Definition. Capability vector and conditional frontier [ftip-00JM]
✍️source

For an architecture record \(A\), optimizer record \(O\), systems record \(S\), evaluation configuration \(\mathfrak e\) from Definition [ftip-00CE], and declared intervention \(\eta \) at cost cap \(C\), define the capability vector

\[ \mathbf K(A,O,S,\mathfrak e,\eta ,C) =\bigl (K_{\rm state},K_{\rm adapt},K_{\rm repr}, K_{\rm plast},K_{\rm port}\bigr ). \]

Its coordinates respectively record state access, adaptive computation, representation and abstraction, plasticity, and portability under the named task, intervention, evaluation law, and cost accounting. A coordinate may be a descriptive field, an estimand, or \(\bot \) when the record is missing, incompatible, or not measured. The tuple is therefore not a common scalar and does not order architectures without a declared comparison rule.

The five coordinates attach to the existing ceiling layers conditionally: representation claims first require a map from \(A\); optimizer-reachable plasticity requires \((A,O)\); state, adaptive computation, and portability claims require the relevant \((A,O,S)\) interface, task family, and deployment regime. A systems-feasible claim also carries \(C\), the cost vector and its declared scalarization. The records in Definition [ftip-00JH], Definition [ftip-00JI], Definition [ftip-00JL], and Definition [ftip-00CE] remain authoritative.

The scalar frontier \(V_A(C)\) of Definition [ftip-00JJ] is a projection onto the declared utility \(J_{\rm ev}\) for one architecture and intervention envelope; it is not a capability coordinate. For a fixed evaluation configuration, a vector-valued study may instead retain the conditional set

\[ \mathcal F_{\mathfrak e}(C) =\operatorname {Pareto}\left \{ \bigl (\mathbf K(A,O,S,\mathfrak e,\eta ,C), J_{\rm ev}(P_{A,\eta }),\mathbf c(P_{A,\eta })\bigr ): c_A(\eta )\leq C \right \}, \]

where Pareto dominance is declared for the reported coordinates. The set is conditional on the task, intervention class, evaluation law, cost vector, and missingness rule; it is not a universal intelligence frontier.

Matched-compute arms in § [ftip-00JU] compare entries only after stating which coordinates are held fixed, which are allowed to vary, and how the cost vector is charged. Cost, numerical stability, and monitorability remain evaluation coordinates alongside the vector, so a lower scalar cost or a stable implementation is not silently counted as a capability increase.

Context

Architecture, optimizer, and systems variables [ftip-00JG]

Architecture, optimizer, and systems fields refine the model/interface description. A comparison depends only on the fields named in its assumptions and conclusion.

Definition. Optional architecture refinement [ftip-00JH]

Let \(M\) denote an existing model intervention. An architecture refinement is a record \(A=(\mathcal X,\mathcal Y,\operatorname {Map}_A)\) whose map realizes the same declared input and output interface. Forgetting \(A\) returns \(M\); no architecture claim is made when the field is omitted.

Definition. Optimizer and systems records [ftip-00JI]

When needed, extend the record by \(O\) for optimizer and state-update rules, and \(S\) for kernels, precision, memory, cache, and scheduling. The triplet \((A,O,S)\) is descriptive; it is not a scalar intelligence score.

Definition. Architecture-indexed evaluation functional [ftip-00JJ]

Fix an architecture \(A\), its base artifact, a declared intervention set \(\mathfrak I_A\), and the common evaluation interface of Convention [ftip-005D]. Each \(\eta \in \mathfrak I_A\) specifies a protocol \(P_{A,\eta }=\operatorname {PostTrain}(A,\eta )\) in that measurable and absolutely integrable evaluation domain, with real performance \(J_{\rm ev}(P_{A,\eta })\). Let \(c_A(\eta )\in [0,+\infty ]\) be its scalar cost under the study's declared accounting rule: a fixed measured cost or an expected nonnegative cost, as specified. For a finite budget \(C\in \mathbb R_{\geq 0}\), define \[ V_A(C)=\sup \left \{ J_{\rm ev}(P_{A,\eta }): \eta \in \mathfrak I_A,\ c_A(\eta )\leq C \right \}\in \overline {\mathbb R}. \] Infinite-cost interventions are infeasible at every such budget.

As in Definition [ftip-005M], the empty feasible set has value \(-\infty \); an unbounded-above feasible score set has value \(+\infty \). The value is real exactly when that set is nonempty and bounded above. Increasing \(C\) enlarges the feasible set, so \(V_A\) is nondecreasing. A finite supremum need not be attained by any intervention.

