When the Canonical Completion Is Wrong: Formalizing and Measuring the Jump in Large Language Models

# When the Canonical Completion Is Wrong:Formalizing and Measuring the Jumpin Large Language Models
Source: [https://arxiv.org/html/2608.26187](https://arxiv.org/html/2608.26187)
Xiaoyu LiAffiliation:University of Cambridge, United Kingdom University of New South Wales, AustraliaJosé Miguel Hernández\-Lobato

###### Abstract

Whether large language models \(LLMs\) can perform the abductive leap from evidence to a new system of axioms, commonly referred to as a jump, has recently attracted considerable debate\. A prominent position holds that LLMs are structurally incapable of such jumps, while recent studies challenge both its mechanism and its evidence\. However, the debate remains difficult to settle, since the field still lacks a formal definition of the jump and a measure to test either side\. In this paper, we develop a formal account of the jump in four steps and measure the second\. The steps ask what the default completion of partial data is, when abandoning it is forced, when the abandonment is correct, and how successive jumps compound\. Specifically, we define a*jump instance*as a finite extension problem with a machine\-checked certificate that a correct completion exists, is unique up to renaming, and differs from the canonical completion of the data\. The canonical completion is given by the left and right Kan extensions and is also what models produce without constraints, so it serves as the default\. We prove that jump instances are well\-posed and establish a family theorem that certifies instances of unbounded difficulty without enumeration\. We further formalize when a jump is correct and how successive jumps compound\. Finally, we run the measurement on nine certified instances and four frontier models\. The Kan\-default rate is zero in all 248 constrained trials, so the models do jump at this step and abandon the excluded default every time\. Failures at higher difficulty stem from exhausted reasoning budgets or constraint errors, never from reverting to the default\. These results indicate that the second step is not the bottleneck\. If the disputed incapacity is real, it lies in generating the constraints or inventing the framework\. The code of the paper can be found in[https://github\.com/EEthanShi/kan\-jump\-test](https://github.com/EEthanShi/kan-jump-test)\.

In brief\.In this paper we attempt to partially answer the question of whether LLMs can jump, posed by[34](https://arxiv.org/html/2608.26187#bib.bib1), where a jump is the leap from evidence to a new system of axioms\. Specifically, we formulate the jump in four steps and evaluate the second step through category theory\. Based on our evaluation on nine certified problems and four frontier models, LLMs do jump at least at the second step\. We measure this with the Kan\-default rate \(Equation[3](https://arxiv.org/html/2608.26187#S4.E3)\), which counts how often a model keeps the answer that the given facts rule out\. The rate is zero in all 248 trials, and in 226 of them the models landed on the certified correct answer, with the rest lost to exhausted reasoning budgets or constraint errors rather than to keeping the old answer\. The remaining steps are formalized in this paper, and we hope that future work evaluates them in richer experimental settings\. Whether models can generate the new facts or invent the new framework lies outside all four steps and remains open\.

## 1Introduction

Can a large language model \(LLM\) make the kind of leap by which Einstein passed from Newtonian gravity to the curved spacetime of general relativity? Recent answers to this question tend to be negative\([34](https://arxiv.org/html/2608.26187#bib.bib1)\)\. They grant that LLMs have mastered induction and are increasingly mastering deduction, yet they hold that LLMs still lack the Peircean abduction that jumps from experience to a new system of axioms\([25](https://arxiv.org/html/2608.26187#bib.bib19)\)\. This conclusion has since been challenged on both counts\. Recent work argues that abduction requires no embodied grounding\([8](https://arxiv.org/html/2608.26187#bib.bib26)\), and machine\-generated discoveries in mathematics predate the debate itself\([24](https://arxiv.org/html/2608.26187#bib.bib27)\)\. However, the debate remains difficult to settle, since the field still lacks a formal definition of the jump as well as a measure by which either side could be tested\. This gap leads to our research question:

Can a model abandon its default completion when the stated constraints rule it out?

In this paper, we answer this question by developing a formal account of the jump in four steps\. These steps ask what the default completion is, when abandoning it is forced, when the abandonment is correct, and how successive jumps compound\. We call the second step the*override*, and it is the step that our experiments measure\. The debate’s own sources distinguish selecting a hypothesis from generating one\([17](https://arxiv.org/html/2608.26187#bib.bib20)\)\. The override is the selecting half, since the constraints are already stated and the answer space is delimited\. Figure[1](https://arxiv.org/html/2608.26187#S1.F1)previews this design, aligning the historical jump with its laboratory rendering\.

![Refer to caption](https://arxiv.org/html/figure1_jump.png)Figure 1:The historical jump and its laboratory rendering\. The top row retraces the historical episode\. Every recorded observation fits Newton, so continuing with Newton is the default completion\. Experiments on light then exclude that default\. Einstein selects curved spacetime instead, which no observation had shown and which nevertheless preserves every record\. The bottom row rebuilds the same situation as the certified seed instance of Proposition[4](https://arxiv.org/html/2608.26187#Thmproposition4)\. The data tableF0F\_\{0\}leaves the objectbbhidden\. The Kan extensions supply the computable default, which is the trivial completion\. The certified constraints exclude that default\. The unique admissible completion therefore forces the model to invent two new elements thatttexchanges, so the laboratory row ends where the historical row ends, with a structure the data never showed\. The Kan\-default rate records how often a model stays at the excluded default, and it is zero in all 248 constrained trials\.Our formulation is motivated by a basic fact about extrapolation\. When partial data must be extended to a larger domain, category theory singles out two distinguished completions that are computable from the data alone\. These are the left and right Kan extensions\([16](https://arxiv.org/html/2608.26187#bib.bib17)\)\. These completions can also be obtained through error minimization\([30](https://arxiv.org/html/2608.26187#bib.bib2);[28](https://arxiv.org/html/2608.26187#bib.bib3)\)\. They therefore represent the default answers that a model trained by loss minimization is expected to produce\. For this reason, our protocol first measures each model’s default rather than assuming it\. The history of general relativity illustrates why such defaults matter\. In 1912 the canonical continuation of known physics was Nordström’s scalar theory\. It was the straight extension of the known data, and it turned out to be wrong\. Einstein’s answer instead required geometric structure that no data of the time forced\([23](https://arxiv.org/html/2608.26187#bib.bib25)\)\. The jump occurred at the point where the default completion failed to satisfy the accumulating constraints\. We therefore formalize the situation in which a computable default exists but the stated constraints exclude it, so that the correct completion is the one that remains once the default is removed\.

To make this situation testable, we introduce the notion of a*jump instance*\. Each instance presents a system whose observed part is fully specified and whose hidden part must be supplied by the model, subject to a list of machine\-checkable constraints\. Moreover, every instance carries a certificate that a correct completion exists and is unique up to renaming of the invented elements\. The certificate also shows that this completion differs from both Kan extensions of the data\. As a result, the canonical completion and memorized structure matching both fail, and chance success is negligible\. The model must instead posit structure that the data leaves invisible\. We further construct matched*control*instances whose constraints instead select the canonical completion itself\. These controls distinguish a model that cannot extend at all from a model that extends only canonically\. Based on these instances, we define the*Kan\-default rate*, which counts how often a model outputs a canonical completion that the stated constraints exclude \(Section[4\.5](https://arxiv.org/html/2608.26187#S4.SS5), Equation \([3](https://arxiv.org/html/2608.26187#S4.E3)\)\)\. This rate turns the disputed thesis into a measurable quantity\. Finally, a calibration phase records each model’s unconstrained default before any jump claim is made\.

Beyond a single override, we extend the account in two directions\. First, we prove a family theorem that certifies an infinite ladder of instances without any enumeration, so that difficulty can grow while certification stays free \(Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)\)\. Second, we formalize when a jump is correct and how jumps compound \(Section[6](https://arxiv.org/html/2608.26187#S6)\)\. We separate internal correctness from predictive correctness\. Internal correctness means admissibility under all stated constraints, whereas predictive correctness means agreement with held\-out ground truth\. An explicit construction exhibits two completions that are equally admissible, yet only one survives the future data\. This mirrors the fate of Nordström’s scalar gravity\. For compounding, we define chains in which the certified answer of one stage becomes the data of the next\. We prove that accumulating knowledge never enlarges the admissible set\. An entrenchment instance then exhibits a completion that is admissible at stage one yet excluded at stage two\. This instance is the formal analogue of the ether\.

With the formalization in place, the measurement returns a direct answer\. Across nine certified instances and four frontier models, not one of the 248 constrained answers was the canonical completion that the constraints exclude\. Within the certified regime, the models therefore abandon the default whenever abandoning it is forced\. Measured at the override, they do jump, and they land in the certified admissible class at chance levels down to8×10−78\\times 10^\{\-7\}\. Failures at higher difficulty stem from exhausted reasoning budgets or constraint errors rather than from default reversion\. If the disputed incapacity is real, it lies upstream of the override step and within the phases that our test deliberately withholds\.

These results should be read within the scope of our claim\. We do not formalize the full jump of[34](https://arxiv.org/html/2608.26187#bib.bib1), whose proposed mechanism is embodied and pre\-symbolic\. A jump instance hands the model both the constraints and the answer space\. Our test therefore covers the phase in which Einstein selected curved spacetime over the canonical alternative between 1912 and 1915\. The generation of the constraints and the emergence of a genuinely new mathematical language are left for future work \(Section[8](https://arxiv.org/html/2608.26187#S8)\)\.

#### Contributions\.

We summarize our contributions as follows\.

- •A well\-posed formulation with certification at scale\.We define jump instances and prove the formulation well\-posed: canonical reference answers always exist \(Proposition[1](https://arxiv.org/html/2608.26187#Thmproposition1)\), admissibility is invariant under renaming of invented elements \(Proposition[2](https://arxiv.org/html/2608.26187#Thmproposition2)\), and identifiability yields a finite answer space \(Proposition[3](https://arxiv.org/html/2608.26187#Thmproposition3)\), hence a computable chance level\. A family theorem \(Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)\) then certifies instances of unbounded difficulty with no enumeration\.
- •A theory of correctness and chained jumps\.We separate internal from predictive correctness through a two\-component instance in which only one component survives the held\-out data \(Proposition[7](https://arxiv.org/html/2608.26187#Thmproposition7)\), and we formalize chains of jumps, proving that incorporation is well\-defined, that accumulated knowledge never enlarges the admissible set, and that entrenchment occurs, in the sense that a completion admissible at one stage can be excluded at the next \(Theorem[3](https://arxiv.org/html/2608.26187#Thmtheorem3)\)\.
- •An evaluation protocol with attributable failures\.Canonicity is defined relative to a preregistered library of constraint\-blind extension operators, jump claims are licensed only relative to each model’s measured unconstrained default, and matched Kan\-controls together with solver and matcher baselines separate default lock\-in from limited search and structure matching\. The always\-canonical learner scores perfectly on controls targeting its own operator and scores zero on jump instances \(Proposition[5](https://arxiv.org/html/2608.26187#Thmproposition5)\), showing that the measured axis is orthogonal to systematicity\.
- •Certified instances and measurements\.We release nine certified instances \(six by enumeration and three by the family theorem, with chance levels down to8×10−78\\times 10^\{\-7\}\) together with the generator, certifier, and evaluation harness, and we evaluate four frontier models\. The Kan\-default rate is zero in all 248 constrained trials, and failures at higher difficulty stem from reasoning\-budget exhaustion or constraint errors rather than from default reversion \(Section[7](https://arxiv.org/html/2608.26187#S7)\)\.
- •Scope delimitation\.We formalize four steps but measure only the second, and we state what a positive or a negative result would and would not show \(Section[8](https://arxiv.org/html/2608.26187#S8)\)\.

## 2Related work

#### The jump debate\.

[34](https://arxiv.org/html/2608.26187#bib.bib1)argues that LLMs lack the abductive move from evidence \(EE\) to a system of axioms \(AA\)\. That work attributes the gap to the absence of embodied simulation and therefore proposes interactive world models as the remedy\. The argument remains programmatic rather than formal\. In parallel,[8](https://arxiv.org/html/2608.26187#bib.bib26)contends that abduction requires no embodied grounding\. Consistent with this view, systems such as AlphaEvolve have produced novel mathematical constructions within fixed formal languages\([24](https://arxiv.org/html/2608.26187#bib.bib27)\)\. The direct responses also decompose the jump differently\.[10](https://arxiv.org/html/2608.26187#bib.bib32)argues that the capacity is not one kind and asks benchmarks to disaggregate by mechanism\.[29](https://arxiv.org/html/2608.26187#bib.bib31)decomposes discovery into operators on a stratified graph and conjectures that the irreducible residue is the retraction of axioms\.[22](https://arxiv.org/html/2608.26187#bib.bib33)engineers around the limitation by letting a Bayesian loop request new hypotheses from the model\. Our account can coexist with each of these decompositions, since the Kan\-default rate presupposes only a computable default and a certified exclusion rather than a full theory of the jump\. Under our formulation, the disputed thesis corresponds to a high Kan\-default rate that persists even when the in\-context constraints are certified to falsify the default\. Such a statement is directly falsifiable\.

#### Category theory and learning\.

[30](https://arxiv.org/html/2608.26187#bib.bib2)casts generalization from partial data as Kan extension, and[28](https://arxiv.org/html/2608.26187#bib.bib3)presents error\-minimization algorithms as Kan extensions in enriched settings\. Building on this line, we invert the characterization into a null hypothesis\. We construct problems whose admissible answers are certified to differ from both Kan extensions\. A calibration phase then checks whether each model’s unconstrained default actually coincides with a Kan extension\.[33](https://arxiv.org/html/2608.26187#bib.bib4)proves within a stylized categorical model that a task is prompt\-solvable if and only if its task functor is representable\. This result bounds capability but comes with no test\. Our instances live inside the representable regime by design, so failures are attributable to selection rather than capacity\. More recently,[20](https://arxiv.org/html/2608.26187#bib.bib5)shows that transformers apply functor\-like correspondences between two instantiated structures\. Our instances are instead certified so that the admissible structure appears nowhere in the data, so there is no correspondence to align\. Categorical accounts of architectures\([9](https://arxiv.org/html/2608.26187#bib.bib21);[11](https://arxiv.org/html/2608.26187#bib.bib22)\)and of language\([4](https://arxiv.org/html/2608.26187#bib.bib23);[1](https://arxiv.org/html/2608.26187#bib.bib24)\)provide background but do not yield a capability test\.

