece types, permutation signs and the subgroup actually generated by facet turns may still impose constraints. Even when the target is nontrivial, conservation requires a generator check; the local abelianization alone does not supply one.

A capability claim in this note names four objects:

1. a starting model artifact \(M_0\), with its accompanying information, libraries, tools and initially available controllers disclosed;
2. a procedure \(A\) that may query models, interact, retain experience and train successor artifacts, with its feedback sources specified;
3. an evaluation rule \(E\), including the task law, retained artifact and deployment procedure being evaluated; and
4. a resource bound \(b\).

The claim concerns the observable outcome of running \(A\) from \(M_0\) and judging the result with \(E\) while respecting \(b\). It is not a claim about an unbounded or unspecified model.

For a comparison across generations, use the complete lineage specification to make these initial resources and permitted operations explicit. Newly generated controllers and learned artifacts belong to the charged process. An externally supplied contribution has a declared producer and acquisition procedure; it is not silently added to the starting model's resources.

The bound \(b\) may contain several coordinates, such as training tokens, rollouts, accelerator work, wall-clock time, tool calls, or evaluation-time samples. Distinguish the complete development budget from the deployment budget used to compare frozen artifacts on fresh problems. Coordinates with different units remain separate unless a declared cost model converts them to a common unit.

This convention follows the experimental separation between training and inference procedures used in [shao2024deepseekmath, secs. 3--4] and [guo2025deepseek, secs. 2--3]. Those studies motivate the objects in a claim; they do not imply that every capability is summarized by one benchmark or one scalar budget.

Holding the model fixed, the comparison isolates how inference procedure and budget alter measured success.

Assume independent draws and a declared perfect checker that recognizes a correct candidate when one appears. The one-draw procedure succeeds with \(J_1=p\). A procedure allowed up to \(k\) draws succeeds whenever not all draws fail: \[ J_k=1-(1-p)^k. \] Both procedures query the same conditional distribution.

Policy performance under a specified interaction and evaluation procedure is standard in [sutton2018reinforcement, Section 3.5]. The formula demonstrates procedural dependence. Independence and a perfect checker are declared toy assumptions, not properties attributed to a frontier language model.

Fix a capability claim in Convention [ftip-0003], a prespecified successful behaviour event \(B\), and a tolerance \(\varepsilon >0\). Let \(p_0(B)\) be the probability of \(B\) under the declared starting procedure and \(p_1(B)\) its probability after the intervention, both measured by the same evaluator. An elicitation witness is an observed increase \[ p_1(B)>p_0(B)\quad \text {with}\quad p_0(B)\geq \varepsilon . \] It witnesses improved access to behaviour that was already observable at the declared starting budget.

The event \(B\), tolerance \(\varepsilon \), sampling procedure, and confidence rule are part of the claim. Changing any of them changes what the witness means.

Reinforcement learning can move probability toward successful traces that a base model already emits; this possibility is tested empirically in [liu2025understanding, secs. 2.3 and 3.3--3.4]. A finite experiment cannot establish that an unrestricted generative model assigns exactly zero probability to a behaviour. The positive threshold \(\varepsilon \) therefore belongs to the declared experiment rather than to an ontological claim about what the model ``contains.''

An elicitation witness depends on the sampling budget, decoding rule, and reward. Bounding the attainable increase in \(p_1(B)\) requires assumptions about the starting probability \(p_0(B)\) and the information supplied by post-training.

Fix a capability claim in Convention [ftip-0003], a prespecified transfer evaluation \(E^{+}\), and thresholds \(0\leq \varepsilon _0<\varepsilon _1\leq 1\). Let \(u_0\) and \(u_1\) be the probabilities of passing \(E^{+}\) before and after the intervention, using the same evaluation protocol and budget. An acquisition witness is the pair of inequalities \[ u_0\leq \varepsilon _0 \qquad \text {and}\qquad u_1\geq \varepsilon _1. \] The transfer evaluation must not be used to select training examples, rewards, or checkpoints.

An acquisition witness is operational. It records a large, independently measured change under one declared procedure; it does not prove that the starting model assigned mathematical probability zero to every successful continuation.

Training success alone does not distinguish learning from memorization, selection, or evaluator exploitation. The base-versus-reinforcement-learning comparisons in [yue2025does, secs. 3--5] motivate this distinction, but their empirical criteria are not a universal definition of acquisition.

The thresholds in Definition [ftip-0007] require enough evaluation samples to certify both inequalities. Whether the witness survives a distribution shift depends on the task law; evaluator leakage invalidates its interpretation as independent evidence.

For a finite set \(X\), write \(|X|\) for its cardinality and \(X^*\) for the set of finite sequences with entries in \(X\). A map \(f:X\to Y\) sends \(x\in X\) to \(f(x)\in Y\); the inverse image of \(A\subseteq Y\) is \(f^{-1}(A)=\{x\in X:f(x)\in A\}\).

Write \[ \Delta (X)=\left \{p:X\to [0,1]:\sum _{x\in X}p(x)=1\right \} \] for the probability simplex on \(X\). If \(p\in \Delta (X)\) and \(Z:X\to \mathbb R\), then \[ \mathbb E_{x\sim p}[Z(x)]=\sum _{x\in X}p(x)Z(x). \] A random variable is a map from the sample set to its value set. Random variables \(Z\) and \(W\) are independent under \(p\) when \(p(Z=z,W=w)=p(Z=z)p(W=w)\) for every pair of values \(z,w\).

For nonfinite interaction spaces, probability laws and conditional kernels require the measurable structure described in Remark [ftip-000C].

Using the notation of Notation [ftip-000A], a finite probability law on \(X\) is an element \(p\in \Delta (X)\). For an event \(A\subseteq X\), its probability is \[ p(A)=\sum _{x\in A}p(x). \] For \(p(A)>0\), the conditional law on \(B\subseteq X\) is \(p(B\mid A)=p(A\cap B)/p(A)\).

A finite vocabulary and a bounded token sequence admit probability laws expressed as ordinary sums, with explicit normalization and conditioning. The finite-sequence interface matches [phuong2022formal, sec. 3, Sequence modelling].

States, observations, and tool outputs need not be finite. In that setting \(\Delta (X)\) is replaced by probability measures on a declared measurable space and conditional laws are probability kernels. Results on finite spaces need additional measurable-space hypotheses before they apply in this setting.

A finite vocabulary is a nonempty finite set \(\mathcal V\) of token identifiers. A token identifier is an atomic symbol for the sequence model; it need not be a word or a character.

When termination is represented in-band, choose a distinguished token \(\mathtt {eos}\in \mathcal V\). Other control tokens, if present, are named as elements of the same vocabulary rather than assumed implicitly.

Let \(\Sigma \) be a finite character or byte alphabet and let \(\mathcal V\) be the vocabulary of Definition [ftip-000D]. A tokenizer is a specified map \(\tau :\Sigma ^*\to \mathcal V^*\). A detokenizer is a specified map \(\delta :\mathcal V^*\to \Sigma ^*\).

A tokenizer package also fixes its text normalization rule \(N\). When the package promises normalized round trips, the relevant condition is \(\delta (\tau (s))=N(s)\) for text \(s\in \Sigma ^*\); it is not the claim that \(\tau \) and \(\delta \) are inverse bijections on all sequences. SentencePiece keeps the segmentation model and detokenization convention together, which is why both maps belong to the model interface.

Byte-pair encoding gives another construction of \(\tau \). Learned merge operations turn an initial symbol sequence into subword units; see [sennrich2016neural, §3.2]. The learned merges and base alphabet are therefore part of the tokenizer specification.

A four-stage toy tokenizer separates token-level round-trip stability from recovery of the original string.

Here underscores display spaces. Declare \(N(A\_\_B)=a\_b\), \(T(a\_b)=(a,b)\), and \(D(a,b)=a\_b\). For \(s=A\_\_B\), \[ D(T(N(s)))=a\_b\neq s, \qquad T(N(D(T(N(s)))))=(a,b). \] The token sequence is stable after normalization although the original case and repeated space are not recovered.

The source-grounded tokenizer interface is Definition [ftip-000E]. The displayed normalization rule belongs only to this finite construction; no such lossy transformation is attributed to SentencePiece.

Let \(\mathcal V\) be the vocabulary of Definition [ftip-000D]. A token sequence of length \(n\geq 0\) is written \(x_{1:n}=(x_1,\ldots ,x_n)\in \mathcal V^n\); positions are one-based. Its length is \(|x_{1:n}|=n\), its prefix before position \(t\) is \(x_{<t}=x_{1:t-1}\), and \(x_{<1}\) is the empty sequence. Juxtaposition \(xy\) denotes concatenation.

If \(\mathtt {eos}\) is distinguished, a completed generated continuation \(y_{1:m}\) ends with \(y_m=\mathtt {eos}\) and contains no earlier \(\mathtt {eos}\). A fixed length limit can also stop generation; the stopping mechanism must state which convention it uses.

Let \(\Theta \) be a parameter set. With the probability notation of Notation [ftip-000A] and the sequence convention of Convention [ftip-000G], a conditional next-token law is a family of maps \[ \pi _\theta :\mathcal V^*\longrightarrow \Delta (\mathcal V), \qquad \theta \in \Theta . \] For a prefix \(x_{<t}\) and token \(a\in \mathcal V\), the number \(\pi _\theta (a\mid x_{<t})\) is the probability assigned to choosing \(a\) at position \(t\).

This is the observable probabilistic interface. It does not yet specify the neural computation that produces the distribution.

Fix the next-token law of Definition [ftip-000H], a prompt \(x\in \mathcal V^*\), and a continuation \(y_{1:m}\in \mathcal V^m\). Its autoregressive continuation probability is \[ P_\theta (y_{1:m}\mid x) =\prod _{t=1}^{m} \pi _\theta \left (y_t\mid x y_{<t}\right ). \] Each factor conditions only on the prompt and earlier continuation tokens.

If generation stops on \(\mathtt {eos}\), the law of a completed continuation is restricted to the sequences specified in Convention [ftip-000G]. If generation instead stops at a length limit, the terminal event and its probability must be recorded separately.

Along one branch of a three-symbol prefix tree, conditional token laws determine both continuation probability and negative log-likelihood.

Let the vocabulary be \(\{a,b,\mathtt {EOS}\}\) and let \(p_i\) be the displayed conditional probability at step \(i\). The continuation \((b,a,\mathtt {EOS})\) has \[ \Pr (b,a,\mathtt {EOS}\mid a) =p_1p_2p_3, \qquad -\log \Pr =-\sum _{i=1}^{3}\log p_i. \]

The multiplication follows the causal product in [phuong2022formal, Section 3]. Its scope is one fixed finite continuation, leaving both the decoding algorithm and the law over all variable-length outputs unspecified.

Let \(\mathcal D_{\rm doc}\) be a measurable document space and \(\mathcal S\) a finite set of data sources. Source \(s\in \mathcal S\) has a document law \(Q_s\) on \(\mathcal D_{\rm doc}\), and mixture weights \(\alpha \in \Delta (\mathcal S)\) use the simplex notation of Notation [ftip-000A]. Let \(T\) be the context length. A token window is \(x_{1:T}\in \mathcal V^T\); a prediction mask is \(m_{1:T}\in \{0,1\}^T\), where \(m_t=1\) means that the token at position \(t\) contributes to the loss.

Let \(\Theta \) be the parameter set used by the next-token law in Definition [ftip-000H]. The model parameters at optimizer update \(k\in \{0,1,\ldots ,K\}\) are \(\theta _k\in \Theta \); \(\theta _0\) is the initialization and \(\theta _K\) the stopped parameter value. A minibatch at update \(k\) is \(\mathcal B_k\). Random document draws, window offsets, masks, batch order, and optimizer noise are part of the training run even when the notation suppresses them.

Using Notation [ftip-0015], a pretraining source mixture is the pair \(((Q_s)_{s\in \mathcal S},\alpha )\). It induces the document law \[ Q_{\alpha }(A)=\sum _{s\in \mathcal S}\alpha _s Q_s(A) \] for every document event \(A\): first sample a source \(s\) with probability \(\alpha _s\), then sample a document from \(Q_s\).

The mixture weights specify sampling frequency. They need not equal the fraction of bytes stored in each source. Resampling, epoch boundaries, or temperature-based weights must therefore be recorded as part of \(\alpha \) or its schedule.

Let \(\dagger \) denote rejection. A document filter is a declared map \[ F:\mathcal D_{\rm doc}\longrightarrow \mathcal D_{\rm doc}\cup \{\dagger \}. \] It may retain a document, return a transformed document, or reject it. The lift \(F_{\#}\) applies \(F\) to a finite corpus and removes every \(\dagger \).

A filter specification includes the features it reads, thresholds, transformation rules, and implementation revision. An acceptance rate alone does not determine the retained document law.

Let \(\mathcal M_{\rm doc}\) be the set of finite multisets of documents. A deduplication operator is a map \[ D:\mathcal M_{\rm doc}\longrightarrow \mathcal M_{\rm doc} \] that selects retained representatives according to a declared duplicate relation. The relation may compare complete documents or spans.

