The originating hypothesis is Russell's: adequate learned models may differ exactly in the cost of realizing core algebra, geometry, analysis and their compositions. This is a serious research proposal if “adequate”, “core”, “cost” and “realizing” are fixed before the results arrive. It is not a theorem supplied by the word transformer. This chapter separates an exact semantics proposition from two unproved hypotheses.
1. A task is more than its answer
Definition (this protocol). A task q consists of input and output spaces X,Y; a mathematical target f:X → Y; an input law mu; an encoding e; supplied initial information s0; an independent verifier V; and a budget B. Specify tolerance in V. Exact integer equality has zero tolerance. A geometric task may require a sign, an orientation or a coordinate system. An analysis task needs a domain, norm and approximation tolerance. No unstated oracle supplies the missing initial values.
For a fixed model/runtime/protocol m and policy p, define the observed performance law to be the joint distribution of verified result, output, accumulated cost and stopping reason. A cost can be a vector: tokens, calls, reference interpreter work, wall time, peak memory. A token count is not a floating-point operation count; an interpreter's instruction count is not hidden neural work. Peak memory combines by maximum, while disjoint sequential work may add. Overlapping timers cannot be summed twice.
An optimal capability is a different object. For example, with a declared policy class P and scalar nonnegative cost C:
Pr[V accepts] >= 1-delta }.
Specify what P may access, whether search and verification are charged, and how failure costs enter. An empty feasible set has infimum infinity; a finite infimum need not be attained. One successful protocol gives an upper bound when it meets the criterion. One failed protocol does not bound all other policies from below. Calling the fastest trial “intrinsic difficulty” would quietly delete the optimization problem.
Recovery uses the same accounting. Searching for an inverse, storing a witness, changing an encoding and executing a repair are different operations with different information and costs. Giving a decoder the answer as side information makes its execution cheap by moving the work into the specification.
2. Two correct-looking marginals walk into a composition
Counterexample (exact finite probability calculation). A primitive is supposed to leave a bit unchanged. Each call costs one. System A flips it using a fresh fair coin each time. System B draws one fair hidden coin at session start and uses that same coin for every call. Both systems have isolated correctness 1/2 and cost one per call.
| flip pattern | A: fresh coins | B: one reused coin | final bit correct? |
|---|---|---|---|
| 00 | 1/4 | 1/2 | yes |
| 01 | 1/4 | 0 | no |
| 10 | 1/4 | 0 | no |
| 11 | 1/4 | 1/2 | yes; both mistakes cancel |
Thus A has final correctness 1/2 and all-step correctness 1/4. B has final correctness 1 and all-step correctness 1/2. Primitive marginal accuracy and cost do not determine either composition statistic. The missing object is the law conditional on the actual history. In B, the first observed flip reveals what will happen next. This table is mathematics, not a finding about a learned model.
The inverse version of this mistake is familiar: an encoder's average reconstruction score does not identify every fibre. The operational version says a compact score profile does not identify every conditional transition. Both replace a family of distinctions with a convenient summary. Convenience needs a validation set.
3. What really does determine compositions
Use common standard Borel state, action, cost and status spaces; finite JSON histories with real timing coordinates are examples. The state must retain enough context for its conditional law to be valid. It can be the entire observable history with the induced posterior over hidden state. A small register vector is sufficient only if that claim is justified. Let
be the conditional law of the next state s', incremental cost c and observation/status l, for action a after history h. A common controller p(da|h) chooses actions. A common stopping rule includes parse rejection, timeout, budget exhaustion and abort. Use absorbing terminal states. The initial law nu includes the inputs, encoding and runtime/cache condition.
Proposition O1 (classical kernel composition). If two systems have equal initial laws, equal conditional kernels at every reachable matched history, and the same controller, cost accumulation and stopping semantics, their finite-horizon joint trace laws agree. Every measurable summary of those traces has the same law.
Proof. The empty histories have the same law nu. If length-t histories have equal laws, integrating the common controller against them gives equal joint laws of history and action. Integrating the equal transition kernels gives equal extended-history laws. The same terminal convention preserves equality. Induction proves the assertion. Applying a measurable output/verification/cost function to equal-law traces preserves equality. No independence of successive calls was assumed.
This determines compositions because the premise contains all relevant conditional behavior. It does not prove that a small experiment can measure it. Nor does equality of terminal answers on one task family imply equality of these kernels.
Proposition O2 (finite-horizon error). Write TV(P,Q)=supA|P(A)-Q(A)|. Suppose initial laws differ by at most epsilon0 and corresponding kernels differ in TV by at most epsilon uniformly on reachable matched histories. For T transitions with the same controller and stopping convention:
Proof. Couple initial states with disagreement probability at most epsilon0. While histories agree, sample the same controller action and couple next outcomes with disagreement probability at most epsilon. The chance of a first disagreement by T is at most epsilon0+T epsilon by a union bound. Coupling bounds TV by that probability. Measurable summaries cannot increase TV. Equivalently, telescope the finite kernel products.
For a loss between 0 and L, the expected-loss difference is at most L times this bound. There is no corresponding guarantee for unbounded wall time without tail or moment control. Error on a primitive calibration distribution does not supply a uniform bound over states reached inside compositions. The quantifier is doing work; do not erase it with a chart.