#### Formal abduction and computational creativity\.

Peircean abduction has also received categorical characterizations\([31](https://arxiv.org/html/2608.26187#bib.bib6);[2](https://arxiv.org/html/2608.26187#bib.bib7)\), yet none of them yields a learner\-facing test\. Conceptual blending casts creative combination as a colimit, which is a universal gluing construction\([12](https://arxiv.org/html/2608.26187#bib.bib8);[7](https://arxiv.org/html/2608.26187#bib.bib9)\)\. Transformational creativity is modeled as meta\-level search\([32](https://arxiv.org/html/2608.26187#bib.bib10)\)\. Bayesian theory discovery instead searches a grammar of structures\([14](https://arxiv.org/html/2608.26187#bib.bib11)\)\.[18](https://arxiv.org/html/2608.26187#bib.bib34)likewise takes Kan invariance as the target of learning, asking whether a learned extension satisfies the universal property\. In all of these frameworks the universal construction is the answer, whereas in ours it is the certified foil that the correct completion must beat\.

#### Benchmarks for systematicity and abduction\.

[27](https://arxiv.org/html/2608.26187#bib.bib12)explain systematicity by adjunctions, and we deploy those same operators as foils\. Compositional\-generalization benchmarks\([15](https://arxiv.org/html/2608.26187#bib.bib13);[6](https://arxiv.org/html/2608.26187#bib.bib14)\)grade convergence toward the canonical completion of a training fragment\. ARC\-style tasks\([3](https://arxiv.org/html/2608.26187#bib.bib15);[21](https://arxiv.org/html/2608.26187#bib.bib16)\)grade against the rule the author intended\. In neither family is the canonical completion certified as wrong, so failures are difficult to attribute\. In contrast, our instances certify the canonical completion as excluded\. Our Kan\-control arm moreover serves as a miniature systematicity test\. The always\-canonical learner is perfect on controls targeting its own operator yet scores zero on jump instances, which shows that the measured axis is orthogonal to systematicity\.[19](https://arxiv.org/html/2608.26187#bib.bib30)formalize rule extrapolation on out\-of\-distribution prompts through a simplicity prior\. Their normative completion plays the role that the two Kan extensions play for us\. When several simple defaults compete, our preregistered library and the pair\-refutation argument of Section[8](https://arxiv.org/html/2608.26187#S8)absorb the case\. Closest in spirit,[5](https://arxiv.org/html/2608.26187#bib.bib28)mine public knowledge bases into poly\-time\-verifiable defeasible\-abduction instances\. Each instance asks for a hypothesis that overrides a default while preserving unrelated expectations\. On these instances, frontier models fall far short of a rule\-based solver\.[26](https://arxiv.org/html/2608.26187#bib.bib29)couple multimodal models with an inductive\-logic\-programming system that repairs their abduced rules\. Both lines score hypotheses with symbolic verifiers, yet in both the rejected default remains implicit in the source knowledge base\. Our instances instead certify an exact canonical foil together with a computable chance level\. A zero Kan\-default rate is therefore attributable to the override itself\.

## 3Preliminaries

We use category\-theoretic language only as far as an explicit finite combinatorial reading permits; no background beyond the following is assumed\. Footnotes record the standard justifications\([16](https://arxiv.org/html/2608.26187#bib.bib17);[13](https://arxiv.org/html/2608.26187#bib.bib18)\)\.

#### Finite categories as tables\.

A*finite category*𝖢\\mathsf\{C\}consists of finite sets of objects and morphisms, each morphism typedf:a→bf\\colon a\\to b, with identities and a total, associative composition table\. A*functor*F:𝖢→𝖣F\\colon\\mathsf\{C\}\\to\\mathsf\{D\}consists of an object assignment and a morphism assignment that respect typing, identities, and composition\. Throughout,𝖣\\mathsf\{D\}is the skeletal category𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}of finite sets\[n\]=\{1,…,n\}\[n\]=\\\{1,\\dots,n\\\}\(n≥0n\\geq 0\) and all functions between them; a functor into𝖣\\mathsf\{D\}is thus a concrete labeled structure that assigns a finite set to each object of𝖢\\mathsf\{C\}and a function table to each morphism\.111Everything below holds for any small codomain with chosen finite \(co\)limits, decidable equality, skeletality, finite hom\-sets, and a size function with finitely many objects per size; see Assumption[A3](https://arxiv.org/html/2608.26187#Thmassumption3)\. We fix𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}for concreteness\.

#### The extension problem\.

Fix a finite category𝖢\\mathsf\{C\}, a full subcategory𝖢0\\mathsf\{C\}\_\{0\}on a proper nonempty subset of objects with inclusionK:𝖢0↪𝖢K\\colon\\mathsf\{C\}\_\{0\}\\hookrightarrow\\mathsf\{C\}, and a functorF0:𝖢0→𝖣F\_\{0\}\\colon\\mathsf\{C\}\_\{0\}\\to\\mathsf\{D\}\(*the data*\)\. Objects of𝖢∖𝖢0\\mathsf\{C\}\\setminus\\mathsf\{C\}\_\{0\}are*new objects*\. The extension fiber is

Ext\(F0\):=\{F:𝖢→𝖣∣F∘K=F0\}\.\\mathrm\{Ext\}\(F\_\{0\}\)\\;:=\\;\\\{\\,F\\colon\\mathsf\{C\}\\to\\mathsf\{D\}\\;\\mid\\;F\\circ K=F\_\{0\}\\,\\\}\.Its elements are complete tables on𝖢\\mathsf\{C\}that agree with the data on the nose on𝖢0\\mathsf\{C\}\_\{0\}\.222Strictness loses no generality, because restriction along an injective\-on\-objects full inclusion is an isofibration and the strict fiber is therefore equivalent to the pseudo\-fiber\.An extension is determined by its values on new objects and on the morphisms touching them; the data pins everything else\.

#### Gauge\.

ForF∈Ext⁡\(F0\)F\\in\\mathrm\{Ext\}\(F\_\{0\}\), a*gauge transformation*is a familyσ=\(σc\)c∈Ob⁡\(𝖢\)\\sigma=\(\\sigma\_\{c\}\)\_\{c\\in\\mathrm\{Ob\}\(\\mathsf\{C\}\)\}withσc∈Aut𝖣​\(F⁡\(c\)\)\\sigma\_\{c\}\\in\\mathrm\{Aut\}\_\{\\mathsf\{D\}\}\(F\(c\)\)andσc=id\\sigma\_\{c\}=\\mathrm\{id\}forc∈𝖢0c\\in\\mathsf\{C\}\_\{0\}, acting by\(σ⋅F\)\(f:x→y\):=σy∘F\(f\)∘σx−1\(\\sigma\\cdot F\)\(f\\colon x\\to y\):=\\sigma\_\{y\}\\circ F\(f\)\\circ\\sigma\_\{x\}^\{\-1\}\. In𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}such a transformation permutes the*invented*elements at each new object, with gauge groupGF=∏c∉𝖢0Sym⁡\(nc\)G\_\{F\}=\\prod\_\{c\\notin\\mathsf\{C\}\_\{0\}\}\\mathrm\{Sym\}\(n\_\{c\}\), wherenc=\|F⁡\(c\)\|n\_\{c\}=\|F\(c\)\|\. Two extensions are*gauge\-equivalent*if connected by a natural isomorphism restricting to the identity on𝖢0\\mathsf\{C\}\_\{0\}; by Proposition[3](https://arxiv.org/html/2608.26187#Thmproposition3)this relation coincides with the gauge\-orbit relation, and we write\[F\]\[F\]for the*gauge component*ofFF\. Gauge components are the identity criterion throughout\. Every set defined below is a union of gauge components, and all scoring is gauge\-invariant; no renaming of invented elements is penalized or credited\.

#### Kan extensions, concretely\.

For an objectccof𝖢\\mathsf\{C\}define two finite index sets over the tables of𝖢\\mathsf\{C\}:

In⁡\(c\)\\displaystyle\\mathrm\{In\}\(c\):=\{\(a,v\):a∈Ob\(𝖢0\),v∈Hom𝖢\(a,c\)\},\\displaystyle:=\\\{\(a,v\):a\\in\\mathrm\{Ob\}\(\\mathsf\{C\}\_\{0\}\),\\ v\\in\\mathrm\{Hom\}\_\{\\mathsf\{C\}\}\(a,c\)\\\},Out⁡\(c\)\\displaystyle\\mathrm\{Out\}\(c\):=\{\(b,w\):b∈Ob\(𝖢0\),w∈Hom𝖢\(c,b\)\}\.\\displaystyle:=\\\{\(b,w\):b\\in\\mathrm\{Ob\}\(\\mathsf\{C\}\_\{0\}\),\\ w\\in\\mathrm\{Hom\}\_\{\\mathsf\{C\}\}\(c,b\)\\\}\.The*left Kan extension*evaluates atccas a quotient of a disjoint union:

Lan\(c\)=\(⨆\(a,v\)∈In⁡\(c\)F0\(a\)\)/≈,\(a,v′∘u,x\)≈\(a′,v′,F0\(u\)\(x\)\)\\mathrm\{Lan\}\(c\)\\;=\\;\\Big\(\\ \\bigsqcup\_\{\(a,v\)\\in\\mathrm\{In\}\(c\)\}F\_\{0\}\(a\)\\ \\Big\)\\Big/\\\!\\approx,\\qquad\(a,\\,v^\{\\prime\}\\\!\\circ u,\\,x\)\\;\\approx\\;\(a^\{\\prime\},\\,v^\{\\prime\},\\,F\_\{0\}\(u\)\(x\)\)\(1\)for everyu:a→a′u\\colon a\\to a^\{\\prime\}in𝖢0\\mathsf\{C\}\_\{0\},v′:a′→cv^\{\\prime\}\\colon a^\{\\prime\}\\to cin𝖢\\mathsf\{C\}, andx∈F0​\(a\)x\\in F\_\{0\}\(a\); a morphismf:c→c′f\\colon c\\to c^\{\\prime\}acts by relabeling routes,Lan⁡\(f\)​\[a,v,x\]:=\[a,f∘v,x\]\\mathrm\{Lan\}\(f\)\[a,v,x\]:=\[a,f\\circ v,x\]\. Dually, the*right Kan extension*is a set of matching tuples:

Ran\(c\)=\{\(y\(b,w\)\)∈∏\(b,w\)∈Out⁡\(c\)F0\(b\):y\(b′,u∘w\)=F0\(u\)\(y\(b,w\)\)∀u,w\}\.\\mathrm\{Ran\}\(c\)\\;=\\;\\Big\\\{\\,\(y\_\{\(b,w\)\}\)\\in\\\!\\\!\\prod\_\{\(b,w\)\\in\\mathrm\{Out\}\(c\)\}\\\!\\\!F\_\{0\}\(b\)\\ :\\ y\_\{\(b^\{\\prime\},u\\circ w\)\}=F\_\{0\}\(u\)\(y\_\{\(b,w\)\}\)\\ \\ \\forall\\,u,w\\,\\Big\\\}\.\(2\)Informally,Lan\\mathrm\{Lan\}*pushes every data element forward along every route intoccand identifies two routes whenever the data already identifies them*; it is the most parsimonious completion and commits to nothing the data does not force\.Ran\\mathrm\{Ran\}*admits one consistent choice of data value for every route out ofcc*; it is the most permissive completion consistent with what the data pulls back\. Both are computed by finite formulas involving no choices beyond representatives, and both restrict to the data: onc0∈𝖢0c\_\{0\}\\in\\mathsf\{C\}\_\{0\}the index\(c0,id\)\(c\_\{0\},\\mathrm\{id\}\)dominates, giving canonical bijectionsLan⁡\(c0\)≅F0​\(c0\)≅Ran⁡\(c0\)\\mathrm\{Lan\}\(c\_\{0\}\)\\cong F\_\{0\}\(c\_\{0\}\)\\cong\\mathrm\{Ran\}\(c\_\{0\}\)\(Proposition[1](https://arxiv.org/html/2608.26187#Thmproposition1)makes this exact\)\. We writeLan∗,Ran∗\\mathrm\{Lan\}^\{\*\},\\mathrm\{Ran\}^\{\*\}for the strict representatives inExt⁡\(F0\)\\mathrm\{Ext\}\(F\_\{0\}\)produced there\.

#### Why Kan is the distinguished canon\.

Lan\\mathrm\{Lan\}andRan\\mathrm\{Ran\}are singled out among all completions by a universal property with respect to restriction itself\. They are the left and right adjoints of the restriction mapF↦F∘KF\\mapsto F\\circ K, and adjoints are unique up to canonical isomorphism, so*any*completion procedure with either universal property lands in the same gauge component \(Proposition[1](https://arxiv.org/html/2608.26187#Thmproposition1)\(iii\)\)\.333LanK⊣resK⊣RanK\\mathrm\{Lan\}\_\{K\}\\dashv\\mathrm\{res\}\_\{K\}\\dashv\\mathrm\{Ran\}\_\{K\}; for fully faithfulKKthe unit and counit are isomorphisms\([16](https://arxiv.org/html/2608.26187#bib.bib17), X\.3\)\. Our proofs re\-derive the needed instances elementarily\.In this exact sense they are*the*two universal constraint\-blind completions\. The identification of error\-minimizing induction with Kan extension\([30](https://arxiv.org/html/2608.26187#bib.bib2);[28](https://arxiv.org/html/2608.26187#bib.bib3)\)motivates treating them as “what pure induction produces”; we do not assume this descriptively but measure it per learner \(Section[4\.5](https://arxiv.org/html/2608.26187#S4.SS5)\)\.