The declaration includes normalization, similarity threshold, cluster construction, and representative tie-breaking. Different choices can remove different training events even when they report the same duplicate rate.

For a protected evaluation collection \(\mathcal E\), an evaluation decontamination operator is a map \[ C_{\mathcal E}:\mathcal M_{\rm doc}\longrightarrow \mathcal M_{\rm doc} \] that removes documents or spans matching \(\mathcal E\) under a declared matching rule.

The protected collection, normalization, match granularity, threshold, and removal policy are part of the operator. Decontamination against one released benchmark does not establish independence from every related task.

Let \(\mathcal C_{\mathrm {raw}}\in \mathcal M_{\rm doc}\) be a finite corpus drawn from the source mixture in Definition [ftip-0016]. Using the declared document filter, deduplication operator, and evaluation decontamination operator, an ordered pretraining data pipeline returns \[ \mathcal C_{\mathrm {clean}} =C_{\mathcal E}\left (D\left (F_{\#}(\mathcal C_{\mathrm {raw}})\right )\right ). \] Changing the order can change the retained corpus. The pipeline therefore records the order and exact revisions in addition to the three operators.

The source mixture and cleaning pipeline determine different parts of the data law. The mixture in Definition [ftip-0016] describes how raw documents are proposed. The construction in Definition [ftip-0017] determines which proposals survive and how they are transformed. Together with all random seeds and thresholds, they induce a cleaned document law \(Q_{\mathrm {clean}}\).

Kaplan et al. describe the concrete WebText2 dataset in [kaplan2020scaling, sec. 2.3]. Hoffmann et al. report the MassiveText sources and mixture in [hoffmann2022training, Appendix A, Table A1]. A finite released corpus is one realization of such choices, and an undocumented change in them is not identified by a loss--compute curve.

Fix the context length \(T\) from Notation [ftip-0015]. A context-window sampler draws cleaned text from the law described in Remark [ftip-0018] and tokenizes it with Definition [ftip-000E]. It then chooses a declared offset and returns a length-\(T\) token sequence \(x_{1:T}\) together with metadata that identifies document boundaries.

If fewer than \(T\) tokens remain, the sampler must specify whether it pads, drops the suffix, or joins another document. These choices change both the distribution of contexts and the prediction mask.

A packed training record is a tuple \(\zeta =(x,m,c,M^{\mathrm {doc}})\). It places tokenized segments \(x^{(1)},\ldots ,x^{(r)}\) into the length-\(T\) token array \(x\), records a prediction mask \(m\), and records a segment identifier \(c_t\in \{0,1,\ldots ,r\}\) at each position. The value \(c_t=0\) denotes padding. Its document mask is \[ M^{\mathrm {doc}}_{ts}= \begin {cases} 0,&c_t=c_s\ne 0\text { and }s\leq t,\\ -\infty ,&\text {otherwise}. \end {cases} \] Replacing the causal mask in Definition [ftip-000S] by this mask prevents one packed document from attending to another. We write \(x_t(\zeta )\), \(m_t(\zeta )\), and \(M^{\mathrm {doc}}(\zeta )\) for the corresponding fields.

The prediction mask \(m_t(\zeta )\) from Notation [ftip-0015] is one only at positions whose target token is retained for training. Padding, a segment's first token when no predecessor is present, and any deliberately excluded control token receive mask value zero.

[kaplan2020scaling], Section 2.2, uses fixed-length sequence batches.

[phuong2022formal], Section 7, gives the causal language-model objective.

The proposed data interface additionally specifies the sampler, segment identifiers, document mask, and prediction mask. Those boundary choices are not determined by the high-level training accounts.

These objects describe one admissible data interface. A concrete training run must replace them with its offset, padding, joining, masking, and packing rules.

Let \(\zeta =(x,m,c,M^{\mathrm {doc}})\) be a packed training record from Definition [ftip-001A]. Write \(\pi _{\theta ,M^{\mathrm {doc}}(\zeta )}(\,\cdot \mid x_{<t}(\zeta ))\) for the next-token law of Definition [ftip-000H] when every decoder attention layer uses the declared document mask. For a predicted position \(t\geq 2\), its masked token negative log-likelihood is \[ \ell _t(\theta ;\zeta ) =-m_t(\zeta )\log \pi _{\theta ,M^{\mathrm {doc}}(\zeta )} \left (x_t(\zeta )\mid x_{<t}(\zeta )\right ), \] A masked position has zero contribution; an unmasked position penalizes the log probability computed under the same segment boundaries that defined the packed example.

Let \(Q_T\) be the law of packed examples induced by Remark [ftip-0018]--Definition [ftip-001A], and assume that its expected number of predicted positions is positive. The population pretraining objective is the expected total token loss divided by the expected number of predicted tokens: \[ L_{\mathrm {pop}}(\theta ) =\frac { \mathbb E_{\zeta \sim Q_T} \left [\sum _{t=2}^{T}\ell _t(\theta ;\zeta )\right ] }{ \mathbb E_{\zeta \sim Q_T} \left [\sum _{t=2}^{T}m_t(\zeta )\right ] }, \] using expectation as fixed in Notation [ftip-000A].

Packing documents into one row requires a block-causal mask so that each token attends only within its own document and to earlier positions there.

For a packed row, set \(M_{ij}=0\) only when tokens \(i,j\) belong to the same document and \(j\leq i\); otherwise set \(M_{ij}=-\infty \). Thus a token in document \(B\) cannot inspect any token in document \(A\), even when \(A\) precedes \(B\) in storage.

The within-document triangle instantiates the causal mask in Definition [ftip-000S]. The block boundary instantiates the packing convention in Definition [ftip-001A]. Document-isolated attention is the declared convention in this packed row, not a universal description of pretraining systems.

For a finite training sample of packed records \(\mathcal D_N=(\zeta ^{(i)})_{i=1}^{N}\) with at least one unmasked target token, the empirical pretraining objective is \[ L_N(\theta ) =\frac { \sum _{i=1}^{N}\sum _{t=2}^{T} \ell _t(\theta ;\zeta ^{(i)}) }{ \sum _{i=1}^{N}\sum _{t=2}^{T}m_t(\zeta ^{(i)}) }. \] This is a per-predicted-token average. Reusing or resampling examples changes the stochastic optimization path even when the displayed finite-sample function is unchanged.

The autoregressive token loss follows [phuong2022formal, Section 7]. Kaplan et al. report language-model loss with parameter and compute variables in [kaplan2020scaling, sec. 2.1]. The population ratio-of-expectations in Definition [ftip-001C] and finite per-token ratio in Definition [ftip-001E] are proposed normalization conventions with explicit prediction masks. Neither cited source defines those exact ratios.

A theorem or experiment using another record weighting, length weighting, or expectation order must state that change rather than reuse these symbols.

At update \(k\), let \(\mathcal B_k\) be the minibatch named in Notation [ftip-0015], and let \(N_k=\sum _{\zeta \in \mathcal B_k}\sum _{t=2}^{T}m_t(\zeta )\) be its number of predicted tokens. For \(N_k>0\), the minibatch gradient estimator is \[ g_k=\frac 1{N_k} \sum _{\zeta \in \mathcal B_k}\sum _{t=2}^{T} \nabla _\theta \ell _t(\theta _k;\zeta ), \] where \(\ell _t\) is defined in Definition [ftip-001B]. Whether \(g_k\) is unbiased for the gradient of Definition [ftip-001C] depends on the window sampler and minibatch weighting. Other sampling schemes require their own bias statement.

Two equal-length records make a full empirical loss directly comparable with a minibatch estimate, while exposing the assumptions behind unbiasedness.

Let \(\ell _{ij}\) denote the token loss at position \(j\) of record \(i\). The full token-average loss is the mean of all \(\ell _{ij}\), while a one-record minibatch averages only the losses from its sampled record. Uniform sampling of equal-length records makes the latter unbiased for the former.

The preceding loss and estimator definitions apply to this equal-length record setup. Equal record lengths make the estimator unbiased here; unequal lengths, padding, dependence, and optimizer noise remain outside its scope.

An optimizer state \(s_k\) is the collection of persistent variables, other than the model parameters \(\theta _k\), that an update rule uses after step \(k\). For Adam, \(s_k\) contains the step index and exponential moving averages of the gradient and squared gradient.

The state is initialized by a declared value \(s_0\). Two runs with the same checkpoint \(\theta _k\) but different optimizer states need not have the same next update.

A parameter update rule is a specified map \(U_k\) that takes the current parameters \(\theta _k\), optimizer state \(s_k\), gradient estimate \(g_k\), and scheduled hyperparameters \(\eta _k\), and returns \[ (\theta _{k+1},s_{k+1}) =U_k(\theta _k,s_k,g_k,\eta _k). \] The learning rate, momentum coefficients, numerical stabilizers, clipping, and weight decay belong to \(\eta _k\) or to the declared form of \(U_k\); they are not determined by the loss alone.

A pretraining run is not determined by minimization of the empirical objective \(L_N\) in Definition [ftip-001E]. It includes the initialization, data order, window offsets, dropout and other model randomness, numerical precision, distributed reduction order, optimizer state, every hyperparameter schedule, and a stopping rule. These objects determine a distribution over final parameters even when the source mixture and nominal objective agree.

The training configuration of [vaswani2017attention, sec. 5.3] and the scaling experiments of [kaplan2020scaling, sec. 2.2] report concrete instances of these choices. An optimization result depends on which randomness it averages over and whether the stopping time is fixed or data-dependent.

A base-model artifact is an executable tuple \[ M_0=(\tau ,\delta ,\mathsf {cfg},\theta _K,\mathsf {num}), \] where \(\tau ,\delta \) are the tokenizer and detokenizer of Definition [ftip-000E]; \(\mathsf {cfg}\) fixes the architecture of Definition [ftip-0012]; \(\theta _K\) is the stopped pretraining checkpoint; and \(\mathsf {num}\) fixes the numerical and inference conventions required to evaluate its next-token law.

The tokenizer model is part of the executable system by Definition [ftip-000E]. A weight file without its vocabulary, architecture configuration, or output convention does not determine the law \(\pi _{\theta _K}\) in Definition [ftip-000H]. Training-data and optimizer provenance may accompany the artifact, but they are not runtime inputs.

Section 6 of [phuong2022formal] describes decoder architecture. Sections 3.1 and 3.5 of [kudo2018sentencepiece] treat the tokenizer as a separate model. Public model releases additionally distribute weights, configuration, and numerical conventions. The proposed artifact model in Definition [ftip-001K] includes these runtime ingredients because none alone determines an executable next-token law.

Training provenance accompanies the artifact but is not itself a runtime argument. Later causal claims must distinguish the executable endpoint from the process that produced it.

An executable-artifact manifest separates runtime components from evidence about how the corresponding weights were trained.

The pinned DeepSeek-V4-Flash-0731 configuration [⧉] declares architecture and numeric-format fields. Sections 2.4 and 4 of the V4 family report [⧉] describe a mixed Muon/AdamW optimizer allocation and pretraining data construction. The exact 0731 data mixture, optimizer schedule, and post-preview training delta remain undisclosed by those sources.

Write \(\mathsf T\) for a task interface, \(\mathcal X_{\mathsf T}\) for its instance set, and \(\mathcal O_{\mathsf T}\) for its candidate-outcome set. A task instance is \(x\in \mathcal X_{\mathsf T}\) and a candidate outcome is \(o\in \mathcal O_{\mathsf T}\). The symbol \(\mu _{\mathsf T}\) denotes a probability law on instances. Equip the instance and outcome sets with declared sigma-algebras; \(\mu _{\mathsf T}\) is a probability measure on the instance sigma-algebra.

Fix an evaluation-resource dimension \(m_{\mathrm {eval}}\geq 1\) and write \(\mathcal B_{\mathrm {eval}}=\mathbb R_+^{m_{\mathrm {eval}}}\) with its coordinatewise order. An evaluation inference protocol is written \(\mathsf I\); its declared evaluation budget is \(b_{\mathrm {eval}}\in \mathcal B_{\mathrm {eval}}\). Its inference-seed set is \(\Omega _I\), with realized seed \(\xi \in \Omega _I\). An evaluator's random-seed set is \(\Omega _E\); its realized seed is \(\omega \in \Omega _E\). It returns utility \(u_{\mathsf T}(x,o;\omega )\in \mathbb R^d\). The dimension \(d\geq 1\) is declared by the evaluation: \(d=1\) gives scalar utility, while \(d>1\) retains several outcomes or costs without an implicit weighting. Equip both seed sets with declared sigma-algebras and \(\mathbb R^d\) with its Borel sigma-algebra. Products below carry the product sigma-algebra; finite or countable discrete spaces may use all subsets. A seed space alone does not specify its sampling law.

A task is a triple \[ \mathsf T=(\mathcal X_{\mathsf T},\mathcal O_{\mathsf T}, \operatorname {Adm}_{\mathsf T}), \] where \(\mathcal X_{\mathsf T}\) is an instance set, \(\mathcal O_{\mathsf T}\) is a candidate-outcome set, and \(\operatorname {Adm}_{\mathsf T}(x,o)\) is a declared well-formedness predicate for an instance-outcome pair.