4. The strong hypothesis: only the bill changes
Unproved research hypothesis H1. After an independently declared adequacy test, compared learned systems realize common semantic kernels Qa and differ only through conditional resource costs. Keep semantic history u separate from the resource ledger r. A precise factorization is:
Shared Q is the substantive assertion. The intended equation “add two numbers” does not imply that learned systems have equal errors, encodings or conditional output laws. Include enough semantic history to make Q legitimate. Defining u to include the cumulative ledger while requiring a common Q would inadvertently require common resource transitions too.
Even if H1 holds, equal success under a finite budget does not follow. A costly system may hit the stopping boundary earlier. A controller that chooses actions using the ledger can choose a different semantic path. Resource accounting and control therefore remain explicit.
Adequacy must precede inspection of the held-out compositions. A frozen primitive accuracy gate can be part of a study, with all excluded configurations reported. Defining an “adequate model” as whichever system already behaves identically on every desired task makes H1 a tautology. That would be an impressively cheap theorem and a useless experiment.
5. The testable hypothesis: a small profile travels
Unproved research hypothesis H2. A small, preregistered operational profile predicts unseen composed-task performance under declared budgets and encodings, with little predictive benefit from model identity after conditioning on the profile. This is weaker than equality of every conditional kernel and still requires evidence.
Freeze the tasks, seeds, prompt versions, profile features, adequacy criterion, predictive rule and tolerances. Hold out composition structures and encodings, not merely fresh numbers in already tuned prompts. Score each held-out instance once per frozen profile and disclose previous exposure. Compare a marginal baseline with a state-aware predictor, and compare the latter with one additionally given model identity. Specify what improvement is large enough to reject the proposed sufficiency. A finite sample can constrain this protocol-level hypothesis; it cannot establish equality on all mathematical tasks.
The product of primitive success probabilities predicts all-step success only under matching conditional distributions and the needed independence assumptions. It is a deliberately misspecified proxy when applied to final-answer correctness. Error cancellation in the coin table already proves why. A successful reference program demonstrates that the executor can implement it. It does not demonstrate that the model can construct it.
Compactness plays a useful but limited role. A compact parameter family with a verified common modulus can be covered at finite resolution. The mixing theorem even gives a common all-word bound in its restricted regime. These facts do not identify a compact predictive profile for learned reasoning. Continuity of model output is not correctness relative to a mathematical target, and an attained compact state set is not a generalization theorem.
6. Broaden the basis; keep the bill itemized
The existing experiment deliberately starts small. A wider test of the originating hypothesis should separate three kinds of task and the compositions between them:
| domain | mathematical target | condition to record |
|---|---|---|
| algebra | integer/rational operations, polynomial evaluation, linear systems | bit length, exact domain, allowed division, output encoding |
| geometry | orientation determinants, distances, affine repair, intersections | coordinate frame, degeneracies, rank, conditioning, tolerance |
| analysis | derivative expressions, bounded integration, fixed-point approximation | function class, norm, regularity, certificates and error budget |
| composition | different graphs with matched primitive counts | dependency depth, intermediate magnitudes, representation changes, observed errors |
A finite arithmetic DSL can express determinant and polynomial derivative evaluations. That is an expressibility statement, not a universal basis theorem for analysis. Approximation algorithms require conditions: “integrate a function” without a function class or access model is not one finite benchmark. A model that selects a stable algorithm may use more primitive steps and recover a better answer. Counting the reference graph alone misses that choice.
A costed latent representation also needs an explicit state correspondence between models. There is no canonical identification between coordinate 17 of two learned spaces. An invertible reparameterization can alter coordinate magnitudes while preserving decoded tasks; a quotient can preserve one task and lose another. Compare declared observable operations or prove a conjugacy/factor map. The fibre criterion says exactly which tasks survive an encoder.
These extensions would connect the mathematical repair examples to operational profiles: find the affine repair, represent it, execute it, verify the invariant, then inspect cost as the ellipse approaches rank collapse. A divergence in condition number can be mathematical, while extra generation cost can be protocol-specific. They should not be folded into one unexplained “difficulty” score.
7. The receipt we actually have
The separate operations instrument implements bounded exact integer add/subtract/multiply probes, affine recurrences, triangle determinants and cubic derivative compositions, with paired JSON/register-table encodings and independent BigInt verification. Whole-task modes are model-only, supplied correct program and one-shot model-authored arithmetic AST. The bounded model-chain mode routes actual parsed primitive outputs into later requests. Invalid outputs stop; no oracle corrects the route.
ei = observed vi - gi(actual observed parent values)
all-step correct = route completed and every ei=0
final correct = route completed and root output = target f(x)
Verification occurs after the route stops. Unattempted nodes remain unobserved. This exposes an external controller's trace, not hidden neural thought.
Evidence status for this edition. The retained automated agent-environment model trial was blocked before tokenization/generation and completed no inference. A separate user-reported browser run did generate model text, then failed the program contract with invalid_json; the interpreter did not execute. That observation has no complete exported audit trace in the retained record. Interpreter controls and synthetic correlated-error fixtures validate software, not learned-model capability. There is no completed, verified learned-model operational-profile experiment establishing H1 or H2.
No model download, inference or API call is performed by this mathematical work. Its local diagrams compute the displayed finite formulas. Opening the separate operations instrument does not itself turn its controls into evidence. Change a prompt, and you have a new condition to measure.