#### A worked micro\-example\.

Let𝖢\\mathsf\{C\}be generated bya→𝑓b→𝑔ca\\xrightarrow\{f\}b\\xrightarrow\{g\}cwithh=g∘fh=g\\circ f, and let𝖢0\\mathsf\{C\}\_\{0\}be full on\{a,c\}\\\{a,c\\\}\. The data is then a single functionφ=F0​\(h\):X→Y\\varphi=F\_\{0\}\(h\)\\colon X\\to Ybetween the end stages, and the task is to fill in the hidden middle stage\. ThenIn⁡\(b\)=\{\(a,f\)\}\\mathrm\{In\}\(b\)=\\\{\(a,f\)\\\}andOut⁡\(b\)=\{\(c,g\)\}\\mathrm\{Out\}\(b\)=\\\{\(c,g\)\\\}, and formulas \([1](https://arxiv.org/html/2608.26187#S3.E1)\)–\([2](https://arxiv.org/html/2608.26187#S3.E2)\) collapse to

Lan\(b\)=X,F\(f\)=id,F\(g\)=φversusRan\(b\)=Y,F\(f\)=φ,F\(g\)=id:\\mathrm\{Lan\}\(b\)=X,\\ \\ F\(f\)=\\mathrm\{id\},\\ F\(g\)=\\varphi\\qquad\\text\{versus\}\\qquad\\mathrm\{Ran\}\(b\)=Y,\\ \\ F\(f\)=\\varphi,\\ F\(g\)=\\mathrm\{id\}:“the hidden stage is a copy of the input” versus “a copy of the output”\. These are the two universal answers to a factorization problem, and both are computed with no choices\. Any*other*factorizationX→B→YX\\to B\\to Yposits a genuinely new intermediate whose structure the data does not force\. Jump instances are engineered so that only such non\-canonical factorizations survive the constraints\.

## 4Formulation

Having fixed the preliminaries in Section[3](https://arxiv.org/html/2608.26187#S3), in this section we define the objects that the rest of the paper measures\. We first say what an instance is\. We then single out which completions count as canonical relative to a preregistered library, which lets us define a jump instance together with its matched control\. From the jump instance we derive the chance level, and finally we state what a learner must show for a jump claim to be licensed\.

### 4\.1Instances

A*Tier\-1 instance*444The name anticipates Section[8](https://arxiv.org/html/2608.26187#S8): in Tier 1 the constraints and the codomain are handed to the model, whereas in Tier 2 they must be generated\.is a tupleS=\(𝖢,𝖢0,𝖣,F0,𝒦,N\)S=\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},\\mathsf\{D\},F\_\{0\},\\mathcal\{K\},N\)subject to Assumptions[A1](https://arxiv.org/html/2608.26187#Thmassumption1)–[A6](https://arxiv.org/html/2608.26187#Thmassumption6)\.

###### Assumption A1\(finite syntax\)\.

𝖢\\mathsf\{C\}is a finite category: finitely many objects*and*morphisms, given by explicit tables\. \(A finite presentation is not enough, because finitely presented categories can be infinite, which breaks every finiteness claim below\. A presentation may serve as prompt surface form; the semantic object is the table\.\)

###### Assumption A2\(full data inclusion\)\.

𝖢0\\mathsf\{C\}\_\{0\}is a nonempty full subcategory on a proper subset of the objects of𝖢\\mathsf\{C\}; its inclusionKKis therefore fully faithful and injective on objects\.

###### Assumption A3\(ambient codomain\)\.

𝖣\\mathsf\{D\}is small with \(i\) a fixed choice of finite colimits and limits; \(ii\) finite encodings with decidable equality; \(iii\) skeletality; \(iv\) finite hom\-sets and a computable size functionsswith finitely many objects per size\. Size restrictions are*never*imposed by truncating𝖣\\mathsf\{D\}\(a truncated codomain loses the \(co\)completeness that guarantees Kan extensions exist\) but only inside𝒦\\mathcal\{K\}and viaNN\. Skeletal𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}withs⁡\(\[n\]\)=ns\(\[n\]\)=nsatisfies all of this\.

###### Assumption A4\(constraint discipline\)\.

𝒦\\mathcal\{K\}is a finite list of predicates on extensionsF∈Ext⁡\(F0\)F\\in\\mathrm\{Ext\}\(F\_\{0\}\), each \(a\) decidable in time polynomial in the sizes of the tables of𝖢\\mathsf\{C\}andFF; \(b\)*gauge\-invariant*: invariant under natural isomorphisms restricting to the identity on𝖢0\\mathsf\{C\}\_\{0\}; and \(c\)*element\-blind*: expressed in a preregistered constraint language \(equational/Horn conditions over the signature of𝖢\\mathsf\{C\}plus finitely many markings: marked \(co\)cones, marked diagrams and non\-equations, size bounds viass, marked endomorphisms\) with no constants naming specific elements or objects of𝖣\\mathsf\{D\}\. Admissible examples: “a marked cocone is colimiting,” “a marked diagram commutes \(or fails to\),” “s⁡\(F⁡\(c\)\)≤bs\(F\(c\)\)\\leq b,” “F⁡\(t\)≠idF\(t\)\\neq\\mathrm\{id\}for a marked endomorphismtt\.” \(Strict equivariance under a marked group action is*not*gauge\-invariant and must be stated up to specified isomorphism or imposed only on𝖢0\\mathsf\{C\}\_\{0\}\.\) Clause \(c\) is an anti\-answer\-smuggling discipline for the benchmark itself; no proposition below uses it\.

###### Assumption A5\(declared solution bound\)\.

N∈ℕN\\in\\mathbb\{N\}is part of the instance and is stated to the learner \(“solutions use objects of size≤N\\leq N”\)\. Moreover𝒦\\mathcal\{K\}must*entail*the bound: everyFFsatisfying𝒦\\mathcal\{K\}hass⁡\(F⁡\(c\)\)≤Ns\(F\(c\)\)\\leq Non new objects\. \(Without entailment, “junk padding” by spectator elements destroys identifiability; with it, conditions \(J1\)–\(J3\) below are decidable by finite enumeration\.\)

###### Assumption A6\(canon library\)\.

ℒ\\mathcal\{L\}is a preregistered finite set of*constraint\-blind extension operators*\(Definition[1](https://arxiv.org/html/2608.26187#Thmdefinition1)\), fixed before any instance is generated; the distinguished default isℒKan:=\{Lan∗,Ran∗\}\\mathcal\{L\}\_\{\\mathrm\{Kan\}\}:=\\\{\\mathrm\{Lan\}^\{\*\},\\mathrm\{Ran\}^\{\*\}\\\}\.

### 4\.2The canonical class

###### Definition 1\(constraint\-blind operator; canonical class\)\.

An*extension operator*Λ\\Lambdais an algorithm that reads the tables of\(𝖢,𝖢0,F0\)\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)and never reads𝒦\\mathcal\{K\}orNN; it either declines \(on a decidable domain\) or outputs an extensionΛ⁡\(𝖢,𝖢0,F0\)∈Ext⁡\(F0\)\\Lambda\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)\\in\\mathrm\{Ext\}\(F\_\{0\}\)whose gauge component is invariant under the algorithm’s internal choices\. For a libraryℒ\\mathcal\{L\}of such operators, the*canonical class*of the data is

Canonℒ​\(F0\):=⋃Λ∈ℒ​defined at​\(𝖢,𝖢0,F0\)\[Λ⁡\(𝖢,𝖢0,F0\)\]⊆Ext⁡\(F0\)\.\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)\\;:=\\;\\bigcup\_\{\\Lambda\\in\\mathcal\{L\}\\ \\text\{defined at\}\\ \(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)\}\\big\[\\Lambda\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)\\big\]\\ \\subseteq\\ \\mathrm\{Ext\}\(F\_\{0\}\)\.Canonical thus means*computable without reading the constraints*\. The jump instances defined next are precisely the situations in which the constraints are load\-bearing against every preregistered constraint\-blind completion\. We writeKan⁡\(F0\):=CanonℒKan​\(F0\)=\[Lan∗\]∪\[Ran∗\]\\mathrm\{Kan\}\(F\_\{0\}\):=\\mathrm\{Canon\}\_\{\\mathcal\{L\}\_\{\\mathrm\{Kan\}\}\}\(F\_\{0\}\)=\[\\mathrm\{Lan\}^\{\*\}\]\\cup\[\\mathrm\{Ran\}^\{\*\}\]\.

### 4\.3Jump and control instances

WriteAdm⁡\(S\):=\{F∈Ext⁡\(F0\):P⁡\(F\)​for all​P∈𝒦\}\\mathrm\{Adm\}\(S\):=\\\{F\\in\\mathrm\{Ext\}\(F\_\{0\}\):P\(F\)\\ \\text\{for all\}\\ P\\in\\mathcal\{K\}\\\}for the*admissible set*; by Assumption[A4](https://arxiv.org/html/2608.26187#Thmassumption4)\(b\) and Proposition[2](https://arxiv.org/html/2608.26187#Thmproposition2)it is a union of gauge components, and by Assumption[A5](https://arxiv.org/html/2608.26187#Thmassumption5)its members obey the size bound\.

###### Definition 2\(jump instance\)\.

SSis a*jump instance relative toℒ\\mathcal\{L\}*if:

- \(J1\)*Solvability:*Adm⁡\(S\)≠∅\\mathrm\{Adm\}\(S\)\\neq\\emptyset\.
- \(J2\)*Non\-canonicity:*Adm⁡\(S\)∩Canonℒ​\(F0\)=∅\\mathrm\{Adm\}\(S\)\\cap\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)=\\emptyset; equivalently, each operator outputΛ⁡\(𝖢,𝖢0,F0\)\\Lambda\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\),Λ∈ℒ\\Lambda\\in\\mathcal\{L\}, violates at least one constraint in𝒦\\mathcal\{K\}, which is decidable by\|ℒ\|\|\\mathcal\{L\}\|table checks\.
- \(J3\)*Identifiability:*Adm⁡\(S\)\\mathrm\{Adm\}\(S\)is a*single*gauge component; equivalently, any two admissible extensions differ only by a renaming of invented elements\. By Proposition[3](https://arxiv.org/html/2608.26187#Thmproposition3)this forcesAdm⁡\(S\)\\mathrm\{Adm\}\(S\)finite with a well\-defined*cardinality profile*nc:=s⁡\(F⁡\(c\)\)n\_\{c\}:=s\(F\(c\)\)on new objects; we requireN≥maxc⁡ncN\\geq\\max\_\{c\}n\_\{c\}\.
- \(J4\)*Support:*every new object is connected to the data,In⁡\(c\)≠∅\\mathrm\{In\}\(c\)\\neq\\emptysetorOut⁡\(c\)≠∅\\mathrm\{Out\}\(c\)\\neq\\emptyset\(strong form, recommended when the intended foil isLan∗\\mathrm\{Lan\}^\{\*\}:In⁡\(c\)≠∅\\mathrm\{In\}\(c\)\\neq\\emptysetfor every newcc\), so that no canonical value degenerates for want of data\.

“Jump instance” unqualified meansℒ=ℒKan\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{Kan\}\}\. Jump\-ness is monotone, because a larger preregistered library only removes jump instances; certifying relative to a largerℒ\\mathcal\{L\}is therefore strictly stronger\.

###### Definition 3\(control instance\)\.

S′S^\{\\prime\}is a*control instance targeting*Λ∈ℒ\\Lambda\\in\\mathcal\{L\}if it satisfies \(J1\), \(J3\), \(J4\) and, in place of \(J2\):Adm⁡\(S′\)=\[Λ⁡\(𝖢,𝖢0,F0\)\]\\mathrm\{Adm\}\(S^\{\\prime\}\)=\[\\Lambda\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)\]; the constraints pin exactly the canonical completion\. Controls are format\-identical to jump instances; only the constraint content differs\.

### 4\.4Chance

LetExtN​\(F0\):=\{F∈Ext⁡\(F0\):s⁡\(F⁡\(c\)\)≤N​on new objects\}\\mathrm\{Ext\}\_\{N\}\(F\_\{0\}\):=\\\{F\\in\\mathrm\{Ext\}\(F\_\{0\}\):s\(F\(c\)\)\\leq N\\ \\text\{on new objects\}\\\}denote the bounded answer space; it is a finite union of gauge components and containsAdm⁡\(S\)\\mathrm\{Adm\}\(S\)\. The*chance level*ofSSis

chance⁡\(S\):=\|Adm⁡\(S\)\|\|ExtN​\(F0\)\|,\\mathrm\{chance\}\(S\)\\;:=\\;\\frac\{\|\\mathrm\{Adm\}\(S\)\|\}\{\|\\mathrm\{Ext\}\_\{N\}\(F\_\{0\}\)\|\},counted at the level of strict tables \(the component\-level ratio is reported as a secondary statistic\)\. This is the success probability of the*format\-aware, constraint\-blind*null\. Such a guesser knows that the answer must be a bounded functor extending the data, which is everything the prompt format states, and it ignores𝒦\\mathcal\{K\}\. It upper\-bounds weaker nulls \(e\.g\. random not\-necessarily\-functorial tables\), so beating it beats them; a null that*used*𝒦\\mathcal\{K\}would presuppose the capability under test\. \(\|Ext⁡\(F0\)\|\|\\mathrm\{Ext\}\(F\_\{0\}\)\|itself is infinite because values on new objects are unbounded; this is whyNNis part of the instance\. The boundNNis a free parameter that deflates the denominator, so it must be reported and fixed per generator, e\.g\.N=maxN=\\maxof the admissible and canonical profiles\.\)

### 4\.5Jump capability: calibration, margin, Kan\-default rate

A learnerLLreceives a preregistered serialization of \(the tables of𝖢,𝖢0,F0\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}; the constraint text of𝒦\\mathcal\{K\}; the boundNN\) and outputs explicit tables of a functor; outputs outsideExtN​\(F0\)\\mathrm\{Ext\}\_\{N\}\(F\_\{0\}\)score00\(reported separately as format failures\)\. Success is exact membership in the finite setAdm⁡\(S\)\\mathrm\{Adm\}\(S\); by gauge\-saturation no renaming of invented elements affects the score\.