Admissibility says that an outcome can be interpreted for the instance; it does not say that the outcome is correct, useful, or safe. Those judgments belong to the evaluation utility introduced later.

For the instance set in Definition [ftip-001O], a task law is a probability law \(\mu _{\mathsf T}\) on \(\mathcal X_{\mathsf T}\). An evaluation sample of size \(N\) is a declared joint law of instances \((X_1,\ldots ,X_N)\) whose marginals are \(\mu _{\mathsf T}\); independent sampling is a further assumption, not part of the term ``task law.''

In an episodic control problem this law includes the distribution of starting situations. In a fixed benchmark it may instead be the uniform law on a held-out finite set. Reweighting instances changes the task law even when the instance files are unchanged.

A task and a task law specify admissible instance--outcome pairs and how instances are sampled. Benchmark datasets usually present an input and expect an answer in a parseable format. Reinforcement-learning formulations instead name states, actions, and reward; see [sutton2018reinforcement, Chapter 3]. The admissibility predicate captures parseability, while the law records how instances are sampled. Neither choice already identifies parsing with success.

One-shot inference and an interactive environment can share the same task interface. Claims about an interaction's temporal structure additionally depend on the environment and the observations available at each decision.

Let \(M\) be an executable model artifact of the form in Definition [ftip-001K]. An evaluation inference protocol is a specified procedure \[ \mathsf I(M,x,b_{\mathrm {eval}};\xi )=(o,c) \in \mathcal O_{\mathsf T}\times \mathcal B_{\mathrm {eval}}, \] where \(x\in \mathcal X_{\mathsf T}\), \(b_{\mathrm {eval}}\in \mathcal B_{\mathrm {eval}}\), and \(\xi \in \Omega _I\) use the notation of Notation [ftip-001N]; \(o\) is a candidate outcome and \(c\in \mathcal B_{\mathrm {eval}}\) is the realized resource-use vector. A valid execution satisfies \(c\preceq b_{\mathrm {eval}}\) coordinatewise.

The protocol fixes prompt construction, decoding, sampling temperature, number of candidates, candidate selection, allowed tools, stopping, and any test-time search. None of these choices is determined by the checkpoint alone.

One next-token law supports many evaluation procedures. Greedy decoding, temperature sampling, majority voting, verifier-guided selection, and multi-round tool use can produce different outcome laws from the same checkpoint. The sampling and selection procedures in [shao2024deepseekmath, secs. 3--4] are concrete examples.

Accordingly, a comparison that changes \(b_{\mathrm {eval}}\) or \(\mathsf I\) does not estimate a training-only effect. A potential frontier therefore depends on evaluation-time compute as well as the trained model.

With the notation of Notation [ftip-001N], an evaluation utility is a declared measurable function \[ u_{\mathsf T}:\mathcal X_{\mathsf T}\times \mathcal O_{\mathsf T}\times \Omega _E\longrightarrow \mathbb R^d, \] where \(\Omega _E\) is the evaluator's measurable seed space. The evaluator applies it only to admissible pairs from Definition [ftip-001O]. A deterministic evaluator is the special case in which the value does not depend on \(\omega \in \Omega _E\).

Binary pass-fail evaluation takes \(d=1\) and values in \(\{0,1\}\). A cumulative return is another scalar case. Vector utility keeps qualities such as correctness, safety, latency, and tool cost distinct until a later rule declares how to compare them.

Whenever an expected utility vector is used, its composite evaluation random variable must be measurable and absolutely integrable in every coordinate under the declared joint law. Thus its expectation lies in \(\mathbb R^d\). Bounded measurable utility is a sufficient specialization; so is finite support for the complete evaluation random variable with finite values on that support. Merely taking a finite real value at each seed is insufficient for an unbounded evaluator. The scalar protocol comparison in Convention [ftip-005D] states its joint law explicitly.

Sutton and Barto define scalar episodic return in [sutton2018reinforcement, Section 3.3]. Benchmarks also use exact-match, test, latency, safety, and cost measurements. A proposed seeded vector utility keeps those coordinates separate, with evaluator randomness included in the declared law.

The vector is not a claim that its coordinates share a natural scale. Scalarization, Pareto order, and worst-group summaries are introduced only after the attainable evaluation set is defined.

One fixed policy can receive different measured success rates when the evaluation horizon changes.

Partition complete trajectories into those that finish within the short horizon, those requiring additional tokens, and failures. The short protocol counts only the first class; the longer protocol may count the first two. Thus the measured score depends on the declared horizon even though the policy is unchanged.

Making an evaluation horizon explicit follows finite-horizon policy evaluation in [sutton2018reinforcement, Chapter 3]. Only truncation is varied here. Larger budgets need not help once decoding rules, tool costs, or selection errors are allowed to change with the budget.

The inference protocol receives only the information designated as public. The public instance \(x\) may include a prompt, tools, and a response schema. Evaluator-only information may include a reference answer, hidden tests, a simulator state, or the evaluator seed \(\omega \). The protocol in Definition [ftip-001R] receives the former and not the latter.

If evaluator-only information influences training, decoding, or candidate selection, the experiment no longer measures performance under the declared information interface. A theorem that assumes an independent evaluator must name the sigma-field or finite data record from which the protocol is excluded.

Let \(\mathcal S,\mathcal O,\mathcal A\) be declared measurable spaces of environment states, public observations, and agent actions. Time is discrete, \(t\in \{0,1,\ldots ,T_{\max }\}\), where \(T_{\max }<\infty \) is a hard horizon. Random states, observations, and actions are \(S_t,O_t,A_t\); lowercase \(s_t,o_t,a_t\) denote realized values. The same convention makes \(h_t\) a realization of \(H_t\) and \(z\) a realization of \(Z\).

The public history available before action \(A_t\) is \(H_t\); the full environment trajectory is \(Z\); and the stopping time is \(\tau _{\mathrm {stop}}\in \{0,1,\ldots ,T_{\max }\}\). Probability kernels are written \(K(dy\mid x)\); on finite spaces this means the probability law \(K(y\mid x)\) and reduces to the notation of Notation [ftip-000A].

With the spaces in Notation [ftip-001X], an initial-state law \(\rho _0(ds_0,do_0)\) is a probability law on \(\mathcal S\times \mathcal O\). It jointly samples the hidden initial state \(S_0\) and the first public observation \(O_0\).

Allowing a joint law covers both a deterministic rendering \(O_0=\operatorname {obs}(S_0)\) and a noisy initial observation. A fixed task instance can be included as a coordinate of \(O_0\); its sampling law is then the task law of Definition [ftip-001Q].

A transition-observation kernel at time \(t\) is a probability kernel \[ K_t(ds_{t+1},do_{t+1}\mid s_t,a_t) \] from the current hidden state and action to the next hidden state and public observation. After \(A_t=a_t\), the environment draws \((S_{t+1},O_{t+1})\) from this kernel.

The factorization \(K_t(ds',do'\mid s,a)=P_t(ds'\mid s,a)\,Q_t(do'\mid s',a)\) for declared kernels \(P_t\) and \(Q_t\) recovers the usual separate state-transition and observation kernels when such a factorization is declared. The joint-kernel form does not assume it.

The kernel in Definition [ftip-001Z] describes both tools and broader environments. A deterministic tool has a measurable update map \(F_t(s,a)=(s',o')\) and the point-mass kernel concentrated at \(F_t(s,a)\). A stochastic environment permits several next state-observation pairs for the same input.

This distinction matters for replay. Re-executing a deterministic tool call under an unchanged state reproduces its observation, whereas a stored rollout from a stochastic environment is one sample from a conditional law. Any analysis that replaces one with the other needs a coupling or concentration assumption.

At time \(t\), the public history is the alternating sequence \[ H_t=(O_0,A_0,O_1,A_1,\ldots ,A_{t-1},O_t). \] Thus \(H_0=(O_0)\). It contains exactly the observations received and actions taken before the next action \(A_t\); it does not contain the hidden state \(S_t\) unless that state was itself revealed as an observation.

The history in Definition [ftip-0021] contains the agent's available information. The hidden state sequence \((S_0,\ldots ,S_t)\), future observations, evaluator-only tests from Remark [ftip-001V], and future random seeds are absent unless the environment has already exposed them through \(O_0,\ldots ,O_t\).

This boundary determines which policies are implementable. It also separates legitimate memory from leakage: a summary of earlier public observations may be stored in the history, while a hidden reference answer may not be supplied under the same declaration.

A non-anticipating policy is a family of probability kernels \[ \pi _t(da\mid H_t), \qquad t=0,1,\ldots ,T_{\max }-1, \] from public histories in Definition [ftip-0021] to the action space \(\mathcal A\). At time \(t\), the action may depend on \(H_t\) and fresh policy randomness, but not on a later observation or an unobserved state.

A decoder-only language model becomes such a policy only after an inference protocol serializes \(H_t\) into tokens, decodes tokens, and parses them as an action. The next-token law in Definition [ftip-000H] alone does not specify those maps.

A stopping rule is a random time \(\tau _{\mathrm {stop}}\in \{0,1,\ldots ,T_{\max }\}\) such that, for each \(t\), the decision whether \(\tau _{\mathrm {stop}}=t\) is determined by the public history \(H_t\) and any declared stopping randomness available by time \(t\). It cannot inspect future observations.

Stopping can be triggered by a terminal environment observation, an agent action, a resource limit, or the hard horizon. The trigger and the owner of the decision are part of the rule.

If \(\tau _{\mathrm {stop}}=n\), the stopped trajectory is \[ Z=(S_0,O_0,A_0,S_1,O_1,A_1,\ldots , A_{n-1},S_n,O_n). \] It contains the environment states as well as the public action-observation record. Its public projection is the history \(H_n\) of Definition [ftip-0021].

For \(n=0\), the trajectory is \((S_0,O_0)\) and contains no action. A terminal utility may depend on the full trajectory when evaluated inside the environment, even though the policy is restricted to its public projection.

For a task \(\mathsf T=(\mathcal X_{\mathsf T},\mathcal O_{\mathsf T}, \operatorname {Adm}_{\mathsf T})\) from Definition [ftip-001O], an interactive realization is

For each \(x\in \mathcal X_{\mathsf T}\), the law \(\rho _0^x\) is on \(\mathcal S_{\mathsf T}\times \mathcal O^{\rm obs}_{\mathsf T}\), and \(K_t^x\) maps a state and action to the next state-observation law for \(0\leq t<T_{\max }\). The outcome map sends a stopped trajectory \(Z\) to \(\operatorname {out}_{\mathsf T}(x,Z)\in \mathcal O_{\mathsf T}\) and satisfies \(\operatorname {Adm}_{\mathsf T}(x,\operatorname {out}_{\mathsf T}(x,Z))\). For one fixed realization, write its three spaces as the unadorned \(\mathcal S,\mathcal O,\mathcal A\) of Notation [ftip-001X].

The agent policy and stopping rule are not environment fields. They are combined with this realization only when an interaction law is formed.

Section 3.1 of [sutton2018reinforcement] specifies the agent--environment interaction. Section 2 of [kaelbling1998planning] separates hidden state from observation. We combine those ingredients with the task interface of Definition [ftip-001O] so each task instance selects an initial law and transition kernels.

The outcome-extraction map is an additional interface choice; neither cited source supplies it in this task-indexed form. This map connects a stopped interaction to the candidate outcome that the independent evaluator will score.

A deterministic three-action episode records each observation before the next policy decision and ends with an explicit terminal observation.

Let \(H_0=(\text {door locked})\). After action \(A_0=\text {take key}\), the environment returns \(O_1=\text {key held}\), so \(H_1=(H_0,A_0,O_1)\). Repeating the same update with unlock and door open, then choosing \(A_2=\text {enter}\), produces \(O_3=\text {terminal success}\) and \(H_3=(H_2,A_2,O_3)\). No later observation is available to an earlier action choice.

The alternating action--observation interface is the finite-history form of the agent setup in [kaelbling1998planning, Section 2]. The trace checks chronology and termination; uncertain state, partial observability, and language-valued actions lie beyond this deterministic instance.

A prefix tree represents token emission as a finite interaction stopped either by an end token or by a hard length cap.

Take vocabulary \(\{a,\mathtt {EOS}\}\). The terminal histories are an immediate end token, an end token after \(a\), and a length-capped history \(aa\). Their probabilities are products of the conditional token laws and sum to one. The last history terminates because the declared maximum length is reached, not because the policy emitted \(\mathtt {EOS}\).

The causal token product is given in [phuong2022formal, Section 3]. The finite representation distinguishes two stopping mechanisms. Token steps remain distinct from tool calls, and unbounded generation is outside the construction.

Sutton and Barto's agent--environment interface in [sutton2018reinforcement, sec. 3.1] supplies alternating states, actions, and rewards. Kaelbling et al. separate hidden state and public observation in [kaelbling1998planning, Section 2]. A stopped interaction is determined by its initial law, transition--observation kernel, policy, and public stopping rule.