Frontier differences and derivatives are ordinary real operations only where the relevant values are finite, with differentiability additionally required for a derivative. Cost ratios use the domain in Definition [ftip-00JP]. The allowed intervention set, evaluation law, and accounting rule are part of the definition, so \(V_A\) is not a universal intelligence function.

Remark. Architecture-neutral projection and gradual complexity [ftip-00JK]

A statement that depends only on the model/interface map is invariant under changes to architecture \(A\), optimizer \(O\), or systems \(S\) that leave that map fixed. A conclusion about one of those fields instead requires assumptions that distinguish its possible values.

Definition. Three ceiling layers [ftip-00JL]

For fixed task and evaluation laws, distinguish the representational ceiling of \(A\), the optimizer-reachable ceiling of \((A,O)\), and the systems-feasible frontier of \((A,O,S,C)\). Each layer is conditional on the objects named; none implies that the next layer attains it.

Definition. Capability vector and conditional frontier [ftip-00JM]
✍️source

\[ \mathbf K(A,O,S,\mathfrak e,\eta ,C) =\bigl (K_{\rm state},K_{\rm adapt},K_{\rm repr}, K_{\rm plast},K_{\rm port}\bigr ). \]

\[ \mathcal F_{\mathfrak e}(C) =\operatorname {Pareto}\left \{ \bigl (\mathbf K(A,O,S,\mathfrak e,\eta ,C), J_{\rm ev}(P_{A,\eta }),\mathbf c(P_{A,\eta })\bigr ): c_A(\eta )\leq C \right \}, \]

Backlinks

Definition. Architecture-family transfer matrix [ftip-00LZ]

A transfer matrix records which interfaces, intervention arms, and budgets permit a comparison. Each row is a study record: its source, estimand, matched controls, evaluation coordinates, and declared scope boundary are explicit. A blank or incompatible cell is an incomparability finding, not a missing score to be imputed. The vector and evaluation coordinates are reported together under the conditional frontier of Definition [ftip-00JM].

Comparison cell	Source	Estimand	Matched controls	Cost / stability / monitorability	Status and scope boundary
Transformer--KDA/full attention	Remark [ftip-00KF], Definition [ftip-00LU], Example [ftip-00LQ]	Difference in utility, cost, and \(K_{\rm state}\) between KDA and full attention	Same task, model scale, recipe, context, seeds, and evaluation law	Report measured cost vector, training/inference stability, and state/failure observability	Open; a finite KDA--Transformer result does not establish an architecture-wide ceiling
Transformer--Mamba	Mamba, Sections 3.1--3.3; § [ftip-00OQ]	Difference in retrieval utility and \(K_{\rm state}\) at the declared budget	Task, scale, recipe, context, and inference budget	Report cost vector, recurrence stability, and state/failure observability	Open; selective compression does not transfer a retrieval result across tasks
Transformer--linear attention	Linear attention, Sections 3.2--3.3; § [ftip-00OQ]	Difference in retrieval utility and \(K_{\rm state}\) for associative summaries	Task family, feature map, context budget, and evaluation law	Report prefix cost, numerical stability, and summary-state observability	Open; associative summaries do not imply full-attention equivalence
Transformer--CTM or looped model	Continuous Thought Machine, Sections 3.1--3.5; § [ftip-00KG]; § [ftip-00OQ]	Difference in utility and \(K_{\rm adapt}\) per charged tick or loop	Input, base model, stopping rule, recipe, and charged internal compute	Report tick/loop cost, training and inference stability, and stopping-signal monitorability	Open; extra passes are charged and do not imply continuous-time behavior
Transformer--BDH	BDH, Section 3.2; § [ftip-00OQ]	Difference in task utility and \(K_{\rm state}\) under the reduced state interface	Task, state parameterization, credit horizon, systems stack, and cost	Report memory and compute cost, optimization stability, and state observability	Open; a reduced state-space proposal is not a production transfer result
Core architecture--JEPA adjunct	I-JEPA, Section 3; § [ftip-00OQ]	Conditional change in \(K_{\rm repr}\) and \(K_{\rm plast}\) after the objective intervention	Same base artifact, target task, predictor budget, data, and optimizer arm	Report intervention cost, training stability, and representation/probe monitorability	Conditional; an objective intervention is not a replacement architecture
I-JEPA--SplitJEPA	SplitJEPA, Sections 3.1--3.4; § [ftip-00OQ]	Subspace identifiability and conditional change in \(K_{\rm repr}\)	Matched invariant correspondences, variant-rank premise, normalization, and encoder budget	Report probe cost, numerical and optimization stability, and subspace monitorability	Conditional; identifiability does not become a general I-JEPA guarantee
Core architecture--liquid dynamics	Liquid time-constant networks, Sections 1--2; § [ftip-00OQ]	Difference in utility and \(K_{\rm adapt}\) under the solver intervention	Input law, solver, step budget, tolerance, task, and cost vector	Report solver cost, numerical stability, and trajectory/failure monitorability	Open; continuous dynamics and solver choices remain intervention coordinates
Core architecture--Neural Operator	Neural Operators, Section 3; § [ftip-00OQ]	Difference in utility and \(K_{\rm repr}\) across discretizations	Function-space task, discretization family, operator layer, and evaluation law	Report training/inference cost, discretization stability, and field-observation monitorability	Open; discretization invariance under one source formulation is not task transfer