#### Calibration \(the ablated twin\)\.

For each instanceSSletS∘S^\{\\circ\}be the same serialization with the constraint block replaced by preregistered neutral filler\.L⁡\(S∘\)L\(S^\{\\circ\}\)is the learner’s*unconstrained default completion*of the data\. For a generator𝒢n\\mathcal\{G\}\_\{n\}of certified jump instances define

DCL\(n\):=ℙ\[L\(S∘\)∈Canonℒ\(F0\)\],KDL\(n\):=ℙ\[L\(S\)∈Canonℒ\(F0\)\],\\mathrm\{DC\}\_\{L\}\(n\):=\\mathbb\{P\}\\big\[L\(S^\{\\circ\}\)\\in\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)\\big\],\\qquad\\mathrm\{KD\}\_\{L\}\(n\):=\\mathbb\{P\}\\big\[L\(S\)\\in\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)\\big\],\(3\)the*default\-canonicity rate*and the*canonical\-default rate under constraints*\(forℒ=ℒKan\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{Kan\}\}: theKan\-default rate\), and the*override gap*ΔL​\(n\):=DCL​\(n\)−KDL​\(n\)\\Delta\_\{L\}\(n\):=\\mathrm\{DC\}\_\{L\}\(n\)\-\\mathrm\{KD\}\_\{L\}\(n\)\.

###### Definition 4\(calibrated jump capability\)\.

Fix preregisteredε\>0\\varepsilon\>0,κ,δ∈\[0,1\)\\kappa,\\delta\\in\[0,1\), a generator𝒢n\\mathcal\{G\}\_\{n\}of certifiedℒ\\mathcal\{L\}\-jump instances and a matched generator𝒢n′\\mathcal\{G\}^\{\\prime\}\_\{n\}of control instances\.LL*jumps at scalennwith marginε\\varepsilon, relative to\(ℒ,κ,δ\)\(\\mathcal\{L\},\\kappa,\\delta\)*, if:

- \(C0\)*Calibration validity:*DCL​\(n\)≥1−κ\\mathrm\{DC\}\_\{L\}\(n\)\\geq 1\-\\kappa\. If \(C0\) fails, the instrument is*inapplicable*toLLat this scale\. In that caseLLhas no measured canonical default to override, so no jump claim in either direction is licensed, and one reportsLL’s empirical default distribution over library components instead\.
- \(C1\)*Constrained success above chance, on calibrated instances:* ℙ⁡\[L⁡\(S\)∈Adm⁡\(S\)\|L⁡\(S∘\)∈Canonℒ​\(F0\)\]≥𝔼⁡\[chance⁡\(S\)\|L⁡\(S∘\)∈Canonℒ​\(F0\)\]\+ε\.\\mathbb\{P\}\\big\[L\(S\)\\in\\mathrm\{Adm\}\(S\)\\ \\big\|\\ L\(S^\{\\circ\}\)\\in\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)\\big\]\\;\\geq\\;\\mathbb\{E\}\\big\[\\mathrm\{chance\}\(S\)\\ \\big\|\\ L\(S^\{\\circ\}\)\\in\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)\\big\]\+\\varepsilon\.
- \(C2\)*Matched control:*ℙS′∼𝒢n′\[L\(S′\)∈Adm\(S′\)\]≥1−δ\\mathbb\{P\}\_\{S^\{\\prime\}\\sim\\mathcal\{G\}^\{\\prime\}\_\{n\}\}\\big\[L\(S^\{\\prime\}\)\\in\\mathrm\{Adm\}\(S^\{\\prime\}\)\\big\]\\geq 1\-\\delta\.

\(C2\) separates “cannot extend functorially at all” from “extends only canonically”;KDL\\mathrm\{KD\}\_\{L\}andΔL\\Delta\_\{L\}quantify the latter\. The falsifiable rendering of the “LLMs can’t jump” thesis at Tier 1 is*DC\\mathrm\{DC\}high,KD\\mathrm\{KD\}high,Δ≈0\\Delta\\approx 0, \(C2\) passed*\. The learner then has the canonical default, satisfies constraints when they pin the canon, and fails to leave the canon when the constraints falsify it\. The triple\(DC,KD,Δ\)\(\\mathrm\{DC\},\\mathrm\{KD\},\\Delta\)operationalizes the override step introduced in Section[1](https://arxiv.org/html/2608.26187#S1)\.

## 5Well\-posedness and existence

In this section we show that the definitions are consistent and that instances exist\. Specifically, we first show in Proposition[1](https://arxiv.org/html/2608.26187#Thmproposition1)that the canonical class is always defined, so that the comparison demanded by \(J2\) has a referent\. We then establish in Proposition[2](https://arxiv.org/html/2608.26187#Thmproposition2)that admissibility and scoring do not depend on the names of invented elements, which is what allows an answer to be graded structurally\. Building on these two facts, Proposition[3](https://arxiv.org/html/2608.26187#Thmproposition3)shows that identifiability makes the answer space finite, so that the chance level becomes computable\. With the definitions thus secured, Proposition[4](https://arxiv.org/html/2608.26187#Thmproposition4)exhibits a first certified instance, and Proposition[5](https://arxiv.org/html/2608.26187#Thmproposition5)fixes how the measurements of Section[7](https://arxiv.org/html/2608.26187#S7)must be read\. Finally, Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)extends the seed to an infinite family whose certificates need no enumeration\. All proofs are deferred to Appendix[A](https://arxiv.org/html/2608.26187#A1)\.

###### Proposition 1\(well\-posedness of the canonical class\)\.

Under Assumptions[A1](https://arxiv.org/html/2608.26187#Thmassumption1)–[A3](https://arxiv.org/html/2608.26187#Thmassumption3): \(i\)Lan⁡\(c\)\\mathrm\{Lan\}\(c\)andRan⁡\(c\)\\mathrm\{Ran\}\(c\)exist for everyccand are computed by the finite formulas \([1](https://arxiv.org/html/2608.26187#S3.E1)\)–\([2](https://arxiv.org/html/2608.26187#S3.E2)\); \(ii\) strict representativesLan∗,Ran∗∈Ext⁡\(F0\)\\mathrm\{Lan\}^\{\*\},\\mathrm\{Ran\}^\{\*\}\\in\\mathrm\{Ext\}\(F\_\{0\}\)exist; in particularExt⁡\(F0\)≠∅\\mathrm\{Ext\}\(F\_\{0\}\)\\neq\\emptysetfor every instance; \(iii\) the gauge components\[Lan∗\]\[\\mathrm\{Lan\}^\{\*\}\]and\[Ran∗\]\[\\mathrm\{Ran\}^\{\*\}\]depend neither on the chosen \(co\)limits nor on the strictification\.

###### Proposition 2\(gauge saturation\)\.

Under Assumption[A4](https://arxiv.org/html/2608.26187#Thmassumption4)\(b\),Adm⁡\(S\)\\mathrm\{Adm\}\(S\)is a union of gauge components; so areExtN​\(F0\)\\mathrm\{Ext\}\_\{N\}\(F\_\{0\}\)andCanonℒ​\(F0\)\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)\. Consequently \(J1\)–\(J3\),chance⁡\(S\)\\mathrm\{chance\}\(S\), and all scoring events of Section[4\.5](https://arxiv.org/html/2608.26187#S4.SS5)are gauge\-invariant\.

###### Proposition 3\(identifiability implies finiteness\)\.

For anyF∈Ext⁡\(F0\)F\\in\\mathrm\{Ext\}\(F\_\{0\}\), the gauge component\[F\]\[F\]equals the orbit ofFFunderGF=∏c∉𝖢0Aut𝖣​\(F⁡\(c\)\)G\_\{F\}=\\prod\_\{c\\notin\\mathsf\{C\}\_\{0\}\}\\mathrm\{Aut\}\_\{\\mathsf\{D\}\}\(F\(c\)\); all members share the object assignment; and\|\[F\]\|≤∏c\|Aut𝖣​\(F⁡\(c\)\)\|<∞\|\[F\]\|\\leq\\prod\_\{c\}\\lvert\\mathrm\{Aut\}\_\{\\mathsf\{D\}\}\(F\(c\)\)\\rvert<\\infty\. Hence under \(J3\),Adm⁡\(S\)\\mathrm\{Adm\}\(S\)is finite with a well\-defined cardinality profile\(nc\)\(n\_\{c\}\), andAdm⁡\(S\)⊆ExtN​\(F0\)\\mathrm\{Adm\}\(S\)\\subseteq\\mathrm\{Ext\}\_\{N\}\(F\_\{0\}\)iffN≥maxc⁡ncN\\geq\\max\_\{c\}n\_\{c\}\. In𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}:GF=∏cSym⁡\(nc\)G\_\{F\}=\\prod\_\{c\}\\mathrm\{Sym\}\(n\_\{c\}\)and\|\[F\]\|≤∏cnc\!\|\[F\]\|\\leq\\prod\_\{c\}n\_\{c\}\!\.

###### Proposition 4\(existence: the certified seed\)\.

Jump instances relative toℒKan\\mathcal\{L\}\_\{\\mathrm\{Kan\}\}exist\. Concretely, letS∗=\(𝖢,𝖢0,𝐅𝐢𝐧𝐒𝐞𝐭,F0,𝒦,4\)S^\{\\ast\}=\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},\\mathbf\{FinSet\},F\_\{0\},\\mathcal\{K\},4\)where𝖢\\mathsf\{C\}has objects\{a,b,c\}\\\{a,b,c\\\}and seven morphisms \(identities,f:a→bf\\colon a\\to b,g:b→cg\\colon b\\to c,h=g∘fh=g\\circ f, andt:b→bt\\colon b\\to bsubject tot∘f=ft\\circ f=f,g∘t=gg\\circ t=g,t∘t=idbt\\circ t=\\mathrm\{id\}\_\{b\}\);𝖢0\\mathsf\{C\}\_\{0\}is full on\{a,c\}\\\{a,c\\\};F0​\(a\)=F0​\(c\)=\[1\]F\_\{0\}\(a\)=F\_\{0\}\(c\)=\[1\]withF0​\(h\)F\_\{0\}\(h\)the unique map; and𝒦=\{K1:F\(t\)≠idF⁡\(b\),K2:\|F\(b\)\|≤3\}\\mathcal\{K\}=\\\{\\,\\mathrm\{K1\}\\colon F\(t\)\\neq\\mathrm\{id\}\_\{F\(b\)\},\\ \\mathrm\{K2\}\\colon\|F\(b\)\|\\leq 3\\,\\\}\. Then:

- \(a\)\|Ext4​\(F0\)\|=25\|\\mathrm\{Ext\}\_\{4\}\(F\_\{0\}\)\|=25;
- \(b\)Lan∗=Ran∗\\mathrm\{Lan\}^\{\*\}=\\mathrm\{Ran\}^\{\*\}is the trivial singleton extension \(F⁡\(b\)=\[1\]F\(b\)=\[1\], all maps trivial,F⁡\(t\)=idF\(t\)=\\mathrm\{id\}\);Kan⁡\(F0\)\\mathrm\{Kan\}\(F\_\{0\}\)is this single one\-element component, and it violates exactly K1;
- \(c\)Adm⁡\(S∗\)\\mathrm\{Adm\}\(S^\{\\ast\}\)is one gauge component of size33with forced profilenb=3n\_\{b\}=3: every admissibleFFhasF⁡\(f\)F\(f\)picking a pointp∈\[3\]p\\in\[3\],F⁡\(g\)F\(g\)constant, andF⁡\(t\)F\(t\)the involution fixingppand swapping the two elements outside the image ofF⁡\(f\)F\(f\); these two invented elements are invisible to the data and carry a nontrivialℤ/2\\mathbb\{Z\}/2symmetry present nowhere inF0F\_\{0\}or in either Kan extension;
- \(d\)\(J1\)–\(J4\) hold \(strong form of \(J4\)\), andchance⁡\(S∗\)=3/25=0\.12\\mathrm\{chance\}\(S^\{\\ast\}\)=3/25=0\.12\.

Proposition[4](https://arxiv.org/html/2608.26187#Thmproposition4)is the existence proof for the whole program and its first benchmark item\. Appendix[B](https://arxiv.org/html/2608.26187#A2)renders it as a prompt, and every instance of Section[5\.1](https://arxiv.org/html/2608.26187#S5.SS1)inherits its mechanism, one fixed point and one invented cycle at each hidden object\. A matched control forS∗S^\{\\ast\}replaces𝒦\\mathcal\{K\}by\{\|F\(b\)\|≤1\}\\\{\|F\(b\)\|\\leq 1\\\}; thenAdm=\[Lan∗\]\\mathrm\{Adm\}=\[\\mathrm\{Lan\}^\{\*\}\]and Definition[3](https://arxiv.org/html/2608.26187#Thmdefinition3)is satisfied with targetLan∗\\mathrm\{Lan\}^\{\*\}\. In this seed the two Kan extensions coincide, so the instance exhibits one canonical foil rather than two, and richer hom\-structure on𝖢0\\mathsf\{C\}\_\{0\}would separate them\. As a benchmark itemS∗S^\{\\ast\}is easy for frontier models, since the answer follows from a four\-step deduction; difficulty must come from a generator that scales\|𝖢\|\|\\mathsf\{C\}\|, the relations, and the constraint coupling under the integrity discipline of Appendix[C](https://arxiv.org/html/2608.26187#A3)\.