The induced interaction law uses four independently declared objects: the initial law \(\rho _0\) in Definition [ftip-001Y], the environment kernels \(K_t\) in Definition [ftip-001Z], the policy \(\pi _t\) in Definition [ftip-0023], and the stopping rule in Definition [ftip-0024]. On finite spaces, the probability of a length-\(n\) trajectory before applying its stopping indicator factors as \[ \rho _0(s_0,o_0) \prod _{t=0}^{n-1} \pi _t(a_t\mid h_t) K_t(s_{t+1},o_{t+1}\mid s_t,a_t). \] The event \(\{\tau _{\mathrm {stop}}=n\}\) selects the stopped trajectories.

This factorization supports later expectations, likelihood ratios, and off-policy reuse. It also exposes an identifiability limit: an outcome distribution by itself generally does not reveal which of the initial law, environment, policy, or stopping rule changed.

Given the base artifact \(M_0=(\tau ,\delta ,\mathsf {cfg},\theta _K,\mathsf {num})\) from Definition [ftip-001K], a fine-tuning run declares training records, a finite update count \(J\geq 0\), a loss \(L_{\rm ft}\), gradient estimators \(g_j^{\rm ft}\), optimizer states \(s_j^{\rm ft}\), and update specifications \((U_j,\eta _j^{\rm ft})\). It starts from \(\theta ^{\rm ft}_0=\theta _K\).

For each \(j=0,\ldots ,J-1\), the update is

Its output artifact replaces \(\theta _K\) by the final \(\theta ^{\rm ft}_J\). Any change to the tokenizer, architecture configuration, numerical convention, or other artifact field must be stated separately. Fine-tuning is not identified with one loss or data type.

In the cited pipeline, supervised fine-tuning (SFT) starts from a pretrained GPT-3 checkpoint and updates it by supervised learning on labeler demonstrations. The resulting policy supplies the starting and reference policy for later stages.

Appendix C.1 reports optimization choices for that construction. It does not define a universal mask, weighting rule, or fine-tuning protocol.

Wei et al. train on instruction-expressed task mixtures in [wei2022finetuned, Section 2 and Figure 2]. Ouyang et al. fine-tune on labeler demonstrations in [ouyang2022training, Section 3.5 and Appendix C.1]. The proposed continued-optimization interface in Definition [ftip-006Q] includes these concrete supervised instances.

Neither cited source defines fine-tuning as one universal objective. A concrete run must state its records, loss and normalization, optimizer, stopping rule, and trainable parameters. Instruction tuning later specializes the records to instruction--response demonstrations.

Let \(\mathcal X_{\mathrm {pr}}\) be the prompt space. For each prompt \(x\in \mathcal X_{\mathrm {pr}}\), let \(\mathcal Y(x)\subseteq \mathcal V^*\) be the set of finite token responses permitted by the training format. We write \(y=(y_1,\ldots ,y_{|y|})\in \mathcal Y(x)\) and \(y_{<t}=(y_1,\ldots ,y_{t-1})\).

A demonstration is denoted \(z=(x,y,\omega )\), where \(\omega \) is its provenance record. A finite multiset of demonstrations is denoted \(D_{\mathrm {sft}}\). For each response position, a loss mask \(m_t(z)\in \{0,1\}\) records whether the token contributes to the supervised objective, and a nonnegative weight \(w_t(z)\) records its declared normalization. Prompt tokens are conditioning context and lie outside the response-position domains of \(m_t(z)\) and \(w_t(z)\).

Using the prompt space of Notation [ftip-002A], a prompt serializer is a declared map \[ s_{\mathrm {pr}}:\mathcal X_{\mathrm {pr}}\longrightarrow \mathcal V^*. \] For token strings \(u,v\in \mathcal V^*\), write \(u\mathbin {\Vert }v\) for their concatenation. A response token \(y_t\) is therefore predicted from the token prefix \(s_{\mathrm {pr}}(x)\mathbin {\Vert }y_{<t}\).

The serializer fixes system text, role markers, separators, and any other prompt-side control tokens. Changing it changes the conditional examples seen by the language model even when the abstract prompts and responses are unchanged.

The next-token law in Definition [ftip-000H] accepts token prefixes, whereas an instruction dataset may store structured prompts. The proposed serializer in Definition [ftip-006V] maps between those types. Ouyang et al. describe SFT on labeler demonstrations in [ouyang2022training, Section 3.5 and Appendix C.1]; their training account supplies a concrete pipeline rather than a universal serializer.

A reproducible objective must retain the serializer revision with the tokenizer and loss mask. Otherwise two runs can share an abstract dataset name while optimizing different token sequences.

Instruction tuning fine-tunes a pretrained language model on a mixture of tasks whose examples are expressed through natural-language instructions and corresponding target outputs. The task description and any input are placed in the prompt; the target output supplies the continuation to be learned.

The construction changes the model parameters by supervised learning. It does not require a learned reward model, preference comparisons, or interaction with an environment.

Using the notation of Notation [ftip-002A], a demonstration record is a triple

where \(\omega \) identifies the declared producer of \(y\), the collection or generation procedure, and any selection rule applied before the record entered \(D_{\mathrm {sft}}\). The response is a training target; its presence in the data does not assert that it is uniquely correct or preferred to every alternative.

The provenance field \(\omega \) distinguishes a demonstration from the process that supplied it. Ouyang et al. collect labeler-written demonstrations in [ouyang2022training, §§3.2, 3.4]. Wei et al. permit a broader task mixture in [wei2022finetuned, §2; Fig. 2].

Human writers, stronger teacher models, filtered self-generations, and programmatically produced targets give different information to the learner. Claims about sample efficiency or capability acquisition must therefore state which producer and selection rule were available.

In the InstructGPT data-collection procedure, a labeler receives a prompt and writes a desired response. The resulting prompt--response pair enters the demonstration data used by the supervised fine-tuning stage. Prompts come from the declared API and labeler-written sources, and the paper records filtering and labeler-selection procedures.

This construction specifies a human demonstration channel. It is distinct from the later comparison channel, in which a labeler ranks model-generated responses rather than writing the target response.

For the language-model policy \(\pi _\theta \) and the notation of Notation [ftip-002A], a masked supervised fine-tuning (SFT) objective has the form

where the nonnegative weights satisfy \(\sum _{z\in D_{\mathrm {sft}}}\sum _{t=1}^{|y|}w_t(z)m_t(z)=1\). The response index \(t\) does not range over serialized prompt tokens. Minimizing this objective increases conditional likelihood on the selected target tokens. The weights and mask are local conventions and must be declared for each implementation.

For a finite demonstration multiset, define the number of selected response tokens by

When \(N_{\mathrm {resp}}>0\), response-token normalization sets \(w_t(z)=N_{\mathrm {resp}}^{-1}\) at every selected position. Each selected token then has equal weight in \(L_{\mathrm {sft}}\). Equal weighting of records or prompts is a different convention because response lengths vary.

The training account alone does not determine this normalization. Any comparison of losses or gradients must retain the mask and normalization that produced them.

Two instruction positions provide conditioning context but lie outside the response-loss index set. The loss is taken on the two generated tokens.

Declare \(\pi (r_1\mid u_1,u_2)=1/2\) and \(\pi (r_2\mid u_1,u_2,r_1)=1/4\). The instruction tokens are not indexed by the response mask. With mask one on each displayed response token, the sum and mean are \[ L_{\rm sum}=-\log \tfrac 12-\log \tfrac 14=\log 8, \qquad L_{\rm mean}=\tfrac 12\log 8\approx 1.040. \]

The supervised objective in Definition [ftip-002F] and the causal factorization in Definition [ftip-000I] specialize to the displayed token strip. Its mask is a local choice on response positions; other instruction-tuning implementations may supervise a different subset of response tokens or declare a larger loss domain explicitly.

Instruction tuning changes conditional likelihood under a declared demonstration distribution. It can teach response format, task interpretation, and behavior represented by the targets, but the training loss alone does not identify which of those effects caused an independent evaluation gain.

Ouyang et al. report supervised fine-tuning and its optimization settings in [ouyang2022training, Section 3.5 and Appendix C.1]. They do not define the exact mask and weighting convention displayed in Definition [ftip-002F]; those choices remain part of the declared objective.

The stage also leaves several questions open. A demonstration does not compare its target with alternatives, a teacher may transmit errors or hidden information, and low loss on the collected prompts need not imply transfer to a new task law. Preference acquisition introduces a different observation: which response a judge selected from a displayed pair.

Use the prompt and response spaces from Notation [ftip-002A]. A comparison query is a displayed triple \(c=(x,y^0,y^1)\) with \(y^0,y^1\in \mathcal Y(x)\). Let \(\mathcal J\) be the declared judge population, and let \(j\in \mathcal J\) identify the human, model, or rule that returns a comparison label \(b_{\mathrm {pref}}\in \{0,1\}\).

When \(b_{\mathrm {pref}}=1\), write \(y^+=y^1\) and \(y^-=y^0\); when \(b_{\mathrm {pref}}=0\), reverse those names. Thus \(y^+\) means selected in this observation, not objectively or uniquely best. The acquisition mechanism may also record a tie, abstention, or invalid comparison, but those outcomes must use an enlarged label space rather than being silently forced into \(\{0,1\}\).

Restrict attention to a comparison for which the judge selects one displayed alternative rather than reporting indifference or incomparability. For a query \(c=(x,y^0,y^1)\) and a judge \(j\), a pairwise comparison observation is the displayed pair together with the judge's selected alternative. In winner--loser notation it is recorded as

The observation reports a choice made under the declared presentation and query procedure. It does not by itself provide a numerical reward, a complete ranking of responses, or a judgment about alternatives that were not shown.

Preference learning may choose which trajectory segments or responses to show to a judge rather than sampling every pair uniformly. The query-selection procedure assigns candidate comparisons a priority derived from the current reward-model ensemble and requests labels for selected pairs.

Because selection depends on the current model and candidate pool, the resulting comparison data reflect an acquisition policy. Active selection can concentrate labels on uncertain pairs, but it does not make the collected comparisons representative of an undeclared target population.

A preference observation depends on the query presented and the judge who evaluates it. Concrete elicitation procedures include [christiano2017deep, Sections 2.2.2 and 2.2.4] and the language-model comparison pipeline of [ouyang2022training, Sections 3.2 and 3.4].

The response generator, pair-selection rule, presentation order, judge population, and aggregation rule can each change the observed law. A later sample-complexity or population claim must condition on those choices rather than treating comparison labels as an unqualified source of ground truth.

A preference data law \(\mathcal D_{\mathrm {pref}}\) is a probability law on pairwise comparison observations \(o=(x,y^+,y^-,j)\) from Definition [ftip-002L]. It includes the randomness of prompt selection, response generation, pair selection, judge selection, and the judge's reported label.

A finite training multiset \(D_{\mathrm {pref}}=(o_1,\ldots ,o_N)\) is sampled or adaptively collected under that law and its acquisition history. When queries are adaptive, the records need not be independent or identically distributed.

A scalar reward model is a parameterized function

whose score is fitted from comparison observations. The score orders or weights responses for a specified training procedure; it is not automatically the independent utility of the response. Ouyang et al. initialize the model from the supervised policy and replace its language-model head with a scalar output for this stage.

Given scalar scores for two alternatives, the Bradley--Terry comparison law assigns the probability

where \(\sigma (a)=(1+\exp (-a))^{-1}\). Conditioning the scores on a language-model prompt is the contextual specialization used in modern reward modeling; see also [rafailov2023direct, Section 3, equation (1)].

For comparison observations sampled from \(\mathcal D_{\mathrm {pref}}\), the pairwise reward-model loss is

The empirical objective replaces the expectation by a declared weighting of the collected comparisons. It fits score differences under the Bradley--Terry law of Definition [ftip-002Q]; it does not observe an absolute reward target for either response.

Fitting a Bradley--Terry predictor is an assumption about representation, not a consequence of observing comparisons. Cyclic choices, context effects, and mixtures of judges can fail to agree with one shared scalar ordering.

Even within the model, scores are not identified absolutely: replacing \(r_\phi (x,y)\) by \(r_\phi (x,y)+c(x)\) leaves every pairwise probability and the loss unchanged. Ouyang et al. give one normalization in [ouyang2022training, §3.5; App. C.2]. Other uses must state their own normalization, judge population, and validation law.

A declared difference between two reward scores maps to one Bradley--Terry preference probability.

Under the Bradley--Terry link, \[ \Pr (y^+\succ y^-\mid x) =\frac {\exp r(y^+)}{\exp r(y^+)+\exp r(y^-)} =\sigma (2-0.5) =\sigma (1.5)\approx 0.8176. \] Reversing the pair gives probability \(1-0.8176=0.1824\).

The fixed scores instantiate Definition [ftip-002Q] and determine the displayed probability. Whether one scalar reward can represent every judge is a separate modeling question.

A reward-model validation law \(\mathcal D_{\mathrm {rm}}^{\mathrm {val}}\) is a held-out law on comparison observations used to evaluate the fitted score model rather than to update \(\phi \). A declared validation statistic may be Bradley--Terry log loss or the probability that \(r_\phi (x,y^+)>r_\phi (x,y^-)\).