A completed row must report its paired observations through § [ftip-00L4] and the common-recipe, equal-tuning, and restricted-envelope arms of § [ftip-00JU]. Until those records are supplied, its status remains open or incompatible.

Mechanism families and capability bottlenecks [ftip-00OQ]

This map asks which mechanism family changes a declared capability bottleneck and what evidence would show that change. It is a comparison aid, not a ranking of architectures. The shared model, task, intervention, cost and evaluation interfaces remain the conditions of every claim. A source description, a measured result and a transfer claim are kept separate.

The five capability dimensions are state access, the information that can remain available across tokens, steps or external memory operations; adaptive computation, conditional allocation of depth, recurrence, search or stopping together with its charged cost; representation and abstraction, the available representation, composition or predictive state; plasticity, the changes reachable by a declared objective, optimizer, parameter tying and post-training recipe; and portability, survival of a result when the model family, task, scale or deployment regime changes. This is distinct from transfer, which evaluates a frozen artifact on new problems under a common deployment budget. They are coordinates for the capability vector, not a single intelligence scalar.

A bottleneck is the declared resource or interface condition that limits one of these dimensions in a comparison. Cost, numerical stability and monitorability remain evaluation coordinates: the ability to observe the relevant state, compute, update or failure signals under the study protocol. A matched-compute comparison fixes the cost accounting before comparing outcomes; its protocol and the architecture frontier remain authoritative elsewhere in this chapter.

The default core panel contains mechanism families that alter the sequence computation or its state interface. Existing Transformer notation and the decoder-only reference are in § [ftip-000K]; the entries below identify the question each family makes testable.

Transformer reference. Attention exposes token-indexed content access through the declared context and serves as the baseline for the matched-compute comparison and interface comparisons.
Selective state-space models. The Mamba construction, Sections 3.1--3.3 makes the state-space parameters input-dependent and computes the resulting time-varying recurrence with a hardware-aware scan. The testable bottleneck is how selective compression trades long-context access and throughput against retrieval and portability boundaries; a source mechanism does not settle that tradeoff.
Linear attention. The linear-attention formulation, Sections 3.2--3.3 uses a feature-map factorization and associative prefix summaries rather than the full softmax attention matrix. The comparison asks which retrieval and memory tasks preserve their evaluation under the changed state interface; it does not identify every hybrid or recurrent model with pure linear attention.
Internal-tick and looped models. The Continuous Thought Machine, Sections 3.1--3.5 uses a discrete internal tick dimension, private neuron-level models that process activation histories, synchronization as a latent representation, and standard cross-attention to the input features. Its loss selects minimum-loss and maximum-certainty ticks, making variable internal compute a testable mechanism. The looped computations in § [ftip-00KG] are a separate family; neither is a continuous-time claim, and every extra tick or loop must be charged in the comparison.
Recurrent or compressed-state mechanisms. The Dragon Hatchling / BDH proposal, Section 3.2 defines a tensor-friendly BDH-GPU state-space system whose attention state admits a reduced representation, alongside local graph dynamics. Its open comparison is whether this state interface supports the target tasks and stable credit assignment at a declared budget; the proposal is not a production result.