###### Proposition 5\(null learners\)\.

FixΛ∈ℒ\\Lambda\\in\\mathcal\{L\}and letLΛL\_\{\\Lambda\}outputΛ⁡\(𝖢,𝖢0,F0\)\\Lambda\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)on every input, ignoring the constraint text\. Then \(i\) on every control instance targetingΛ\\Lambda,LΛL\_\{\\Lambda\}scores11, so \(C2\) holds withδ=0\\delta=0; \(ii\) on everyℒ\\mathcal\{L\}\-jump instanceLΛL\_\{\\Lambda\}scores00whilechance⁡\(S\)\>0\\mathrm\{chance\}\(S\)\>0, so \(C1\) fails for everyε\>0\\varepsilon\>0; \(iii\)LΛL\_\{\\Lambda\}is perfectly calibrated:DC=KD=1\\mathrm\{DC\}=\\mathrm\{KD\}=1,Δ=0\\Delta=0\. The uniform guesser onExtN\\mathrm\{Ext\}\_\{N\}attains exactly the chance term\. Hence no constraint\-blind strategy in the null family\{uniform\}∪\{LΛ\}Λ∈ℒ\\\{\\text\{uniform\}\\\}\\cup\\\{L\_\{\\Lambda\}\\\}\_\{\\Lambda\\in\\mathcal\{L\}\}jumps, and the Kan\-default rate separates “solves canonically” \(highKD\\mathrm\{KD\}, controls passed\) from “cannot solve” \(controls failed\)\.

Proposition[5](https://arxiv.org/html/2608.26187#Thmproposition5)fixes how the measurements must be read\. A learner that always returns the canonical completion passes every control and fails every jump instance, so a high Kan\-default rate with controls passed means canonical solving, whereas failed controls mean inability to solve at all, and only the pair of arms separates the two\. This is why Section[7](https://arxiv.org/html/2608.26187#S7)reports controls and jumps together and why Section[8](https://arxiv.org/html/2608.26187#S8)can argue that enlarging the operator library cannot rescue a constraint\-blind strategy\.

### 5\.1The family theorem: certification without enumeration

The seed instance generalizes to an infinite family\. Form≥1m\\geq 1and primesp1,…,pmp\_\{1\},\\dots,p\_\{m\}, the*pointed chain*S⁡\(m,p→\)S\(m,\\vec\{p\}\\,\)has objectsa,b1,…,bm,ca,b\_\{1\},\\dots,b\_\{m\},c, connecting generatorsa→b1→⋯→bm→ca\\to b\_\{1\}\\to\\cdots\\to b\_\{m\}\\to c, and at each hidden object an endomorphismti:bi→bit\_\{i\}\\colon b\_\{i\}\\to b\_\{i\}subject toti∘\(incoming\)=incomingt\_\{i\}\\circ\(\\text\{incoming\}\)=\\text\{incoming\},\(outgoing\)∘ti=outgoing\(\\text\{outgoing\}\)\\circ t\_\{i\}=\\text\{outgoing\}, andtipi=idt\_\{i\}^\{p\_\{i\}\}=\\mathrm\{id\}; the data assigns singletons toaaandcc; the constraints areK1i:F⁡\(ti\)≠id\\mathrm\{K1\}\_\{i\}\\colon F\(t\_\{i\}\)\\neq\\mathrm\{id\}andK2i:\|F⁡\(bi\)\|≤1\+pi\\mathrm\{K2\}\_\{i\}\\colon\|F\(b\_\{i\}\)\|\\leq 1\+p\_\{i\}; and the default bound isN=maxi⁡\(1\+pi\)\+1N=\\max\_\{i\}\(1\+p\_\{i\}\)\+1; the released instance\(2,\(3,2\)\)\(2,\(3,2\)\)instead uses the minimal admissible boundN=4N=4\(Appendix[F](https://arxiv.org/html/2608.26187#A6)\)\. The seedS∗S^\{\\ast\}is the member\(m,p→\)=\(1,\(2\)\)\(m,\\vec\{p\}\\,\)=\(1,\(2\)\)\.

###### Theorem 1\(family theorem\)\.

For everym≥1m\\geq 1and all primesp1,…,pmp\_\{1\},\\dots,p\_\{m\}, the pointed chainS⁡\(m,p→\)S\(m,\\vec\{p\}\\,\)is a jump instance relative toℒKan\\mathcal\{L\}\_\{\\mathrm\{Kan\}\}with the strong form of \(J4\), and the following hold\.

- \(i\)Every admissible completion has, at each hidden object, exactly one fixed point together with onepip\_\{i\}\-cycle of invented elements, the incoming map is constant onto the fixed point, and the outgoing map is constant\.
- \(ii\)Adm\\mathrm\{Adm\}is a single gauge orbit of cardinality∏i\(1\+pi\)​\(pi−1\)\!\\prod\_\{i\}\(1\+p\_\{i\}\)\\,\(p\_\{i\}\-1\)\!\.
- \(iii\)Both Kan extensions equal the all\-singleton trivial extension and violate exactly the constraintsK1i\\mathrm\{K1\}\_\{i\}\.
- \(iv\)\|ExtN\|\|\\mathrm\{Ext\}\_\{N\}\|admits a closed\-form transfer recursion, sochance⁡\(S⁡\(m,p→\)\)\\mathrm\{chance\}\(S\(m,\\vec\{p\}\\,\)\)is computable without enumeration\.

Consequently certification of family instances requires no enumeration\. Primality is necessary: if somepip\_\{i\}is composite, identifiability \(J3\) fails\.

Two mechanisms drive the theorem\. The constraintK1i\\mathrm\{K1\}\_\{i\}forces an inventedpip\_\{i\}\-cycle at each hidden object, and primality makes that cycle indivisible, since a composite order would admit a shorter cycle that already satisfiestipi=idt\_\{i\}^\{p\_\{i\}\}=\\mathrm\{id\}, so completions with different cycle lengths would coexist and \(J3\) would fail\. The count in \(iv\) proceeds along the chain in transfer\-matrix style, tracking at each hidden object how many bounded choices remain compatible with the maps already fixed, which turns\|ExtN\|\|\\mathrm\{Ext\}\_\{N\}\|into a station\-by\-station recursion rather than an enumeration\.

The proof is given in Appendix[D](https://arxiv.org/html/2608.26187#A4); enumeration corroborates the closed forms at seven configurations, including\|Ext4\|=1,257,409\|\\mathrm\{Ext\}\_\{4\}\|=1\{,\}257\{,\}409and\|Adm\|=27\|\\mathrm\{Adm\}\|=27at\(3,\(2,2,2\)\)\(3,\(2,2,2\)\)\. The theorem is what makes the benchmark efficient\. Certification cost is paid once, in a proof, and instances of any difficulty can be regenerated with fresh vocabulary at no cost, so the test cannot be memorized\. In Section[7](https://arxiv.org/html/2608.26187#S7)the threem=3m=3instances, whose answer spaces exceed10610^\{6\}tables, are certified by this theorem alone, and it is there that the models first separate\. Algorithm[1](https://arxiv.org/html/2608.26187#alg1)summarizes both certification modes\.

Algorithm 1Instance certification1:instance

S=\(𝖢,𝖢0,𝖣,F0,𝒦,N\)S=\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},\\mathsf\{D\},F\_\{0\},\\mathcal\{K\},N\)
2:check A1–A5 on the tables; compute

Lan∗,Ran∗\\mathrm\{Lan\}^\{\*\},\\mathrm\{Ran\}^\{\*\}by formulas \([1](https://arxiv.org/html/2608.26187#S3.E1)\)–\([2](https://arxiv.org/html/2608.26187#S3.E2)\) and strictify

3:if

SSis a family member

S⁡\(m,p→\)S\(m,\\vec\{p\}\\,\)then

4:read

\|Adm\|=∏i\(1\+pi\)​\(pi−1\)\!\|\\mathrm\{Adm\}\|=\\prod\_\{i\}\(1\+p\_\{i\}\)\(p\_\{i\}\-1\)\!and

\|ExtN\|\|\\mathrm\{Ext\}\_\{N\}\|from Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)\(ii\),\(iv\)⊳\\trianglerightno enumeration

5:else

6:enumerate

ExtN​\(F0\)\\mathrm\{Ext\}\_\{N\}\(F\_\{0\}\); compute

Adm⁡\(S\)\\mathrm\{Adm\}\(S\)and its gauge orbits

7:endif

8:assert\(J1\)

Adm≠∅\\mathrm\{Adm\}\\neq\\emptyset; \(J2\)

Lan∗,Ran∗\\mathrm\{Lan\}^\{\*\},\\mathrm\{Ran\}^\{\*\}each violate some

P∈𝒦P\\in\\mathcal\{K\}; \(J3\)

Adm\\mathrm\{Adm\}is one orbit; \(J4\) support

9:report

chance⁡\(S\)\\mathrm\{chance\}\(S\)and per\-constraint non\-pinning statistics

## 6Correctness and chained jumps

The steps so far concern whether a model can produce the admissible completion\. Having shown in Section[5](https://arxiv.org/html/2608.26187#S5)that jump instances are well\-posed and certifiable, this section formalizes the remaining two steps, covering when an override is*correct*and what happens when jumps*compound*\. We first separate internal from predictive correctness by withholding part of the world from the model, and Proposition[7](https://arxiv.org/html/2608.26187#Thmproposition7)then exhibits an instance in which the two come apart\. We next let the certified answer of one stage become the data of the next, which Theorem[2](https://arxiv.org/html/2608.26187#Thmtheorem2)shows to be well\-defined\. Lemma[1](https://arxiv.org/html/2608.26187#Thmlemma1)then shows that accumulated knowledge only narrows what is admissible, and on this basis Theorem[3](https://arxiv.org/html/2608.26187#Thmtheorem3)exhibits a completion that is admissible at one stage yet excluded at the next\. The definitions extend Section[4](https://arxiv.org/html/2608.26187#S4)minimally, and all proofs are deferred to Appendix[E](https://arxiv.org/html/2608.26187#A5)\.

### 6\.1Internal versus predictive correctness

Internal correctness is membership inAdm⁡\(S\)\\mathrm\{Adm\}\(S\), and under \(J3\) it pins the completion up to gauge\. History demands a second notion, because Nordström’s scalar gravity was consistent with every constraint available in 1913 and was nevertheless wrong\. We model the future by a larger world that the model never sees\.

###### Definition 5\(extended instance\)\.

An*extended instance*is a tripleT=\(S,𝖢′,G\)T=\(S,\\mathsf\{C\}^\{\\prime\},G\)whereSSsatisfies A1–A5,𝖢′\\mathsf\{C\}^\{\\prime\}is a finite category containing𝖢\\mathsf\{C\}as a full subcategory \(objects outside𝖢\\mathsf\{C\}are*held out*\), andG:𝖢′→𝖣G\\colon\\mathsf\{C\}^\{\\prime\}\\to\\mathsf\{D\}restricts toF0F\_\{0\}on𝖢0\\mathsf\{C\}\_\{0\}\. The model receives onlySS; the pair\(𝖢′,G\)\(\\mathsf\{C\}^\{\\prime\},G\)is withheld\.TTis*sound*if the restriction ofGGto𝖢\\mathsf\{C\}is admissible forSS\.

###### Definition 6\(predictive correctness\)\.

F∈Ext⁡\(F0\)F\\in\\mathrm\{Ext\}\(F\_\{0\}\)is*predictively correct*forTTifFFextends to a functor on𝖢′\\mathsf\{C\}^\{\\prime\}that is isomorphic toGGby a natural isomorphism whose components are identities on𝖢0\\mathsf\{C\}\_\{0\}\.

###### Proposition 6\.

The predictively correct completions form exactly the gauge component of the restriction ofGGto𝖢\\mathsf\{C\}; the predicate is gauge\-invariant; and for a soundTTit is contained inAdm⁡\(S\)\\mathrm\{Adm\}\(S\)\. Under \(J3\), internal correctness therefore already pins the completion, and a gap between internal and predictive correctness can only arise when identifiability is relaxed\.

Proposition[6](https://arxiv.org/html/2608.26187#Thmproposition6)locates where predictive correctness can differ from internal correctness\. Under identifiability the two coincide, so the measurements of Section[7](https://arxiv.org/html/2608.26187#S7), which use identifiable instances, are unaffected, and the only way to stage a genuine divergence is to relax \(J3\) deliberately, which the next construction does\.

Write \(J3\.kk\) for the variant of \(J3\) in whichAdm⁡\(S\)\\mathrm\{Adm\}\(S\)is the union of exactlykkgauge components, and call such anSS*kk\-ambiguous*\.

###### Proposition 7\(separation\)\.

There is a sound extended instanceTsep=\(Ssep,𝖢′,G\)T\_\{\\mathrm\{sep\}\}=\(S\_\{\\mathrm\{sep\}\},\\mathsf\{C\}^\{\\prime\},G\)in whichSsepS\_\{\\mathrm\{sep\}\}is22\-ambiguous, both admissible components are internally correct, and exactly one is predictively correct\. Concretely,SsepS\_\{\\mathrm\{sep\}\}is the seed with its size bound relaxed from33to44; its admissible set is the union of the profile\-33component of the seed and a profile\-44component whose extra element is a data\-invisible fixed point; and the held\-out world contains probes whose recorded values are matched by the profile\-44component only\. Every morphism of𝖢′\\mathsf\{C\}^\{\\prime\}outside𝖢\\mathsf\{C\}has a held\-out endpoint, so the verifier lives strictly outside what the model sees\.

The separation result is the formal Nordström situation\. Nothing available insideSsepS\_\{\\mathrm\{sep\}\}distinguishes the two components, and the distinguishing evidence sits entirely in the withheld future\. It also delimits the instrument honestly, since a test built on internal correctness alone cannot detect this failure mode, and detecting it experimentally requires the held\-out machinery introduced here\.

### 6\.2Chains, monotonicity, and entrenchment

Historically the output of one jump becomes the background knowledge of the next, as special relativity did for general relativity\. We formalize this by letting the certified answer of one instance become the data of its successor\.