The validation population and acquisition procedure are part of the quantity. Accuracy on comparisons drawn from the same collection process does not establish calibration under a new judge population or agreement with an independent task-success criterion.

Ouyang et al. hold out comparison data and report reward-model validation accuracy and loss in [ouyang2022training, Section 3.5 and Appendix C.2]. Representing the validation population by a probability law makes the estimand depend explicitly on the judge population and acquisition procedure.

This law belongs to reward-model validation. It is not the independent task evaluation used later to compare post-training protocols.

Three strict comparisons arranged in a directed cycle obstruct representation by a single scalar ordering.

Declare \(A\succ B\), \(B\succ C\), and \(C\succ A\). If a scalar \(r\) represented all three comparisons by strict inequalities, then \[ r(A)>r(B)>r(C)>r(A), \] which implies \(r(A)>r(A)\), a contradiction. Thus no real-valued score can represent this cycle by ordinary greater-than.

Scalar pairwise reward modeling is used in [christiano2017deep, Section 2]; the finite cycle marks one assumption needed for that reduction. No empirical cycle is attributed to a particular dataset or population, and the obstruction concerns scalar orderings rather than all preference models.

The ratio and policy-objective formulas in this section specialize the measurable policy interface of Definition [ftip-0023] to a declared finite or countable discrete action space. Thus \(\pi (a\mid h)\) denotes probability mass, not an unspecified density. Whenever \(\log \pi _\theta (A_t\mid H_t)\) is displayed, its sampled mass is assumed positive.

For decision times \(t=0,\ldots ,N-1\), write \(\mathsf H_t:=H_t\) for the public history defined in Definition [ftip-0021], \(A_t\) for its action, and \(R_{t+1}\in \mathbb R\) for the next reward. Let \(\gamma \in [0,1]\) be the discount factor and \(G_t\) the return from time \(t\).

Write \(\pi _b\) for the behavior policy that produced a stored action, \(\pi _\theta \) for the current trainable policy, and \(\pi _{\mathrm {ref}}\) for a fixed reference policy. Their value, action-value, and advantage functions are denoted \(V^\pi \), \(Q^\pi \), and \(A^\pi \); an estimated advantage is \(\widehat A_t\). The current-to-behavior likelihood ratio is denoted \(\rho _t(\theta )\).

In a response-level episode, \(\pi _\theta (y\mid x)\) abbreviates the completed-continuation probability \(P_\theta (y\mid s_{\mathrm {pr}}(x))\) of the response-task interface, using the declared prompt serializer; the same convention applies to \(\pi _b\) and \(\pi _{\mathrm {ref}}\). In an interactive episode these symbols instead denote the action laws on public histories defined in Definition [ftip-0023].

In the response-level specialization used for language-model RLHF, a prompt \(x\) is the initial context, a complete response \(y\sim \pi _\theta (\mathord \cdot \mid x)\) is the sampled action, and a scalar score supplies the terminal reward. The episode terminates after that response.

This contextual-bandit view is sufficient for sequence-level reward-model training. Token generation may still be exposed as multiple policy decisions when an optimizer assigns token-level likelihood ratios or advantages. A later agentic setting also permits environment observations and tool actions between model responses.

A scalar reward \(R_{t+1}\in \mathbb R\) is the numerical signal received after action \(A_t\) and before the next decision. Its probabilistic law may depend on the current history, action, and environment transition.

Reward specifies the training objective of the reinforcement-learning problem. It need not equal an independent evaluator's utility, and a terminal reward need not identify which earlier action caused the outcome.

For an episode ending at time \(N\), the return from decision time \(t\) is

When all intermediate rewards vanish, every return is determined by the terminal reward, up to discounting. Return is a random variable under the policy and environment law; its expectation defines value.

For a policy \(\pi \), the value function and action-value function are

whenever the conditional expectations exist. Both quantities depend on the reward, transition law, horizon, discount, and the policy followed after the conditioned decision.

For a policy \(\pi \), its advantage function is

The advantage compares an action with the policy's average continuation value at the same information state. Policy-gradient implementations replace it by an estimator \(\widehat A_t\); the estimator and the mathematical advantage are not interchangeable without assumptions on bias and variance.

Subtracting a baseline that does not depend on the sampled action can reduce the variance of a policy-gradient estimator without changing its expected gradient under the source conditions. A learned value function, a group mean, and a leave-one-out mean are different baselines and have different finite-data dependencies.

An estimated advantage may use sampled returns, bootstrapped values, or a mixture of both. Its error enters the update even when the likelihood-ratio calculation is exact.

The behavior policy \(\pi _b\) is the policy that generated a sampled action. The current policy \(\pi _\theta \) is the policy whose parameters are being optimized. Proximal policy optimization (PPO) collects a batch under an old policy and then compares candidate current policies with that data.

A reference policy \(\pi _{\mathrm {ref}}\) is instead held fixed to define a regularizer or preference objective; see [rafailov2023direct, Section 3, equation (3)]. It need not equal the behavior policy that generated a later rollout.

For an action \(A_t\) sampled from \(\pi _b\) at information state \(\mathsf H_t\), the current-to-behavior likelihood ratio is

The ratio is defined on sampled actions for which \(\pi _b(A_t\mid \mathsf H_t)>0\). A bounded or clipped sampled ratio does not by itself control policy probabilities at unvisited histories.

For a nonempty finite index set \(I\) and real values \((Z_i)_{i\in I}\in \mathbb R^I\), write \[ \widehat {\mathbb E}_{i\in I}[Z_i] =\frac 1{|I|}\sum _{i\in I}Z_i. \] When the subscript is only \(t\), the index set is the declared collection of sampled decision times in the current batch.

PPO uses empirical expectation notation for its sampled surrogates; see Section 2, equations (1)--(2) of [schulman2017proximal]. The batch and its weighting must still be declared; the hat does not assert unbiasedness.

For samples collected under the declared behavior law and estimated advantages \(\widehat A_t\), the empirical policy-gradient surrogate is the following sample average:

Its gradient is the familiar score-function estimator. The sampled states, actions, and advantage estimator are held fixed while differentiating this surrogate; changing the data-collection law defines a different estimator.

On a three-step episode, a discounted return and a baseline-relative advantage can be computed separately.

Take discount \(\gamma =1/2\). The return attached to the first action is \[ G_0=r_1+\gamma r_2+\gamma ^2 r_3 =0+\tfrac 12+\tfrac 14\cdot 2=1. \] If the declared baseline is \(V(s_0)=0.4\), then the corresponding advantage estimate is \(\widehat A_0=G_0-V(s_0)=0.6\).

The calculation instantiates the return and advantage in Definition [ftip-0030] and Definition [ftip-0032]. It illustrates one baseline choice. Low variance and correct attribution of terminal reward to an internal decision require additional assumptions or evidence.

A terminal response score can be copied into returns for many token decisions, but that bookkeeping does not reveal which token caused the score.

Monte Carlo returns, learned values, generalized advantage estimation, and group-relative normalization make different bias, variance, and dependence choices. Any theorem about an update must name the estimator actually used, the behavior policy that supplied its samples, and the horizon over which its credit signal is propagated.

An alignment-training claim is a capability claim \((M_0,A,E_{\rm align},b)\) in the sense of Convention [ftip-0003]. The procedure \(A\) changes model parameters and declares its training observations, loss or reward, and update rule. The independently fixed criterion \(E_{\rm align}\) states the behavior relative to which the word ``alignment'' is used.

Calling a procedure alignment training does not assert that its criterion is complete or that optimizing its training signal improves an independent evaluation.

Ouyang et al. combine demonstrations, comparison-trained rewards, and PPO in [ouyang2022training, Sections 3.1--3.5]. Direct Preference Optimization replaces the learned-reward PPO stage with a preference loss in [rafailov2023direct, Section 4]. The broader alignment-training description is a proposed way to compare such routes; it does not identify their observations or objectives with one another.

The declared behavioral criterion is part of the claim, while the independent evaluation remains a separate object. Their agreement is an empirical or theoretical claim, not a consequence of the word alignment.

The InstructGPT pipeline first performs supervised fine-tuning on labeler demonstrations. It then samples response pairs, collects labeler rankings, and fits a scalar reward model. Finally, it samples responses from the trainable policy and applies PPO using the reward-model score together with a penalty relative to the supervised policy.

The three stages consume different records and optimize different losses. Calling their composition RLHF does not make supervised targets, pairwise preferences, learned rewards, and policy-gradient samples the same kind of feedback.

The RLHF pipeline makes three logically separate choices. The comparison data specify which judgments were observed; the reward model specifies how those observations are represented and generalized; the policy optimizer specifies how sampled actions change the model.

An alignment criterion lies outside this chain unless it is identified with the training reward by assumption. A reward model can predict its held-out comparisons while failing under policy-induced distribution shift, and PPO can increase the learned reward while independent utility stays fixed or falls.

For probability mass functions \(p\) and \(q\) on a finite or countable set \(\mathcal Y\), the Kullback--Leibler divergence of \(p\) relative to \(q\) is \[ D_{\mathrm {KL}}(p\Vert q) =\sum _{y\in \mathcal Y}p(y)\log \frac {p(y)}{q(y)}. \] We use \(0\log (0/q)=0\). If \(p(y)>0\) and \(q(y)=0\) for some \(y\), the value is \(+\infty \). On a countable set, the negative part of the displayed series is finite; a divergent positive part gives the value \(+\infty \). Since \(\mathcal V^*\) is countable, the definition applies to the response laws of Notation [ftip-002A].

The order of the arguments matters. In particular, finiteness of the displayed forward divergence requires \(p\) to be absolutely continuous with respect to \(q\).

The KL-regularized reward objective appears as equation (3) in Section 3 of [rafailov2023direct]. Behavior-policy ratios, clipped ratios, and KL penalties enter alignment objectives for different purposes.

A coefficient multiplying \(D_{\mathrm {KL}}\) declares an optimization tradeoff. It is neither a guarantee that every sampled ratio is small nor an independent evaluation of the resulting policy.

Let \(\mu \) be a prompt law, \(r(x,y)\) a scalar reward, and \(\pi _{\mathrm {ref}}\) a fixed reference policy. For \(\beta >0\), the KL-regularized RLHF objective, using forward KL, is

The prompt law, reward, reference policy, and coefficient are part of the objective. The forward KL requires the optimized policy to be absolutely continuous with respect to the reference wherever the objective is finite.

For a prompt \(x\), sampled response \(y\), learned reward score \(r_\phi (x,y)\), and fixed reference policy \(\pi _{\mathrm {ref}}\), the sequence-level KL-shaped reward is

Autoregressive factorization writes the logarithmic term as a sum of token-level log ratios. Ouyang et al. also mix a pretraining-gradient term into their reported PPO objective; that auxiliary term is separate from the shaped reward displayed here.

Given an approximate value function \(V\), with \(V(\mathsf H_N)=0\) at a terminal information state, define the temporal-difference residual

For \(\lambda \in [0,1]\), the finite-episode generalized advantage estimator is

The parameters \(\gamma \) and \(\lambda \) trade temporal reach against the variance and approximation error induced by bootstrapping. The estimator also depends on the fitted value function.

For a clipping parameter \(\epsilon _{\mathrm {clip}}\in (0,1)\), define \(\operatorname {clip}_{\epsilon _{\mathrm {clip}}}(u) =\min (1+\epsilon _{\mathrm {clip}},\max (1-\epsilon _{\mathrm {clip}},u))\). The PPO clipped surrogate is

Clipping changes the sampled surrogate when the ratio moves outside the declared interval in a direction favored by the estimated advantage. It is not a hard bound on the KL divergence of the complete updated policy.

PPO implementations may optimize the combined sampled objective

Here \(L_t^{\mathrm {CLIP}}\) is the per-sample integrand of the clipped surrogate in Definition [ftip-003F], \(c_1,c_2\geq 0\), \(\widehat G_t\) is the declared value target, and, on the declared finite or countable discrete action set,

The combined objective is used only when this nonnegative countable sum is finite, with \(0\log 0=0\). For a continuous action law, an implementation must instead declare a reference measure and the corresponding density-based entropy convention.

The value loss and entropy bonus change the shared-parameter update in addition to the clipped policy term. Their coefficients, target construction, and action space are part of the optimization method.

A PPO iteration collects a batch of trajectories under the behavior policy, computes returns or advantage estimates, and then performs several epochs of minibatch optimization on the same collected batch. The updated policy becomes the behavior policy for a later collection round.

The number of trajectories, epochs, minibatches, and optimizer steps governs how often one batch is reused. The clipped objective does not determine those protocol choices by itself.

Positive and negative advantages select different branches of the PPO clipped minimum.

With \(\epsilon =0.2\), write \(\bar r=\operatorname {clip}(r,0.8,1.2)\) and \(L(r,A)=\min (rA,\bar rA)\). For positive advantage and \(r=1.4\), the gain is capped at \(2.4\). For negative advantage and \(r=0.6\), the clipped term \(-1.6\) is smaller than \(-1.2\), so the objective retains the penalty.