The adjunct panel changes an objective, temporal law or function-space bias without being interchangeable with the core architecture families. I-JEPA, Section 3 is an encoder/predictor training architecture as well as an embedding-space prediction objective: its context, target and predictor use ViT backbones, the target encoder is updated by exponential moving average, and masked target blocks are predicted from a context block. That predictive objective leaves a specific representation limit: retaining or recovering the latent state does not by itself identify which directions are invariant and which vary across matched observations. SplitJEPA, Sections 3.1--3.4 (the project overview) makes the boundary explicit—predictive recovery can leave a global orthogonal mixing—and resolves it under stationary Gaussian predictive dynamics, a full-rank second moment of variant differences across invariant correspondences, and representation normalization, by identifying invariant and variant subspaces up to within-block isometries without an observation decoder. Here SplitJEPA is a structured JEPA adjunct and a conditional identifiability result, not a general guarantee for I-JEPA. Liquid time-constant networks, Sections 1--2 supply input-dependent continuous dynamics and a numerical ODE-solver interface; the Neural Operator formulation, Sections 1 and 3 supplies a function-space lens with discretization-invariant operator layers. The Zoology study, Sections 3--5 provides controlled recall analyses of sequence mixers. These sources motivate hypotheses and controls; they do not establish a common capability ordering or make an adjunct a replacement architecture. The JEPA comparability interface specifies fields that must be fixed for a JEPA-like comparison; it is not the source of I-JEPA's architecture. Combining CTM's internal dynamics with I-JEPA's encoder/predictor package is a proposed joint intervention, not an established result, and would require separate attribution of both components.

For each panel row, a study should name the changed state or compute interface, the bottleneck dimension, the task and intervention, the evidence type, and the transport boundary. Existing architecture-indexed ceilings and matched compute, published paired measurements (§ [ftip-00KU] and § [ftip-00L4]), conditional frontiers (§ [ftip-00LH]) and study design (§ [ftip-00LS]) provide those controls. Blank, incompatible and unmeasured cells remain findings rather than scores to be filled by intuition.

Remark. Open comparison questions [ftip-00M2]

The matrix in Definition [ftip-00LZ] is a protocol map, not a result table. The following probes keep the five capability coordinates separate and leave an unknown cell visible until the required records exist. Each probe names its source, estimand, controls, and evaluation coordinates so that a missing record cannot be mistaken for a zero score.

State access. Sources: Mamba, Sections 3.1--3.3 and linear attention, Sections 3.2--3.3. Estimand: the paired difference in retrieval utility and \(K_{\rm state}\). Controls: Transformer, Mamba, and linear-attention systems on a predeclared retrieval and continuation suite at matched parameter, context, recipe, and cost budgets. Evaluation: report cost, recurrence or summary stability, and state observability. Scope boundary: a long-context score does not transfer to another task family without its task and transport record.
Adaptive computation. Source: Continuous Thought Machine, Sections 3.1--3.5 and the looped-model contract in § [ftip-00KG]. Estimand: the paired change in utility and \(K_{\rm adapt}\) per charged internal tick or loop. Controls: common input, base model, recipe, stopping rule, and charged internal compute. Evaluation: report tick/loop cost, training and inference stability, and stopping-signal monitorability. Scope boundary: an observed adaptive gain remains conditional on that stopping law.
Representation and abstraction. Sources: I-JEPA, Section 3 and SplitJEPA, Sections 3.1--3.4. Estimand: subspace identifiability and the conditional change in \(K_{\rm repr}\). Controls: fixed correspondences, full-rank variant differences, normalization, encoder budget, and matched downstream probes. Evaluation: report probe cost, optimization stability, and subspace monitorability. Scope boundary: the conditional subspace result is separate from downstream task transfer; failure of a premise is an incompatible cell, not a zero score.
Plasticity. Source: the objective and optimizer interfaces in Definition [ftip-00JM] and the I-JEPA intervention description in § [ftip-00OQ]. Estimand: the paired change in \(K_{\rm plast}\) after a declared post-training intervention. Controls: common recipe and equal-tuning arms with data, optimizer, trial count, stopping, and feedback fixed or explicitly varied. Evaluation: report intervention cost, training stability, and update-signal monitorability. Scope boundary: an intervention effect is limited to the declared base artifact, task, and objective; it does not transfer across architectures without a new paired record.
Portability. Sources: the Mamba, BDH, and Neural Operator method descriptions in § [ftip-00OQ]: Mamba, BDH, and Neural Operators. Estimand: survival of the declared utility and \(K_{\rm port}\) across one transfer edge. Controls: source model, task family, model scale, deployment regime, evaluation law, and cost vector retained as paired fields. Evaluation: report migration cost, stability after the shift, and failure monitorability. Scope boundary: evidence for one transfer edge does not order unmeasured edges; blank edges remain unmeasured.
Adjunct interactions. Sources: liquid time-constant networks, Sections 1--2, Neural Operators, Section 3, and I-JEPA, Section 3. Estimand: the joint change in the declared capability coordinates after an explicit intervention. Controls: declared core model, objective, temporal law, function-space bias, matched controls, and separate single-intervention arms. Evaluation: report joint and marginal costs, numerical/training stability, and interaction monitorability. Scope boundary: a joint effect is limited to the declared core, task, temporal law, and function-space regime; it does not transfer across regimes without separate intervention records.