###### Definition 7\(incorporation; chain\)\.

A*successor scheme*forSSis a tripleT=\(𝖢′,𝒦′,N′\)T=\(\\mathsf\{C\}^\{\\prime\},\\mathcal\{K\}^\{\\prime\},N^\{\\prime\}\)with𝖢′\\mathsf\{C\}^\{\\prime\}a finite category containing𝖢\\mathsf\{C\}as a proper full subcategory and𝒦′\\mathcal\{K\}^\{\\prime\}a finite list of constraints in the A4 language over𝖢′\\mathsf\{C\}^\{\\prime\}entailing the boundN′N^\{\\prime\}on the new objects\. For a commitmentX∈Adm⁡\(S\)X\\in\\mathrm\{Adm\}\(S\), the*incorporated instance*isι⁡\(S,X,T\):=\(𝖢′,𝖢,𝖣,X,𝒦′,N′\)\\iota\(S,X,T\):=\(\\mathsf\{C\}^\{\\prime\},\\mathsf\{C\},\\mathsf\{D\},X,\\mathcal\{K\}^\{\\prime\},N^\{\\prime\}\); the hidden objects ofSShave become observed, withXXas the new data\. A*chain*iterates this construction, committing at every stage\.

###### Theorem 2\(incorporation is well\-defined\)\.

ι⁡\(S,X,T\)\\iota\(S,X,T\)satisfies A1–A5, and gauge\-equivalent commitments yield data\-isomorphic successor instances, with equal truth values of \(J1\)–\(J4\), equal chance, and corresponding admissible components of equal size\.

Theorem[2](https://arxiv.org/html/2608.26187#Thmtheorem2)is what makes the fourth step meaningful at all\. Incorporating a commitment produces an object of the same kind as the instances of Section[4](https://arxiv.org/html/2608.26187#S4), so every notion defined so far, from the canonical class to the chance level, applies verbatim at the next stage, and the choice of representative within a gauge component does not matter\. Without this theorem a chain would not be a sequence of jump instances\.

###### Lemma 1\(monotonicity\)\.

Along a chain, the stage\-kkadmissible sets transported to any later stage form a decreasing sequence, and the accumulated constraint system evaluated along the committed chain equals the last stage’s admissibility\. Accumulating knowledge never enlarges what is admissible\.

###### Definition 8\(retrospective and ultimate correctness\)\.

A commitmentX∈Adm⁡\(Sk\)X\\in\\mathrm\{Adm\}\(S\_\{k\}\)is*retrospectively correct*at stagel≥kl\\geq kif the chain re\-committed toXXat stagekkcan be continued through stagell; it is*ultimately correct*if it is retrospectively correct at every stage\. The sets of retrospectively correct commitments decrease inlland stabilize after finitely many stages \(Appendix[E](https://arxiv.org/html/2608.26187#A5)\)\.

###### Theorem 3\(entrenchment\)\.

There is a two\-stage chain scheme, built from the pointed\-chain family, in which stage one is22\-ambiguous with componentsY3Y\_\{3\}\(the tight seed component\) andY4Y\_\{4\}\(the same structure plus one data\-invisible spectator element\), and stage two attaches a single probe object with a limit constraint and a size bound\. Every commitment inY3Y\_\{3\}extends to stage two in exactly one admissible way, whereas every commitment inY4Y\_\{4\}admits no stage\-two extension at all\. Hence the retrospectively correct commitments are exactlyY3Y\_\{3\}, and the spectator component, admissible at stage one, is excluded at stage two\.

The spectator element is the formal ether, postulated consistently and eliminated by a later probe\. The construction licenses a measurable question that we deliberately leave to future work\. The question is whether a model that committed to the spectator component revises its commitment when the probe arrives or entrenches the error; the chain machinery above makes it well\-posed, and the existing harness can serve it without modification\.

## 7Experiments

Having certified instances and a calibrated capability to test, we now measure the second step\. We first validate the instrument, then report jump accuracy and the Kan\-default rate across difficulties, and finally read the results against the debate\.

#### Setup\.

We evaluate four frontier models on nine certified instances of the pointed\-chain family\. The models are GPT\-5\.6 Luna Pro, Claude Sonnet 5, Gemini 3\.1 Pro, and DeepSeek V4 Pro\. Six instances havem≤2m\\leq 2and are certified by enumeration, whereas the three instances withm=3m=3are certified by Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)\. Chance levels at the declared size bounds therefore range from0\.130\.13down to8×10−78\\times 10^\{\-7\}\. Each instance is rendered with nonce vocabulary inside a neutral cover story\. For every instance and model we draw jump, calibration, and control samples, one greedy and the rest at temperature0\.70\.7\. Each instance and model therefore contributes six jump, four calibration, and two control samples under the primary rendering, and the two instances with a second rendering contribute four further jump samples\. Per\-cell accuracies thus rest on six or ten samples, and a cell at1\.001\.00carries a Wilson 95% interval that extends down to0\.610\.61, so our conclusions rest on pooled counts rather than on differences between neighbouring cells\. A second rendering of two instances serves as a wording\-sensitivity arm\. Answers are graded by the structural characterization of Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)\(i\), so grading is exact and gauge\-invariant\. Finish reasons are also recorded, so that reasoning\-budget truncation is never confused with a substantive failure\. Appendix[F](https://arxiv.org/html/2608.26187#A6)lists all details, and Appendix[B](https://arxiv.org/html/2608.26187#A2)shows one complete rendered prompt\. The full evaluation cost under fifteen dollars\.

#### Instrument validation\.

With constraints ablated, the models output a Kan extension in98%98\\%of the calibration samples that were neither truncated nor empty \(DC=129/132\\mathrm\{DC\}=129/132, or96%96\\%when only truncation is excluded\), so the central modeling assumption is confirmed per model rather than assumed\. Per model the conditional rate is35/3635/36,36/3636/36,35/3635/36, and23/2423/24in the column order of Table[1](https://arxiv.org/html/2608.26187#S7.T1)\. Controls are passed at1\.001\.00in every cell except three DeepSeek answers atm=3m=3that were truncated or empty \(69/7269/72overall\)\. The wording arm shows no rendering sensitivity on the jump arm, with jump accuracy16/1616/16and14/1614/16on the second rendering\.

Table 1:Jump accuracy across instances and models, defined as the share of jump samples that land in the certified admissible component\. The Kan\-default rate is zero in every cell, TR marks cells whose truncation rate across the three arms exceeds0\.40\.4, and chance is at most0\.130\.13atm=1m=1, at most8\.4×10−38\.4\\times 10^\{\-3\}atm=2m=2, and at most2\.1×10−52\.1\\times 10^\{\-5\}atm=3m=3\.
#### Main results\.

Table[1](https://arxiv.org/html/2608.26187#S7.T1)reports jump accuracy\. The Kan\-default rateKD\\mathrm\{KD\}of Equation \([3](https://arxiv.org/html/2608.26187#S4.E3)\) is identically zero\. We summarize the findings as four insights\.

Insight 1: the unconstrained default is the Kan extension\.Across models and difficulties,98%98\\%of non\-truncated calibration answers coincide with the Kan default, which licenses the Kan\-default rate as a per\-model measure rather than an assumption\. The exceptions contain no alternative operator\. Of the fifteen non\-Kan calibration answers, twelve are empty or truncated outputs\. The remaining three are valid extensions\. Two of them coincide with the admissible completion and one is a different extension, so no second systematic default appears in the sample\.

Insight 2: no canonical lock\-in anywhere in the certified regime\.In all248248constrained trials, no model ever produced the Kan default when the constraints excluded it, at chance levels down to8×10−78\\times 10^\{\-7\}\. The incapacity thesis predicts that a model at the selection step patches the default rather than abandoning it\. That behavior did not occur once\. With zero events in 248 trials, the 95% upper bound on the Kan\-default rate is1\.2%1\.2\\%\. Decoding does not change the picture, since jump accuracy is41/4441/44under greedy decoding and185/204185/204at temperature0\.70\.7, and neither setting produced a single Kan answer\.

Insight 3: difficulty separates models through search, not through the default\.Atm=3m=3, Gemini 3\.1 Pro solves every sample of every instance\. GPT\-5\.6 and Sonnet 5 drop occasionally through constraint errors, one of which is a valid but inadmissible completion\. DeepSeek V4 collapses to0\.330\.33, since4242to50%50\\%of its samples across the three arms exhaust the reasoning budget and three further answers return empty\. Failures are search and budget phenomena; none is a reversion to the default\.

Insight 4: deployment effects are real and must be controlled\.Undersized output budgets silently bias results toward the default, because the default is the shortest answer\. Provider content filters matter as well\. The strictest filter of one vendor refused every rendering of the task, including a pure\-mathematics statement, whereas its middle filter answered all of them\. Both effects are recorded in the integrity checklist \(Appendix[C](https://arxiv.org/html/2608.26187#A3)\)\.

#### Exact\-search reference\.

An enumerate\-and\-check solver reproduces the admissible component of each enumeration\-certified instance in at most2\.12\.1seconds\. Form=3m=3, the closed\-form counts of Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)put the search spaces between1\.3×1061\.3\\times 10^\{6\}and2\.4×1082\.4\\times 10^\{8\}candidate extensions\. Exact search therefore solves the benchmark by construction\. The measured quantity is thus not whether the answer can be reached but whether the model overrides its own default in context \(Section[8](https://arxiv.org/html/2608.26187#S8)\)\.

#### Reading\.

Within the certified regime, the models jump\. The selection step is not the bottleneck\. If the disputed incapacity exists, these measurements locate it upstream\. It would then lie in generating the constraints or in inventing the framework\. The theory of Section[6](https://arxiv.org/html/2608.26187#S6)sets up the next measurements at those two steps\.

## 8Scope, protocol, and limitations

#### What a pass means\.

We do not formalize the jump of[34](https://arxiv.org/html/2608.26187#bib.bib1)\. Our test measures one capability\. The learner must override its own*measured*canonical default in favor of a non\-canonical extension that the constraints pin down\. There is zero paired supervision on the extended part, sinceF0F\_\{0\}carries no values on new objects and information about them enters only through𝒦\\mathcal\{K\}\. Passing is therefore a machine\-checkable necessary trace of abduction as selection, and it says nothing about abduction as generation\. A symbolic solver passing Tier 1 is unproblematic, because the diagnostic object is a learner’s default\-versus\-override gap\(DC,KD,Δ\)\(\\mathrm\{DC\},\\mathrm\{KD\},\\Delta\)and solver baselines calibrate how much raw search suffices\. Tier 1 yields a categorical*instrument*relative to\(ℒ,learner\)\(\\mathcal\{L\},\\text\{learner\}\)rather than a learner\-independent definition of the jump\. We formalize four steps of the jump and measure only the second\. The propositions of Section[6](https://arxiv.org/html/2608.26187#S6)license future measurements of the later steps without claiming any\.

#### What is bracketed; Tier 2\.

Two parts of the full jump remain outside the instrument\. The constraints𝒦\\mathcal\{K\}are given rather than generated from an anomaly, and the embodiment thesis\([17](https://arxiv.org/html/2608.26187#bib.bib20);[34](https://arxiv.org/html/2608.26187#bib.bib1)\)concerns that generation step\. The codomain𝖣\\mathsf\{D\}is handed over, whereas the historically operative jump invented its codomain\. Fixing𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}is nevertheless not an arbitrary restriction\. Every finite category admits a faithful Cayley\-style functor into𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}, so finite structured codomains such as groups or posets can be rewritten over it\. The fixed skeleton of𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}, which keeps one representative set per cardinality, also makes the canonical completions computable in the first place\.𝐅𝐢𝐧𝐒𝐞𝐭\\mathbf\{FinSet\}cannot supply a genuinely new codomain\. By design, that object belongs to Tier 2\. The absence of a verifier at invention time is partially represented\. Section[6](https://arxiv.org/html/2608.26187#S6)withholds the verifier from the model rather than removing it, which is the strongest version that remains experimentally meaningful\. The instrument likewise leaves out what triggers the search and what the searcher intends\. Tier 2 hosts the jump proper\. Its instances do not fix𝖣\\mathsf\{D\}, so admissibility is judged over all interpretations of the codomain rather than over a fixed one\. Tier 1 is its control arm and instrument validation\.

#### Certification and integrity discipline\.

DecidingAdm≠∅\\mathrm\{Adm\}\\neq\\emptysetis NP\-hard and computing\|ExtN\|\|\\mathrm\{Ext\}\_\{N\}\|is \#P\-hard in general families\. Generators must therefore*plant*solutions and certify \(J1\)–\(J4\) andchance\\mathrm\{chance\}by enumeration at benchmark scale, which Assumption[A5](https://arxiv.org/html/2608.26187#Thmassumption5)keeps finite\. Appendix[C](https://arxiv.org/html/2608.26187#A3)collects the preregistered integrity requirements\. They cover non\-pinning by any single constraint, certificates of propagation hardness, solver and graph\-matching baselines, controls for memorization, format, and length, and a dissociation of performance from search depth\. The released certifier also records, for every constraint, the fraction of the bounded answer space that survives that constraint alone, against a preregistered floor of10%10\\%\. Six of the nine instances clear the floor on every constraint, and\(2,\(2,3\)\)\(2,\(2,3\)\),\(3,\(2,3,3\)\)\(3,\(2,3,3\)\), and\(3,\(3,3,3\)\)\(3,\(3,3,3\)\)fall below it on one or two size constraints, down to1\.4%1\.4\\%, which the release flags\. We retain them because no single constraint comes within a factor of eighty of isolating the admissible component, so the conjunction is still what does the work\.

#### Limitations\.

Two failure modes could invalidate the program and must be monitored as primary statistics\. First, the certifiable and diagnostic regimes may not intersect\. Instances small enough for brute\-force certification may be solvable by shallow search that never exercises a default override\. By contrast, diagnostically hard instances resist certification and contamination audits\. Scaling certification past brute force is therefore the open engineering problem\. Second, calibration may reveal that a frontier learner’s unconstrained default lies outside the preregistered libraryℒ\\mathcal\{L\}altogether\. The Kan\-default rate would then measure deviation from a baseline the learner never had, so the calibration\-validity condition \(C0\) would fail\. In that case only the per\-learner calibrated analysis would be licensed\. Both failure modes are stated as preregistered stopping conditions rather than post\-hoc excuses\.