The pointwise surrogate comes from [schulman2017proximal, Section 3]. The two sign cases check only its arithmetic; they provide no monotone-improvement result for a neural policy or finite minibatch.

One prompt passes successively through supervised tuning, comparison-based reward modeling, and PPO in the InstructGPT-style RLHF pipeline.

For the displayed illustrative prompt, suppose the comparison-trained reward model scores the preferred response by \(0.8\). If the candidate policy assigns it twice the probability assigned by the reference policy and the declared KL coefficient is \(\beta =0.1\), its shaped scalar is \[ 0.8-\beta \log \frac {\pi (y_B\mid x)} {\pi _{\rm ref}(y_B\mid x)} =0.8-0.1\log 2\approx 0.731. \]

The stage ordering is grounded in [ouyang2022training, Sections 3.1--3.5]; the numbers are a schematic calculation rather than reported InstructGPT hyperparameters. The diagram records the interface used in that pipeline. Other RLHF systems may order or replace the stages differently.

RLHF is a feedback-and-training pipeline; PPO is one optimizer used within that pipeline. PPO specifies a sampled update surrogate; it does not specify whose preferences are collected, what the reward means, or which independent behavior counts as aligned.

Changing the comparison population or reward model can change the alignment target while leaving PPO unchanged. Conversely, replacing PPO by another optimizer changes update geometry without necessarily changing the declared preference data or evaluation criterion.

Let \(\beta >0\) be the KL coefficient, \(\pi _{\mathrm {ref}}\) a fixed reference policy, and \(\pi _\theta \) a trainable policy with support contained in that of the reference on the responses under study. Define the reference-relative log likelihood

and, for a comparison observation, the paired difference

Direct preference training starts from a fixed reference policy and a dataset of preferred and dispreferred responses. It evaluates a binary cross-entropy loss formed from the trainable policy's reference-relative log likelihoods and updates the policy directly.

This stage does not separately fit a scalar reward network, sample online rollouts for PPO, or train a value function. Its loss nevertheless comes from a particular reward-model and KL-regularized optimization derivation.

For a reward \(r(x,y)\), reference policy \(\pi _{\mathrm {ref}}\), and \(\beta >0\), the optimizer of the per-prompt KL-regularized reward objective has the form

where

normalizes the policy on the response space. The identity assumes that the normalizer is finite and uses the same reference policy as the underlying objective.

Rearranging the optimal-policy identity of Definition [ftip-003O] gives

The final term depends on the prompt but not on the response. It cancels from pairwise reward differences, matching the additive non-identifiability of Bradley--Terry scores described in Remark [ftip-002S].

Substituting the reward--policy reparameterization into the Bradley--Terry law yields

where \(\Delta _\theta \) is defined in Notation [ftip-003M]. The prompt-dependent normalizer cancels because the two responses share the same prompt.

For comparison observations from \(\mathcal D_{\mathrm {pref}}\), the Direct Preference Optimization loss is

The empirical loss is evaluated on a fixed comparison dataset. The temperature, reference policy, record weights, and handling of ties or abstentions are part of the training specification.

Four declared policy probabilities determine the log-ratio margin in one DPO loss term.

Let \(\pi _\theta (y^+\mid x)=0.6\), \(\pi _{\rm ref}(y^+\mid x)=0.3\), \(\pi _\theta (y^-\mid x)=0.2\), and \(\pi _{\rm ref}(y^-\mid x)=0.4\). With \(\beta =1/2\), the DPO logit is \[ z=\beta \left [\log 2-\log (1/2)\right ] =\tfrac 12\log 4=\log 2. \] Hence \(\sigma (z)=2/3\) and the one-pair negative log-likelihood is \(-\log \sigma (z)=\log (3/2)\approx 0.405\).

Substitution into Equation (7) of [rafailov2023direct, Section 4] verifies one loss term. Consistency of the pairwise data, suitability of the reference policy, and improvement in independent utility are separate questions.

The derivation uses a fixed reference policy, a finite KL-regularized optimum, and a Bradley--Terry model for pairwise preferences. The relevant policy probabilities must be positive wherever their logarithmic ratios are evaluated.

The algebra eliminates the prompt-dependent reward offset, not every source of reward-model misspecification. Heterogeneous or nontransitive preferences, adaptive data collection, support mismatch, and finite-sample optimization remain separate questions.

DPO optimizes a policy directly on a fixed preference dataset. Reward-model PPO instead fits an explicit scalar reward predictor, generates policy rollouts, estimates advantages, and applies a policy-gradient update. DPO therefore removes the separately represented reward and online PPO loop from that training stage.

The methods still share ingredients: comparison data, a reference policy, a KL coefficient or temperature, and assumptions connecting comparisons with latent reward. Neither method defines preference optimization in general, and their data and compute requirements are not causally interchangeable.

An outcome-supervised reward model assigns a score to a complete response and is trained from labels attached to the final outcome. For a prompt \(x\) and completed response \(y\), write its score as

The training label states whether the completed solution reaches the declared outcome. It need not identify the first invalid step or distinguish a sound derivation from an answer reached for an unsound reason.

Suppose a response is segmented into reasoning steps \(y=(s_1,\ldots ,s_m)\). A process-supervised reward model produces a score for each declared prefix,

and is fitted from step-level labels. The segmentation rule and label semantics are part of the supervision: a score after step \(k\) need not be a decomposition of an outcome score for the whole response.

Outcome supervision uses one completed-response label, whereas process supervision supplies labels at declared intermediate steps. The latter can localize feedback, but it also requires a segmentation, a step-labeling criterion, and additional annotations.

Neither label type is automatically causal credit. A process label may be assigned after a reviewer has seen the whole response, and a terminal outcome may be predicted reliably from an early prefix without identifying which action should change.

Let \(\mathcal F_t\) be the sigma-algebra generated by the public history \(\mathsf H_t\) available through decision time \(t\). A feedback variable \(Z_t\) is online measurable at time \(t\) when it is \(\mathcal F_t\)-measurable and is delivered before the protocol chooses its next action.

A label computed only after observing a later action or terminal outcome is not online measurable at the earlier time, even if it is subsequently attached to that earlier prefix in the training data.

The index attached to a label does not determine when its information becomes available. The process-supervision construction in [lightman2023verify, Section 2.6] supplies step-indexed training labels, but it does not require every such label to be available while the response is being generated.

This distinction determines which interventions are admissible. An online agent may act on currently measurable feedback; an offline learner may use a post-hoc label for a past prefix; a causal claim needs assumptions connecting either signal to the consequences of the earlier action.

A four-step arithmetic trace can receive a correct terminal label while retaining an incorrect intermediate process label.

The final answer \(19\) is correct, so outcome supervision supplies \(f_{\rm out}=1\). The declared process labels are \((f_1,f_2,f_3,f_4)=(1,1,0,1)\) and expose the corrected arithmetic error at step three. The labels are attached to the displayed prefixes, which remain unchanged by those annotations.

The two supervision regimes follow the outcome- and process-supervised reward-model constructions in [lightman2023verify, the outcome- and process-supervised reward-model subsections]. The two observations are thereby separated. Noise-free labels and a general policy advantage for process supervision are not consequences of the finite trace.

Let \(\mathcal E\) be an evidence space. Write \(\mathsf V\) for a declared verifier, \(e_{\mathrm {ver}}\in \mathcal E\) for its evidence, \(d_{\mathrm {ver}}\in \{0,1\}\) for its decision, and \(R_{\mathrm {ver}}\in \mathbb R\) for the reward derived from that result.

For a prompt \(x\) and group size \(g\geq 2\), write \(\mathbf Y_x=(Y^{(1)},\ldots ,Y^{(g)})\) for a rollout group sampled from the behavior policy. Let \(R_i\) be the reward of member \(i\), \(\bar R\) the group mean, \(S_R\) the group standard deviation, and \(\widehat A_i\) its group-relative advantage.

Reinforcement learning with verifiable rewards is a post-training method that samples responses from a language-model policy, computes rewards with a deterministic check of the response, and applies a reinforcement-learning update to increase expected checked reward, optionally with regularization to a reference policy.

In the cited construction, a correct response receives reward \(\alpha \) and an incorrect response receives \(0\). Equation (8) gives this two-valued form, and the paragraph immediately after it sets \(\alpha =10\). The checked predicate, reward scale, prompt law, rollout procedure, and policy optimizer remain separate parts of the method.

Concrete RLVR methods include the PPO-based construction in [lambert2024tulu3, Section 6], group relative policy optimization (GRPO) in [shao2024deepseekmath, Section 4.1], R1-Zero in [guo2025deepseek, Section 2.2], and Decoupled Clip and Dynamic sAmpling Policy Optimization (DAPO) in [yu2025dapo, Sections 2--3].

A proposed common description groups methods that acquire policy samples, evaluate a declared verifiable signal, assign credit, and update the policy. A specific claim must still name its verifier, group sampling, estimator, clipping, normalization, reference-policy treatment, and length rule.

For prompts and responses from Notation [ftip-002A], a deterministic verifier correctness indicator is a declared map

Here \(v(x,y)=1\) means that the response satisfies the checked predicate. The map includes the parsing and execution rules needed to reproduce the decision. A stochastic judge or learned reward model is a different feedback mechanism unless its randomness and decision procedure are separately exposed.

A verifier evidence record is a tuple

where \(\iota _{\mathrm {ver}}\) identifies the verifier and its version, \(o\) is the reproducible output used by the check, and \(d_{\mathrm {ver}}\in \{0,1\}\) is the resulting decision. Examples of \(o\) include a parsed exact answer, a test log, or a proof-checker result.

The evidence record preserves how the decision was obtained. It does not assert that the checked predicate is complete for the task's independent success criterion.

The same correctness decision can arise from different parsers, tests, proof kernels, or failure modes. Tülu 3 uses a scaled two-valued reward \(\alpha \) or \(0\), with \(\alpha =10\), in [lambert2024tulu3, Section 6, equation (8), and the immediately following paragraph]; it does not identify that reward with a complete record of the checking procedure.

Evidence enables later audits of false positives, false negatives, version changes, and shortcut exploitation. Reproducibility of the check still does not establish that the checked predicate equals the intended capability.

Given a deterministic correctness indicator \(v\) and reward scale \(\alpha >0\), the verifiable reward in the cited construction is

The paragraph after Equation (8) sets \(\alpha =10\). Another method may choose a different scale or combine several checked components, but that transformation is part of its reward design. A reward is verifiable relative to the implemented check, not relative to every property an independent evaluator may care about.

Rule evidence compares a parsed answer or format with a declared target, as in [guo2025deepseek, Section 2.2]. Test evidence executes a candidate in a sandbox and records test outcomes. Proof evidence submits a term to a specified checker and records acceptance or an error.

These routes expose different predicates and costs. A rule can ignore the derivation, a test suite can omit behavior, and a proof checker establishes only the proposition and trusted-kernel boundary it was given. Their rewards must not be merged without retaining which evidence source fired.

Fix a prompt \(x\), group size \(g\geq 2\), and behavior policy \(\pi _b\). A rollout group is the indexed family

DeepSeekMath samples several outputs for the same prompt so their rewards can define a group-relative baseline. The producing policy, decoder, group size, and conditional sampling dependence are part of the rollout law.

For rollout-group rewards \(R_1,\ldots ,R_g\), define the group mean and group standard deviation by

Both statistics are prompt-local and depend on every member of the sampled group. The source leaves the standard-deviation divisor symbolic and does not add a numerical stabilizer. An implementation must state those conventions and its treatment of zero-variance groups.

When \(S_R>0\), the group-relative advantage of rollout member \(i\) is

The estimator compares responses sampled for the same prompt and does not require a learned value function. Its members are statistically coupled through \(\bar R\) and \(S_R\); it is not an independent estimate of the population advantage of each response.

Outcome-supervision GRPO samples a group of responses for one prompt, assigns each response an outcome reward, standardizes those rewards within the group, and uses the resulting group-relative advantage across the response's token log-likelihood terms.

The construction removes a separately learned critic but retains a behavior policy, likelihood ratios, a reference-policy term in the source formulation, and explicit normalization choices.

Process-supervision GRPO supplies rewards at declared reasoning steps and forms advantages from the subsequent step rewards assigned to each token. The policy objective therefore receives credit that can vary within one response rather than repeating one terminal outcome advantage at every token.

This construction requires a step segmentation and a process reward model. Its estimator is not obtained merely by relabeling an outcome-supervision rollout group.

GRPO names a family of group-relative policy updates rather than one fully determined estimator. Outcome and process supervision assign different reward structures. Implementations may also differ in token versus response normalization, KL terms, clipping intervals, importance weights, and treatment of groups with zero reward variance.

A result about one variant must retain its producing policy and all of those conventions. Sharing the group-relative baseline does not make the resulting gradients identical.

Clip-higher replaces the symmetric upper and lower PPO clipping widths by separate constants. It permits a larger upward likelihood-ratio movement for tokens with positive advantage while retaining a smaller lower-side interval.

The change alters which sampled terms are saturated by the surrogate. It does not impose a policy-wide trust region or guarantee preservation of low-probability actions.