These probes do not establish a common ordering of architecture families. They specify what would count as evidence for a declared bottleneck change, while cost, numerical stability, monitorability, and transfer remain separate evaluation coordinates. Paired observations should use § [ftip-00L4], the study arms of § [ftip-00LS], and the capability vector of Definition [ftip-00JM].

Definition. Architecture-indexed evaluation functional [ftip-00JJ]

Definition. Evaluation configuration record [ftip-00CE]

An evaluation configuration record is a finite record \(\mathfrak e\) that resolves the task and instance law, environment and tool interfaces, executable model artifact, serializer and inference policy, harness configuration, compaction and refinement policies, retry rule, completion gate, evaluator, seed law, and componentwise resource limits.

Together with the independent evaluation interface of Convention [ftip-005D], the record determines the law of a stopped evaluation run and its realized cost vector. A reported score is the pair consisting of the statistic and the exact record \(\mathfrak e\); a scalar without its record is an under-specified family of measurements.

Matched-compute comparison protocol [ftip-00JU]

Definition. Common-recipe comparison arm [ftip-00JV]

A common-recipe arm fixes training data, optimizer family, schedule, post-training procedure and feedback, training and post-training budgets, inference budget, and evaluation protocol and draws. Interface-compatible parameter choices are declared in advance as part of each permitted architecture package. The comparison varies that package under the common recipe; any further deviation is recorded as a separate comparison coordinate. Its score difference measures performance under this recipe, not the best attainable result for either architecture.

Definition. Equal-tuning-budget comparison arm [ftip-00JW]

An equal-tuning-budget arm permits architecture-specific tuning under a predeclared protocol that fixes the tuning data and its volume, trial count, selection rule, stopping rule, and tuning cost for both architectures. The cost uses the same declared accounting rule. The resulting comparison measures practical performance under this equal optimization effort; it does not determine best possible training or a representation-only limit.

Definition. Restricted-envelope comparison arm [ftip-00JX]

A restricted-envelope arm declares an allowed intervention class \(\mathcal E_A\subseteq \mathfrak I_A\) for each architecture, specifying the permitted training, post-training, and inference procedures. The classes expose exclusions, search budgets, seeds, and stopping rules. Restrict the evaluation functional of Definition [ftip-00JJ] to \(\mathcal E_A\) and compare the resulting suprema under the same finite cost cap \(C\geq 0\), common evaluation law, and common declared cost accounting.

The measurable, integrable evaluation domain and extended-real conventions of Definition [ftip-00JJ] apply: an empty feasible class has value \(-\infty \), an unbounded-above feasible score set has value \(+\infty \), and a finite supremum need not be attained. The allowed class is part of the comparison, so its envelope is a restriction-dependent quantity, not an unrestricted architectural ceiling.

Definition. Model-instance comparison evidence record [ftip-00JY]

A model-instance record contains architecture/checkpoint identity, parameter count, data and post-training recipe, optimizer, precision, hardware, cache policy, training and inference cost vectors, task suite, and independent evaluation seed. A missing field makes an architecture attribution unknown.

Remark. Systems optimizations are measured interventions [ftip-00JZ]

FlashAttention, KV-cache layouts, quantization, batching, and kernels can change feasible cost. If they approximate, truncate, or alter precision of the mathematical computation, the record must mark that as a systems intervention rather than as a pure architecture comparison.

Theorem. Matched-protocol difference is an estimand [ftip-00K0]

Given \(n\geq 1\) evaluation draws, let \(\widehat V_A(C)\) be the sample mean of finite real evaluation scores for architecture \(A\) at a finite cost cap \(C\geq 0\). Under a fixed evaluation law and common-recipe arm, the finite difference \(\widehat V_A(C)-\widehat V_B(C)\) is a well-defined empirical estimand.

It does not identify a causal architecture effect when data, tuning, or systems coordinates differ. The conclusion follows directly from the declared record.

Remark. Conditions for applying a toy model [ftip-00K1]

A toy theorem applies to a concrete model only if an explicit map from its state and interface to the model preserves the theorem's assumptions, including its cost accounting. Without such a map, the conclusion concerns only the abstract model; its validity for the concrete system is unknown.

Definition. Optimizer and systems records [ftip-00JI]

Definition. Optional architecture refinement [ftip-00JH]

Definition. Three ceiling layers [ftip-00JL]