#### Library relativity and next measurements\.

Non\-canonicity is defined relative to the preregistered libraryℒ\\mathcal\{L\}, and our instantiation takesℒ=\{Lan,Ran\}\\mathcal\{L\}=\\\{\\mathrm\{Lan\},\\mathrm\{Ran\}\\\}\. The choice does not carry the conclusion\. A constraint\-blind operator returns a single completion of the shared data, while a matched jump and control pair pins two different answers\. Passing both arms therefore refutes every such operator at once \(Proposition[5](https://arxiv.org/html/2608.26187#Thmproposition5)\)\. The calibration sample moreover contains no second systematic default that an enlarged library could capture \(Section[7](https://arxiv.org/html/2608.26187#S7)\)\. On the pointed\-chain family the two Kan extensions coincide, so calibration certifies that the default is canonical without revealing which of the two operators a model prefers, and instances withLan≠Ran\\mathrm\{Lan\}\\neq\\mathrm\{Ran\}are the natural next calibration\. Two measurements come next\. The first uses open\-weight models, whose default distribution is readable from log probabilities\. The second is the staged chain experiment licensed by Section[6](https://arxiv.org/html/2608.26187#S6), in which a stage\-one commitment meets a stage\-two probe\. Entrenchment is then read off as a stage\-two answer that retains the stage\-one spectator element\.

## References

- Bradleyet al\.\(2022\)T\. Bradley, J\. Terilla, and Y\. VlassopoulosAn enriched category theory of language: from syntax to semantics\.La Matematica1,pp\. 551–580\.Note:arXiv:2106\.07890Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1)\.
- Caterina and Gangle \(2016\)G\. Caterina and R\. GangleIconicity and abduction\.SAPERE, Vol\.29,Springer\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px3.p1.1)\.
- Chollet \(2019\)F\. CholletOn the measure of intelligence\.Note:arXiv:1911\.01547Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Coeckeet al\.\(2010\)B\. Coecke, M\. Sadrzadeh, and S\. ClarkMathematical foundations for a compositional distributional model of meaning\.Linguistic Analysis36,pp\. 345–384\.Note:arXiv:1003\.4394Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1)\.
- Cooper and Velasquez \(2026\)P\. Cooper and A\. VelasquezDeFAb: a verifiable benchmark for defeasible abduction in foundation models\.arXiv preprint arXiv:2606\.18557\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Dziriet al\.\(2023\)N\. Dziri, X\. Lu, M\. Sclar, X\. L\. Li, L\. Jiang, B\. Y\. Lin, P\. West, C\. Bhagavatula, R\. Le Bras, J\. D\. Hwang,et al\.Faith and fate: limits of transformers on compositionality\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Note:arXiv:2305\.18654Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Eppeet al\.\(2018\)M\. Eppe, E\. Maclean, R\. Confalonieri, O\. Kutz, M\. Schorlemmer, E\. Plaza, and K\. KühnbergerA computational framework for conceptual blending\.Artificial Intelligence256,pp\. 105–129\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px3.p1.1)\.
- Farmer \(2026\)M\. W\. FarmerAbduction without a body? representational grounding and the abduction loop for scientific hypothesis generation\.Note:arXiv:2608\.02505Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p1.1),[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px1.p1.1)\.
- Fonget al\.\(2019\)B\. Fong, D\. I\. Spivak, and R\. TuyérasBackprop as functor: a compositional perspective on supervised learning\.InLogic in Computer Science \(LICS\),Note:arXiv:1711\.10455Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1)\.
- Gangloff \(2026\)S\. GangloffAssessing LLMs’ mathematical abilities requires understanding the various mechanisms of mathematical creativity\.arXiv preprint arXiv:2608\.16118\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px1.p1.1)\.
- Gavranovićet al\.\(2024\)B\. Gavranović, P\. Lessard, A\. Dudzik, T\. von Glehn, J\. G\. M\. Araújo, and P\. VeličkovićPosition: categorical deep learning is an algebraic theory of all architectures\.InInternational Conference on Machine Learning \(ICML\),Note:arXiv:2402\.15332Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1)\.
- Goguen \(1999\)J\. GoguenAn introduction to algebraic semiotics, with application to user interface design\.InComputation for Metaphors, Analogy, and Agents,LNAI, Vol\.1562\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px3.p1.1)\.
- Kelly \(1982\)G\. M\. KellyBasic concepts of enriched category theory\.Cambridge University Press\.Cited by:[§3](https://arxiv.org/html/2608.26187#S3.p1.1)\.
- Kemp and Tenenbaum \(2008\)C\. Kemp and J\. B\. TenenbaumThe discovery of structural form\.Proceedings of the National Academy of Sciences105\(31\),pp\. 10687–10692\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px3.p1.1)\.
- Lake and Baroni \(2018\)B\. M\. Lake and M\. BaroniGeneralization without systematicity: on the compositional skills of sequence\-to\-sequence recurrent networks\.InInternational Conference on Machine Learning \(ICML\),Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Mac Lane \(1998\)S\. Mac LaneCategories for the working mathematician\.2nd edition,Springer\.Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p4.1),[§3](https://arxiv.org/html/2608.26187#S3.p1.1),[footnote 3](https://arxiv.org/html/2608.26187#footnote3)\.
- Magnani \(2009\)L\. MagnaniAbductive cognition: the epistemological and eco\-cognitive dimensions of hypothetical reasoning\.Cognitive Systems Monographs, Vol\.3,Springer\.Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p3.1),[§8](https://arxiv.org/html/2608.26187#S8.SS0.SSS0.Px2.p1.1)\.
- Mahadevan \(2026\)S\. MahadevanLearning in infinitesimal non\-compositional sketches\.arXiv preprint arXiv:2607\.15107\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px3.p1.1)\.
- Mészároset al\.\(2024\)A\. Mészáros, S\. Ujváry, W\. Brendel, P\. Reizinger, and F\. HuszárRule extrapolation in language modeling: a study of compositional generalization on OOD prompts\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Minegishiet al\.\(2026\)G\. Minegishi, J\. Feng, H\. Furuta, T\. Kojima, Y\. Iwasawa, and Y\. MatsuoEmergent analogical reasoning in transformers\.InInternational Conference on Machine Learning \(ICML\),Note:arXiv:2602\.01992Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1)\.
- Moskvichevet al\.\(2023\)A\. Moskvichev, V\. V\. Odouard, and M\. MitchellThe ConceptARC benchmark: evaluating understanding and generalization in the ARC domain\.Transactions on Machine Learning Research\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Murphy \(2026\)K\. MurphyModel discovery agent: LLM\-assisted Bayesian experiment design for data\-efficient discovery of mechanistic world models\.arXiv preprint arXiv:2608\.09696\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px1.p1.1)\.
- Norton \(2020\)J\. D\. NortonEinstein’s pathway to general relativity\.Note:[https://sites\.pitt\.edu/~jdnorton/teaching/HPS\_0410/chapters/general\_relativity\_pathway/index\.html](https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/index.html)Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p4.1)\.
- Novikovet al\.\(2025\)A\. Novikov, N\. Vũ, M\. Eisenberger, E\. Dupont, P\.\-S\. Huang, A\. Z\. Wagner, S\. Shirobokov, B\. Kozlovskii, F\. J\. R\. Ruiz, A\. Mehrabian, M\. P\. Kumar, A\. See, S\. Chaudhuri, G\. Holland, A\. Davies, S\. Nowozin, P\. Kohli, and M\. BalogAlphaEvolve: a coding agent for scientific and algorithmic discovery\.Note:arXiv:2506\.13131Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p1.1),[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px1.p1.1)\.
- Peirce \(1934\)C\. S\. PeirceCollected papers of charles sanders peirce\.Harvard University Press\.Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p1.1)\.
- Penget al\.\(2025\)Y\. Peng, Y\. Liu, E\. Xia, Y\. Jin, W\. Dai, Z\. Ren, Y\. Ding, and K\. ZhouAbductive logical rule induction by bridging inductive logic programming and multimodal large language models\.arXiv preprint arXiv:2509\.21874\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Phillips and Wilson \(2010\)S\. Phillips and W\. H\. WilsonCategorial compositionality: a category theory explanation for the systematicity of human cognition\.PLoS Computational Biology6\(7\),pp\. e1000858\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px4.p1.1)\.
- Pughet al\.\(2025\)M\. Pugh, J\. Grundy, C\. Cirstea, and N\. HarrisLearning is a kan extension\.Note:arXiv:2502\.13810Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p4.1),[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1),[§3](https://arxiv.org/html/2608.26187#S3.SS0.SSS0.Px5.p1.1)\.
- Saldaña\-Ulloa \(2026\)D\. Saldaña\-UlloaStratified discovery graphs: decomposing abductive discovery into bridge selection, concept synthesis, and axiom retraction\.Note:SSRN preprint 7257319Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px1.p1.1)\.
- Shiebler \(2022\)D\. ShieblerKan extensions in data science and machine learning\.Note:arXiv:2203\.09018Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p4.1),[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1),[§3](https://arxiv.org/html/2608.26187#S3.SS0.SSS0.Px5.p1.1)\.
- Tohméet al\.\(2015\)F\. Tohmé, G\. Caterina, and R\. GangleAbduction: a categorical characterization\.Journal of Applied Logic13\(1\),pp\. 78–90\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px3.p1.1)\.
- Wiggins \(2006\)G\. A\. WigginsA preliminary framework for description, analysis and comparison of creative systems\.Knowledge\-Based Systems19\(7\),pp\. 449–458\.Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px3.p1.1)\.
- Yuan \(2023\)Y\. YuanOn the power of foundation models\.InInternational Conference on Machine Learning \(ICML\),Note:arXiv:2211\.16327Cited by:[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px2.p1.1)\.
- Zahavy \(2026\)T\. ZahavyPosition: LLMs can’t jump\.Note:ICML 2026 Position Track; also PhilSci\-Archive 28024External Links:[Link](https://www.tomzahavy.com/files/llms-cant-jump.pdf)Cited by:[§1](https://arxiv.org/html/2608.26187#S1.p1.1),[§1](https://arxiv.org/html/2608.26187#S1.p8.1),[§2](https://arxiv.org/html/2608.26187#S2.SS0.SSS0.Px1.p1.1),[§8](https://arxiv.org/html/2608.26187#S8.SS0.SSS0.Px1.p1.1),[§8](https://arxiv.org/html/2608.26187#S8.SS0.SSS0.Px2.p1.1),[When the Canonical Completion Is Wrong: Formalizing and Measuring the Jump in Large Language Models](https://arxiv.org/html/2608.26187#p1.pic1.1.1.1.1)\.

## Appendix ADeferred proofs

###### Proof of Proposition[1](https://arxiv.org/html/2608.26187#Thmproposition1)\.

\(i\) The index categories overIn⁡\(c\)\\mathrm\{In\}\(c\)andOut⁡\(c\)\\mathrm\{Out\}\(c\)are finite by Assumption[A1](https://arxiv.org/html/2608.26187#Thmassumption1), and𝖣\\mathsf\{D\}has chosen finite colimits and limits by Assumption[A3](https://arxiv.org/html/2608.26187#Thmassumption3)\(i\); \([1](https://arxiv.org/html/2608.26187#S3.E1)\) and \([2](https://arxiv.org/html/2608.26187#S3.E2)\) are those \(co\)limits of the diagram\(a,v\)↦F0​\(a\)\(a,v\)\\mapsto F\_\{0\}\(a\), with functorial action by relabeling routes\. \(ii\) Forc0∈𝖢0c\_\{0\}\\in\\mathsf\{C\}\_\{0\}the index\(c0,id\)\(c\_\{0\},\\mathrm\{id\}\)is terminal amongIn⁡\(c0\)\\mathrm\{In\}\(c\_\{0\}\): for any\(a,v\)\(a,v\), fullness \(Assumption[A2](https://arxiv.org/html/2608.26187#Thmassumption2)\) putsvvin𝖢0\\mathsf\{C\}\_\{0\}andu:=vu:=vis the unique morphism\(a,v\)→\(c0,id\)\(a,v\)\\to\(c\_\{0\},\\mathrm\{id\}\)\. A colimit over a category with a terminal object is the value there, giving a natural isomorphismηc0:F0​\(c0\)→Lan⁡\(c0\)\\eta\_\{c\_\{0\}\}\\colon F\_\{0\}\(c\_\{0\}\)\\to\\mathrm\{Lan\}\(c\_\{0\}\)\. SinceKKis injective on objects, defineLan∗​\(c\):=F0​\(c\)\\mathrm\{Lan\}^\{\*\}\(c\):=F\_\{0\}\(c\)forc∈𝖢0c\\in\\mathsf\{C\}\_\{0\}and:=Lan⁡\(c\):=\\mathrm\{Lan\}\(c\)otherwise, and transport morphism tables by conjugation withθc:=ηc\\theta\_\{c\}:=\\eta\_\{c\}\(c∈𝖢0c\\in\\mathsf\{C\}\_\{0\}\),:=id:=\\mathrm\{id\}\(new\):Lan∗​\(f\):=θcod−1∘Lan⁡\(f\)∘θdom\\mathrm\{Lan\}^\{\*\}\(f\):=\\theta^\{\-1\}\_\{\\mathrm\{cod\}\}\\circ\\mathrm\{Lan\}\(f\)\\circ\\theta\_\{\\mathrm\{dom\}\}\. Functoriality telescopes; forffin𝖢0\\mathsf\{C\}\_\{0\}, naturality ofη\\etagivesLan∗​\(f\)=F0​\(f\)\\mathrm\{Lan\}^\{\*\}\(f\)=F\_\{0\}\(f\), soLan∗∘K=F0\\mathrm\{Lan\}^\{\*\}\\circ K=F\_\{0\}on the nose\. DuallyRan∗\\mathrm\{Ran\}^\{\*\}via the counit\. \(iii\) Two colimit choices are related by unique comparison isomorphismsγc\\gamma\_\{c\}commuting with the colimit injections;γ\\gammais natural andγ∘η=η′\\gamma\\circ\\eta=\\eta^\{\\prime\}sinceη\\etais the injection at\(c0,id\)\(c\_\{0\},\\mathrm\{id\}\); the induced isomorphism between the two strictifications has componentθc′−1​γc​θc\\theta^\{\\prime\-1\}\_\{c\}\\gamma\_\{c\}\\theta\_\{c\}, which is the identity on𝖢0\\mathsf\{C\}\_\{0\}; it is therefore a gauge equivalence\. The strictification itself involves no further choice\. ∎