DAPO dynamic sampling filters prompt groups whose sampled responses all receive the same reward and continues sampling until the training batch contains the declared number of groups with nonzero reward variation.

The intervention changes the distribution of prompts and rollouts entering the update. It can prevent zero-advantage groups from consuming an optimizer batch, but the retained batch is no longer an unconditioned sample from the original prompt law.

The token-level loss aggregates active token terms across the complete minibatch and divides by the number of active tokens. A longer response therefore contributes in proportion to its active token count rather than receiving the same total weight as every shorter response.

This normalization changes the empirical gradient even when the sampled responses, rewards, advantages, and likelihood ratios are unchanged. It must be distinguished from averaging a separately normalized loss over responses.

Overlong reward shaping introduces a soft penalty near the maximum response length before the hard truncation boundary. The penalty increases over a declared buffer region rather than assigning the same abrupt terminal penalty to every response that reaches the limit.

The construction changes both the reward value and which length-related behavior receives gradient. The maximum length, buffer width, and penalty schedule are part of the reward design.

Clip-higher changes the saturated region of the policy surrogate, dynamic sampling conditions which prompt groups enter a batch, token-level aggregation changes response-length weighting, and overlong shaping changes the reward near a truncation boundary. These are four different interventions.

An observed training improvement cannot be assigned to a generic ``algorithm'' without an ablation that holds the other sampling, estimator, reward, and compute choices fixed. The interventions also change which rollouts and gradients are observed, so their effects need not add linearly.

Three sampled responses pass through exact verification, group credit, and a proposed update in one finite RLVR round.

The exact-answer checker gives mean reward \(\bar r=2/3\). Using the declared unnormalized centered credit \(\widehat A_i=r_i-\bar r\) yields \((1/3,-2/3,1/3)\). An optimizer may use these numbers to propose a parameter change, but the tuple itself is not yet an accepted update.

The RLVR protocol is Definition [ftip-0044]. Its group-credit objects are Definition [ftip-004B], Definition [ftip-004C], and Definition [ftip-004D]. Together they specialize to this finite order of operations. Centered, unscaled credit and an exact-answer checker are local choices rather than requirements on other RLVR algorithms.

A rollout group is normalized to relative advantages and then passed to a clipped surrogate objective.

For a rollout group with rewards \(r_i\), form a group mean and scale and set \(\widehat A_i=(r_i-\bar r)/s\). For current-to-behaviour ratios \(\rho _i\) and clip interval \([1-\epsilon ,1+\epsilon ]\), define each pointwise term by

The mean of the resulting terms is the group contribution to the update; its value depends on the sampled rewards, ratios, and clip width.

Group-relative normalization is grounded in [shao2024deepseekmath, Section 4.1], while clipping is grounded in [schulman2017proximal, Section 3]. The normalization convention and ratios must be declared by the training protocol; a positive surrogate mean alone gives no guarantee of improved held-out performance after an optimizer step.

Fix the token vocabulary and strings of the tokenizer interface, and the action space \(\mathcal A\) from the declared interactive task realization. An action serializer is a pair of partial maps

The encoder gives an action a textual representation. The decoder recognizes a completed representation and returns the executable action. A string outside the decoder's domain is malformed; it is not silently promoted to an environment action. For every action in the encoder's domain, the declared pair satisfies \(\operatorname {dec}(\operatorname {enc}(a))=a\).

An action serializer maps generated text to an executable action. Section 2 of ReAct: Synergizing reasoning and acting in language models[yao2023react] writes actions and observations in a textual trajectory. The proposed serializer interface in Definition [ftip-004P] additionally distinguishes malformed calls, constrained grammars, and environment actions.

An improvement can therefore come from better reasoning, better serialization, or a more permissive parser. These are separate intervention coordinates.

A ReAct interaction alternates model-generated reasoning and actions with observations returned by an external interface. The observation is appended to the next model context, so subsequent actions may depend on earlier tool results.

A decoded action belongs to \(\mathcal A\) in the declared interactive task realization; the transition--observation kernel of Definition [ftip-001Z] returns the next observation. The cited construction does not by itself fix a sandbox, reward, or training algorithm.

Fix an interactive realization \(\mathsf {Env}_{\mathsf T}\) from the declared task interface, a measurable evidence space \(\mathcal E_{\rm box}\), and a cost dimension \(m\geq 1\). A sandbox specifies an allowed action set \(\mathcal A^{\rm box}_{\mathsf T}\subseteq \mathcal A_{\mathsf T}\) that is measurable, a cost bound \(b_{\rm box}\in \mathbb R_+^m\), and, for each \(x\in \mathcal X_{\mathsf T}\) and \(0\leq t<T_{\max }\), an execution kernel

It also specifies, for each \(x\in \mathcal X_{\mathsf T}\), a reset kernel

For an allowed action, the state--observation marginal of \(K_t^{\rm box,x}\) is the kernel \(K_t^x\) in the task realization; the state--observation marginal of the reset kernel is \(\rho _0^x\). The remaining coordinates record reproducible evidence and a realized cost increment. An admitted stopped trajectory must have componentwise cumulative cost at most \(b_{\rm box}\).

The sandbox semantics include permissions, kernel and reset versions, evidence encoding, and cost accounting. A tool name or API schema alone does not determine these fields.

A sandbox restricts the environment through its executor, reset rules, permissions, and evidence. ReAct [yao2023react, Section 2] supplies the action--observation pattern. Software-agent systems surveyed in [wang2024agents, Section 2.3.2] add repositories, compilers, tests, and other digital tools. The combined sandbox interface is a proposed model of these choices.

Two protocols that share prompts but change reset or permission semantics need not induce the same trajectory law.

Let \(\mathcal M_{\rm exec}\) be the space of executable model artifacts, \(\mathcal Z_{\rm opt}\) the optimizer-state space, and \(\mathcal R_{\rm ag}\) a declared space of typed agentic round records. At training round \(n\), an agentic post-training state is a tuple \[ \Sigma _n=(M_n,s_n,\mathcal L_n) \in \mathcal M_{\rm exec}\times \mathcal Z_{\rm opt} \times \mathcal R_{\rm ag}^{n}, \] where \(M_n\) is the executable model artifact, \(s_n\) is the optimizer state, and \(\mathcal L_n=(r_0,\ldots ,r_{n-1})\) is an append-only training ledger.

The round-record schema records the task and instance draw, policy and inference stamp, environment version, stopped trajectory, feedback and credit outputs, update decision, proposed and committed next artifact and optimizer state, random seeds, operator versions, and realized cost vector. The tuple separates executable state from optimizer state and from evidence about how that state was reached.

At round \(n\), fix a task \(\mathsf T_n\), its law \(\mu _{\mathsf T_n}\), and a versioned interactive realization \(\mathsf {Env}^{\upsilon _n}_{\mathsf T_n}\). Let \(\mathcal I_{\rm art}\) and \(\mathcal I_{\rm inf}\) be declared identifier spaces. Let \(a_n\in \mathcal I_{\rm art}\) be an immutable content identifier resolving the artifact \(M_n\), and let \(\iota _n\in \mathcal I_{\rm inf}\) record all serializer, decoding, tool, and stopping settings. The policy--inference stamp \(\nu _n=(a_n,\iota _n)\in \mathcal I_{\rm art}\times \mathcal I_{\rm inf}\) resolves a non-anticipating policy \(\pi _n\). Let \(\Omega _n^{\rm task}\), \(\Omega _n^{\rm roll}\), \(\Omega _n^{\rm feed}\), \(\Omega _n^{\rm credit}\), and \(\Omega _n^{\rm update}\) be declared seed spaces. The rollout seed contains both policy-decoding and environment randomness. Write \(\Omega _n=\Omega _n^{\rm task}\times \Omega _n^{\rm roll}\times \Omega _n^{\rm feed}\times \Omega _n^{\rm credit}\times \Omega _n^{\rm update}\), and let \(\omega _n=(\omega _n^{\rm task},\omega _n^{\rm roll}, \omega _n^{\rm feed},\omega _n^{\rm credit},\omega _n^{\rm update})\) be sampled from a declared joint law on \(\Omega _n\). Write \(\mathcal Z_{\mathsf T_n}\) for the stopped trajectories admitted by the versioned realization.

For declared feedback and credit spaces \(\mathcal F_n\) and \(\mathcal G_n\), decision space \(\mathcal D_n=\{\mathsf {accept},\mathsf {reject}\}\), accounting dimension \(m\geq 1\), and exact implementation versions \(v_n^{\rm S}\), \(v_n^{\rm R}\), \(v_n^{\rm F}\), \(v_n^{\rm C}\), \(v_n^{\rm U}\), and \(v_n^{\rm A}\), the round specifies maps

The accounting map receives the starting state, proposed state, decision, and realized seeds, so update work and rejected proposals remain observable in the cost record.

They produce

The task sampler pushes its seed law forward to \(\mu _{\mathsf T_n}\). Conditional on \(X_n\), the rollout sampler pushes its seed law forward to the interaction law of Remark [ftip-0028] under \(\mathsf {Env}^{\upsilon _n}_{\mathsf T_n}\) and \(\pi _n\).

If \(d_n=\mathsf {accept}\), set \((M_{n+1},s_{n+1})=(\widetilde M_{n+1},\widetilde s_{n+1})\); if \(d_n=\mathsf {reject}\), set \((M_{n+1},s_{n+1})=(M_n,s_n)\). The realized cost \(c_n\) includes proposal work even after rejection. With \(v_n^{\rm ops}=(v_n^{\rm S},v_n^{\rm R},v_n^{\rm F},v_n^{\rm C}, v_n^{\rm U},v_n^{\rm A})\), append

The record belongs to \(\mathcal R_{\rm ag}\) from Definition [ftip-004U] and determines \(\Sigma _{n+1}\) without rewriting an earlier record.

An agentic post-training run fixes a finite round count \(N_{\rm ag}\geq 0\), an initial state \(\Sigma _0\), a fixed accounting dimension \(m\geq 1\), a componentwise budget \(b_{\rm ag}\in \mathbb R_+^m\), and the round specifications of Definition [ftip-0073]. Put \(\Omega _{\rm ag}=\prod _{n=0}^{N_{\rm ag}-1}\Omega _n\), using a singleton for the empty product. Once those specifications and their exact versions \(v_{\rm ag}\) are fixed, the run is the map

Write a realized round-seed tuple as

The run recursively constructs \(\Sigma _1,\ldots ,\Sigma _{N_{\rm ag}}\); the seed tuple and state sequence are empty beyond \(\Sigma _0\) when \(N_{\rm ag}=0\). Its cumulative realized cost is \[ c_{\rm ag}=\sum _{n=0}^{N_{\rm ag}-1}c_n. \] The empty sum is \(0\in \mathbb R_+^m\) when \(N_{\rm ag}=0\). The map returns \((M_{N_{\rm ag}},\mathcal L_{N_{\rm ag}},c_{\rm ag})\). A run claiming the hard budget must declare a pre-admission or stopping rule that guarantees \(c_{\rm ag}\preceq b_{\rm ag}\).

The run declaration includes the task sampler, environments, serializers, rollout inference settings, feedback and credit maps, update maps, stopping rule, seed laws, implementation versions, and resource account. The round records retain every realized seed, version, decision, and cost. Leaving one of these fields implicit defines a family of runs rather than one reproducible intervention.

A proposed interactive extension of the response-level RLVR cycle in § [ftip-0042] uses the environment of § [ftip-001W] and the action boundary of Definition [ftip-004P]. It retains intermediate observations and environment state instead of treating a whole response as one indivisible action.

This change introduces new questions about partial observability, long-horizon credit, environment versioning, recovery after interruption, and the cost of external execution. It does not assert that multi-turn training is uniformly better than response-level training.

A failing test observation triggers a second edit, whose passing tool evidence determines the terminal result.

The public history records both edits and both test outputs. The first tool call returns the evidence string expected 9, got 6; the second returns exit status \(0\) and 1 passed. The success rule reads the second test result rather than the model's assertion that its edit is correct.

The action--observation alternation follows the tool-interleaving pattern in [yao2023react, Section 2]. The transparent sandbox interaction certifies only the displayed test result; completeness of the suite and satisfaction of unstated intent remain unverified.

Two coupled panels place resumable generation beside sandbox evidence in a single rollout record.

A write-ahead log and cached state allow a resumable rollout; a sandbox execution contributes a separately recorded evidence tuple. The two records should remain distinguishable when lifecycle cost is audited.

Token-granular write-ahead logging with key--value cache recovery and DeepSeek Elastic Compute (DSec), the report's named sandbox interface, are described in Sections 5.2.3 and 5.2.5 of the DeepSeek-V4 family report [⧉]. The cited sources do not establish that DeepSeek-V4-Flash-0731 used this job, resource limit, rollout schedule, or sandbox configuration.

A policy stamp \(\nu \) is a content-addressed description of an executable policy: model artifact, tokenizer, action serializer, decoding settings, and implementation revision. Resolving the stamp produces the action law \(\pi ^\nu (\,\cdot \mid \mathsf H_t)\) for every public history in its declared domain.

Two stamps are equal only when every component that may change the action law is equal. A display name or checkpoint step is therefore insufficient.

Off-policy methods distinguish the behavior and current policies, while language-model artifacts also depend on tokenization and decoding. Section 3 of When to stop reusing: Dynamic gradient gating for sample-efficient RLVR[miao2026when] records the behavior/current distinction. The additional artifact and implementation fields in the proposed policy stamp make the executable law reproducible.

The stamp identifies an executable law; it does not claim that two implementations with different stamps must behave differently on every task.

An immutable rollout record is a tuple

where \(q\) is the sampled task, \(\nu _b\) is the behavior-policy stamp, \(\tau \) is the stopped trajectory of Definition [ftip-0025], and \(e\) is the verifier evidence of Definition [ftip-0047]. The tuple records facts fixed when the rollout finishes.

Current-policy likelihood ratios, clipping decisions, reuse counts, and acceptance decisions are excluded because they may change on a later update attempt.

An immutable historical record prevents an update attempt from rewriting the facts attached to its reused samples. The sample-reuse setup in [miao2026when, Section 3] retains behavior-policy rollouts while recomputing current-policy ratios. Dynamic Gradient Gating (DGG), defined in [miao2026when, Section 5 and Algorithm 1], is one such reuse protocol, while the complete immutable tuple is a proposed interface for such protocols.

Recorded facts and estimates recomputed for an attempted update are different objects; errors in the latter do not alter the former.

Let \(\mathcal R\) be the set of immutable rollout records and \(\mathcal F\) a typed set of feedback events. A feedback operator is a possibly randomized kernel

A feedback event may contain an outcome score, step labels, a preference, or an abstention. Its type records which observations were available when it was produced.

Let \(R\) be a rollout record with \(\tau _{\rm stop}\) actions, and let \(f\) be its feedback event. A credit-assignment rule returns indexed training targets

Each \(c_t\) is attached to an action or token position declared by the rule. It may be a return, an advantage estimate, a binary label, or a structured target. The rule is distinct from the feedback source that supplied \(f\).

Terminal returns, process labels, preference observations, and verifier rewards determine what information is observed. A credit rule determines where that information enters an objective. Treating them as separate inputs is a proposed common description of these different methods.

Separating the interfaces permits questions about delayed, noisy, or misallocated credit without changing the underlying evidence.

At update index \(n\), a replay pool \(\mathcal R_n\) is a finite multiset of immutable rollout records eligible for selection. Its management rule specifies insertion, eviction, and any partition by task or behavior stamp.

The pool is persistent state of the training protocol. It is not the minibatch selected for one update.

Selecting \(R\in \mathcal R_n\) under a current policy stamp \(\nu _n\) produces an attempt-local replay annotation

where \(\rho \) is the declared collection of behavior/current likelihood ratios, \(\kappa \) records clipping status, \(m\) is the record age, and \(u\) is its prior reuse count. Every component is computed for this attempt and may change at the next one.

The proposed replay interface separates a persistent pool from annotations computed for an update attempt. DGG in [miao2026when, Section 3; Section 5 and Algorithm 1] recomputes policy ratios and a gradient diagnostic while reusing a rollout batch. Those quantities depend on the current policy and therefore belong to the attempted update rather than the historical record.

The split also exposes the cost of selecting, scoring, and rejecting reused records.

Reusing one historical rollout recomputes current-policy annotations without mutating the stored record.

Let the stored record be \(R=(q,\nu _b,\tau ,e)\) and suppose its behaviour policy assigned the recorded action probability \(0.25\). Under the current stamp the same action has probability \(0.50\), so this attempt records importance ratio \(0.50/0.25=2\). Its age \(3\) and prior-use count \(2\) are also attempt-time metadata; none of these three values rewrites \(R\).

The behaviour/current-policy distinction is required by the reuse analysis in [miao2026when, Section 3]. The finite annotation makes provenance explicit. Statistical safety of ratio \(2\), age \(3\), or a second reuse is left undecided.

Given a current parameter--optimizer state \((\theta _n,z_n)\), selected records, their attempt annotations, and credit targets, an update proposal is a candidate next state

The diagnostic record \(\delta _n^{\rm diag}\) contains the quantities required by the declared acceptance rule. It is distinct from the accept--reject decision \(d_n\) recorded for the proposal. Constructing the proposal incurs update cost even if the proposal is later rejected.

Dynamic gradient gating computes the language-model-head gradient before the optimizer step. It compares the change in squared Frobenius norm with a trailing-window Z-score. When the score crosses its threshold after a reused update, the method discards that gradient and returns to fresh rollouts.

The gate is an empirical stopping rule for reuse. The source does not prove that its Z-score is a calibrated divergence test or a general safety certificate.

A gate may reject a computed gradient even though its computation has already consumed resources. The DGG definition in Definition [ftip-005A] supplies the motivating pre-optimizer discard operation. Other protocols may inspect held-out loss, divergence, or resource limits.

The generic proposal interface does not transfer DGG's empirical detector to those other diagnostics. Each gate needs its own assumptions and calibration evidence.

Two candidate reused-gradient updates fall on opposite sides of a declared DGG-style gate.

Declare a trailing-window increment mean \(\mu =1\), standard deviation \(s=0.5\), and toy threshold \(z_\star =2\). Proposal A has increment \(1.5\), so \(Z_A=(1.5-1)/0.5=1\) and is committed. Proposal B has increment \(2.5\), so \(Z_B=3\) and is rejected before an Adam transition. Rejection therefore leaves both parameters and optimizer moments unchanged in this protocol.

The monitored last-layer gradient energy, trailing-window Z-score, and pre-optimizer rejection order are modeled on [miao2026when, Section 5 and Algorithm 1]; the numeric threshold and window statistics above are locally declared. Acceptance is a reuse heuristic; it supplies neither a safety certificate nor a guarantee of improved independent evaluation.

NExt forms global and local parameter differences from saved low-rank adaptation (LoRA) checkpoints, approximates each matrix difference by leading singular factors, and trains a predictor on those representations to estimate a future difference. It adds a scaled predicted difference to a checkpoint; the reported experiments then resume RLVR updates.

The construction replaces some realized rollout-and-update steps with a learned parameter jump. It does not introduce a new verifier signal.

The NExt construction Low-rank optimization trajectories modeling for LLM RLVR acceleration[chen2026lowrank] motivates a distinction between following a predictable parameter path and acquiring information from new rollouts. Low-rank dominance of saved differences does not imply linear future motion, invariance across parameterizations, or preservation of optimizer state.

A cost comparison includes checkpoint storage, decomposition, predictor training, extrapolation, and recovery updates. Bounding independent-evaluation regret additionally requires stability assumptions connecting parameter forecasts to the evaluated outcome.

Leading rank-one projectors encode a checkpoint direction without inheriting the arbitrary signs of singular vectors.

For either singular-vector representation \((u,v)\) or \((-u,-v)\), the projectors \(P_u=uu^\top \) and \(P_v=vv^\top \) are unchanged. A trajectory forecaster can extrapolate the observed projector-aligned deltas to a next checkpoint, after which a measured recovery update can correct the forecast without changing the represented one-dimensional subspace.

The history-to-delta-to-recovery order is a toy illustration of NExt, described in [⧉]. NExt models signed singular value decomposition (SVD) factors rather than the projector construction used here. Accordingly, the calculation supplies neither a theorem about checkpoint trajectories nor evidence of capability acquisition.

The executable model artifact, active context, explicitly managed computation, and persistent harness state are distinct parts of an agent. Replayability depends on the rule that materializes state from events, as well as the recorded inputs and execution versions.

Example. One artifact and two harness policies [ftip-00CB]

AGENTDRAFTED

The same executable artifact can be paired with two persistent states and therefore two resolved harness policies. The harness state changes while the executable model remains fixed.

Different policies here establish only that harness state is an intervention coordinate. An independent evaluation is still required to compare their outcomes.

A long-horizon score is attached to a configuration, not merely to a model name. Delegation also creates descendant work that must remain visible in the resource account. Reproducible evaluation therefore requires a specified configuration, and resource accounting includes the work of its descendants.

Example. A finite delegated run and its lineage cost [ftip-00CL]

AGENTDRAFTED

Use two cost coordinates, such as output tokens and tool calls. The root delegates to two children, and one child delegates once more.

The cost of a child subtree is its local cost plus the costs of all descendants. Therefore

\[ C_{\rm lin}(\mathcal T)=c_r+C_{\rm lin}(\mathcal T_a)+C_{\rm lin}(\mathcal T_b). \]

Charging only the root hides descendant work; charging a descendant again as a separate parent-local cost double counts it.

A research harness cannot place its complete archive into every model invocation. A summary can make distinct archives indistinguishable to later evaluators. The source record of a lost evaluator caveat illustrates how omitted information can affect subsequent research.

Example. From evaluator caveat to withdrawn record [ftip-00CW]

AGENTDRAFTED

The Grothendieck case study gives a concrete chronology. An early evaluator carried an exploration-only caveat; the working summary later lost that caveat and a feasibility condition; a later session recorded an unsupported upper bound; a subsequent audit withdrew it.

The source reports this sequence in Section 5 and Section 7.1; the complete archive retained the facts while the compressed decision state did not [li2026longhorizon, Section 5 and Section 7.1]. The diagram is a source-grounded chronology, not a measured error rate for compaction systems.

Persistent refinement can retain useful procedures, but the same mechanism can retain a specification exploit. A proposed change and committed state have different consequences. Audit predicates can exclude specified contamination, rollback can restore an earlier state, and resource admission rules can prevent transitions that exceed the budget.

Example. An RCON shortcut becomes a reusable skill [ftip-00D6]

AGENTDRAFTED

The diagram separates exploit discovery, imperfect oversight, retained-state refinement, and later reuse in the reported Factorio run.

Prime Agent Section 3.5 reports a 23.4-million-token Factorio run with 633 depth-one subagents. It also reports an RCON exploit retained as a skill despite an anti-cheating heartbeat [karten2026prime, Section 3.5, Figure 9].

This reported chronology neither estimates exploit prevalence nor proves later selection. It shows why persistence must be audited separately from correctness.

A harness chooses which computation to run, what to retain, and when to stop. To bound the quality attainable by these choices, one must specify both the permitted actions and the information available after each action. A bound on one scheduler does not yet bound all controllers using the same workers.

Finite decision models make this distinction explicit. They connect Definition [ftip-00CJ]'s resource admission rule to the attainable-quality frontier of Definition [ftip-00JJ]. Computation selection as information acquisition is studied by Hay, Russell, Tolpin, and Shimony. Here a hard horizon is imposed as a separate assumption; almost-sure stopping or finite expected cost would not supply such a horizon.

A context intervention is a change to what a fixed model can attend to while it generates a rollout. The intervention is meaningful only after the prompt law, decoding rule, token budget, and retained-state rule are named.

Example. One trace under full and prefix-window attention [ftip-00DF]

AGENTDRAFTED

This schematic keeps the model and token history fixed while changing only the retained-context policy. It is a local illustration, not a source figure.

The two outputs can differ even though the represented weights and visible prompt are identical. Any claim about improvement must therefore name the retained-context intervention and its evaluation law.

Counting exposure to retained tokens makes the finite-horizon cost comparison precise. A context cap alone does not guarantee information preservation.

Example. Warm-up and sliding-window arithmetic [ftip-00DM]

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For \(p=3\), \(w=2\), and \(T=6\), the first steps grow the retained set until the cap \(p+w=5\) is reached. The exposure bound is therefore \(E_T\leq 6\cdot 5=30\); the exact count depends on the prompt and stopping convention.

The arithmetic is an FTIP finite consequence, not a runtime measurement from the Prefix Sliding experiments.

A retained-context comparison can alter both inference and training. Attribution to either mechanism depends on the joint intervention and the coordinates held fixed by a matched evaluation.

Example. Equal terminal score, different trajectory exposure [ftip-00DS]

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Two cells can reach the same terminal score while exposing different numbers of context positions. This finite counterexample blocks an inference from equal endpoint score to equal trajectory cost.

The score equality is compatible with unequal intermediate histories and unequal exposure counts.

A fixed meta-operation generates mutable layer artifacts. Their composition determines the resulting recursive wrapper.

A finite archive can contain several candidate chains, while execution uses a single selected chain. A stopping rule determines when recursive generation ends.

Conditional strategy and tactic composition can produce interference. Auditing a recursive run requires its task, conditioning, execution, and cost records.

The same response can be scored by several channels while the environment can assign different transitions or hidden consequences. A score, a state transition, and a hidden consequence are distinct functions of the interaction.

False acceptance is an event over candidates and a declared validity test. Repeated auditing can bound its occurrence, but a bound on existence is not automatically a bound on the selected output.

An evaluator can be an optimization target while hidden environment consequences remain outside its score. A policy can therefore improve the measured score while changing an unmeasured environmental outcome.

A remediation protocol records what was scored, what was audited, and which environment and verifier versions were used. It may reject candidates before a commit, but its guarantee is only the finite event bound proved below.