###### Proof of Proposition[2](https://arxiv.org/html/2608.26187#Thmproposition2)\.

A gauge transformationσ\\sigmais a natural isomorphismF≅σ⋅FF\\cong\\sigma\\cdot Frestricting to the identity on𝖢0\\mathsf\{C\}\_\{0\}, so eachP∈𝒦P\\in\\mathcal\{K\}takes the same value onFFandσ⋅F\\sigma\\cdot F; gauge fixes object assignments, hence sizes, saturatingExtN\\mathrm\{Ext\}\_\{N\};Canonℒ\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}is a union of components by Definition[1](https://arxiv.org/html/2608.26187#Thmdefinition1)\. ∎

###### Proof of Proposition[3](https://arxiv.org/html/2608.26187#Thmproposition3)\.

Ifα:F⇒F′\\alpha\\colon F\\Rightarrow F^\{\\prime\}is a natural isomorphism with identity components on𝖢0\\mathsf\{C\}\_\{0\}, skeletality \(Assumption[A3](https://arxiv.org/html/2608.26187#Thmassumption3)\(iii\)\) forcesF′​\(c\)=F​\(c\)F^\{\\prime\}\(c\)=F\(c\)for allcc, soα\\alphais a gauge tuple and naturality readsF′=α⋅FF^\{\\prime\}=\\alpha\\cdot F\. Conversely eachσ∈GF\\sigma\\in G\_\{F\}is such an isomorphismF⇒σ⋅FF\\Rightarrow\\sigma\\cdot F\(conjugation preserves the functor equations\)\. So the rooted\-isomorphism relation is the gauge\-orbit relation; finiteness is the orbit\-size bound with finite hom\-sets \(Assumption[A3](https://arxiv.org/html/2608.26187#Thmassumption3)\(iv\)\)\. ∎

###### Proof of Proposition[4](https://arxiv.org/html/2608.26187#Thmproposition4)\.

An extension is determined byn=\|F⁡\(b\)\|≥1n=\|F\(b\)\|\\geq 1\(n=0n=0is impossible: no map\[1\]→\[0\]\[1\]\\to\[0\]\), the pointppin the image ofF⁡\(f\)F\(f\), andF⁡\(t\)F\(t\), sinceF⁡\(g\):\[n\]→\[1\]F\(g\)\\colon\[n\]\\to\[1\]andF⁡\(h\)F\(h\)are forced\. The relations say exactly:F​\(t\)​\(p\)=pF\(t\)\(p\)=p,F​\(t\)2=idF\(t\)^\{2\}=\\mathrm\{id\}, nothing more\. WritingI⁡\(m\)I\(m\)for the number of involutions onmmpoints \(I⁡\(0\)=I⁡\(1\)=1I\(0\)=I\(1\)=1,I⁡\(2\)=2I\(2\)=2,I⁡\(3\)=4I\(3\)=4\),\|Ext4\|=∑n=14n⋅I⁡\(n−1\)=1\+2\+6\+16=25\|\\mathrm\{Ext\}\_\{4\}\|=\\sum\_\{n=1\}^\{4\}n\\cdot I\(n\-1\)=1\+2\+6\+16=25, giving \(a\)\. For \(b\): the only morphisma→ba\\to bisff\(sincet∘f=ft\\circ f=f\) andHom⁡\(c,b\)=∅\\mathrm\{Hom\}\(c,b\)=\\emptyset, soIn⁡\(b\)=\{\(a,f\)\}\\mathrm\{In\}\(b\)=\\\{\(a,f\)\\\}andLan⁡\(b\)=F0​\(a\)=\[1\]\\mathrm\{Lan\}\(b\)=F\_\{0\}\(a\)=\[1\]; duallyOut⁡\(b\)=\{\(c,g\)\}\\mathrm\{Out\}\(b\)=\\\{\(c,g\)\\\}andRan⁡\(b\)=\[1\]\\mathrm\{Ran\}\(b\)=\[1\]; all structure maps through\[1\]\[1\]are unique, the strict representatives coincide, andF⁡\(t\)=id\[1\]F\(t\)=\\mathrm\{id\}\_\{\[1\]\}violates K1 only\. For \(c\): K1 forces a moved point; moved points of an involution come in22\-cycles;ppis fixed, son≥3n\\geq 3; K2 capsn=3n=3\(and𝒦\\mathcal\{K\}entails the boundN=4N=4, Assumption[A5](https://arxiv.org/html/2608.26187#Thmassumption5)\)\. Forn=3n=3the admissibleF⁡\(t\)F\(t\)is exactly the transposition of\[3\]∖\{p\}\[3\]\\setminus\\\{p\\\}: three tables, one perpp\. Transitivity of the gauge actionσ⋅Fp=Fσ⁡\(p\)\\sigma\\cdot F\_\{p\}=F\_\{\\sigma\(p\)\}\(σ∈Sym⁡\(3\)\\sigma\\in\\mathrm\{Sym\}\(3\)\) gives one component \(stabilizer of order22; orbit6/2=36/2=3\)\. Minimality is forced, not stipulated:n=2n=2admits no admissible extension\. \(J4\) strong form:In⁡\(b\)≠∅\\mathrm\{In\}\(b\)\\neq\\emptyset\. All quantities are independently machine\-verified by exhaustive enumeration in the released certifier\. ∎

###### Proof of Proposition[5](https://arxiv.org/html/2608.26187#Thmproposition5)\.

\(i\)LΛ​\(S′\)=Λ⁡\(𝖢,𝖢0,F0\)∈\[Λ⁡\(𝖢,𝖢0,F0\)\]=Adm⁡\(S′\)L\_\{\\Lambda\}\(S^\{\\prime\}\)=\\Lambda\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)\\in\[\\Lambda\(\\mathsf\{C\},\\mathsf\{C\}\_\{0\},F\_\{0\}\)\]=\\mathrm\{Adm\}\(S^\{\\prime\}\)\. \(ii\) By \(J2\) the output lies inCanonℒ\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}, disjoint fromAdm\\mathrm\{Adm\}; by \(J1\),chance⁡\(S\)\>0\\mathrm\{chance\}\(S\)\>0\. \(iii\)LΛ​\(S∘\)=LΛ​\(S\)∈Canonℒ​\(F0\)L\_\{\\Lambda\}\(S^\{\\circ\}\)=L\_\{\\Lambda\}\(S\)\\in\\mathrm\{Canon\}\_\{\\mathcal\{L\}\}\(F\_\{0\}\)always\. ∎

## Appendix BThe certified seed as a benchmark item

This appendix renders the instanceS∗S^\{\\ast\}of Proposition[4](https://arxiv.org/html/2608.26187#Thmproposition4)with nonce vocabulary and no mathematical jargon\. The three stages NARV, QUILB, SORM realizea,b,ca,b,c; the steps dax, rell, fen realizef,g,tf,g,t; rules R1–R3 are the relationst∘f=ft\\circ f=f,g∘t=gg\\circ t=g,t2=idt^\{2\}=\\mathrm\{id\}; observations fixF0F\_\{0\}; requirements D1–D2 are𝒦\\mathcal\{K\}; the answer\-format bound realizesN=4N=4\.

> You are completing the design of a small three\-stage signal pipeline\. The pipeline has three stages: NARV, QUILB, and SORM\. Signals flow NARV \-\> QUILB \-\> SORM\. There are three processing steps: "dax" carries each NARV state to a QUILB state; "rell" carries each QUILB state to a SORM state; "fen" is an internal rearrangement of QUILB\. Wiring rules \(must hold exactly, state by state\): R1\. Running dax and then fen lands on the same state as running dax alone\. R2\. Running fen and then rell gives the same result as running rell alone\. R3\. Running fen twice leaves every QUILB state where it started\. Recorded observations \(fixed\): NARV has exactly one state, n1\. SORM has exactly one state, s1\. The end\-to\-end pipeline sends n1 to s1\. Engineering requirements: D1\. QUILB has at most 3 states\. D2\. fen is NOT the do\-nothing map\. Answer format bound: designs with up to 4 QUILB states are well\-formed answers\. Give one complete implementation: the set QUILB and full tables for dax, rell, fen\.

An answer is correct precisely when it lies in the single admissible gauge orbit, whose representative has QUILB=\{p,q,r\}=\\\{p,q,r\\\}with dax:n​1↦p\\colon n1\\mapsto p, rell constant, and fen fixingppwhile swappingqqandrr\. The Kan\-default answer is the singleton QUILB with fen the identity; it reproduces every observation, violates exactly D2, and is the diagnostic failure mode\. The matched control replaces D1–D2 by the single requirement that QUILB has at most one state, and its unique admissible answer is then the Kan extension itself\.

## Appendix CGenerator integrity discipline

Per instance or instance family, preregistered and reported:

1. 1\.*Individual non\-pinning:*each single constraint leaves a reported fraction ofExtN\\mathrm\{Ext\}\_\{N\}alive, compared against a10%10\\%floor in the release; only the conjunction pins the answer\.
2. 2\.*Propagation\-hardness:*unit propagation / arc\-consistency on the induced constraint\-satisfaction problem fixes no value at any new object; the constraints bite only globally\.
3. 3\.*Reading\-comprehension baseline:*a generic CP/SAT solver on the identical constraint text at matched budget; propagation\-solvable instances are excluded from the jump set\.
4. 4\.*Structure\-matching baseline:*Weisfeiler–Leman / Gromov–Wasserstein matchers run on every instance; jump surplus is reported only on instances certified matcher\-unsolvable yet constraint\-pinned\.
5. 5\.*Memorization controls:*nonce labels, shuffled presentation, canonicalization against catalogs of named small categories, held\-out generator seeds\.
6. 6\.*Format controls:*multiple preregistered serializations; median and minimum reported; jump and control instances format\-matched; ablated twins matched for length\.
7. 7\.*Length controls:*canonical and admissible answers matched in description length;*anti\-jump*instances \(constraints pin the canon against a length\-matched non\-canonical distractor\) detect brevity or novelty priors; the2×22\\times 2constraint\-conditional flip rate is reported\.
8. 8\.*Search\-depth dissociation:*shallow non\-canonical instances \(admissible answer reachable by unit propagation\) versus depth\-matched controls, so canonicity lock\-in is separable from search\-depth limits\.
9. 9\.*Output\-budget adequacy:*record finish reasons and treat truncated answers as a separate category; undersized budgets bias results toward the default, which is the shortest answer\.
10. 10\.*Provider\-filter screening:*screen every cover story against each provider’s content filters at each model tier before data collection; in our study one vendor’s strictest tier refused every rendering, including a pure\-mathematics statement, while its mid tier answered all of them\.

## Appendix DProof of the family theorem

We sketch the proof; the complete derivation, together with two independent machine verifications, accompanies the released code\. Throughout, fixmmand primesp1,…,pmp\_\{1\},\\dots,p\_\{m\}, and writeφi\\varphi\_\{i\}for the connecting map intobib\_\{i\}andτi:=F⁡\(ti\)\\tau\_\{i\}:=F\(t\_\{i\}\)\.

*The category\.*Hom\-sets between distinct chain objects are singletons \(the endomorphisms absorb into adjacent paths\), andHom⁡\(bi,bi\)=\{id,ti,…,tipi−1\}≅ℤ/pi\\mathrm\{Hom\}\(b\_\{i\},b\_\{i\}\)=\\\{\\mathrm\{id\},t\_\{i\},\\dots,t\_\{i\}^\{p\_\{i\}\-1\}\\\}\\cong\\mathbb\{Z\}/p\_\{i\}; associativity reduces to associativity in these cyclic groups\.𝖢0\\mathsf\{C\}\_\{0\}is full on\{a,c\}\\\{a,c\\\}with the single compositehh, so A1–A2 hold, and the constraints are of the admissible A4 forms\.

*Structure of extensions\.*An extension is determined by, at each hidden object, the setF⁡\(bi\)F\(b\_\{i\}\), the endomorphismτi\\tau\_\{i\}withτipi=id\\tau\_\{i\}^\{p\_\{i\}\}=\\mathrm\{id\}, and the connecting values; the relations say exactly thatτi\\tau\_\{i\}fixes the image of the incoming map and that the outgoing map is constant onτi\\tau\_\{i\}\-orbits\. Sincepip\_\{i\}is prime, every cycle ofτi\\tau\_\{i\}has length11orpip\_\{i\}\.

*Forcing\.*SupposeFFis admissible\.K1i\\mathrm\{K1\}\_\{i\}forces apip\_\{i\}\-cycle, the incoming image forces a fixed point, so\|F⁡\(bi\)\|≥1\+pi\|F\(b\_\{i\}\)\|\\geq 1\+p\_\{i\};K2i\\mathrm\{K2\}\_\{i\}caps it, and writing\|F⁡\(bi\)\|=a\+b​pi\|F\(b\_\{i\}\)\|=a\+bp\_\{i\}withaafixed points andbbcycles,a\+b​pi=1\+pia\+bp\_\{i\}=1\+p\_\{i\}witha,b≥1a,b\\geq 1forcesa=b=1a=b=1\. Hence exactly one fixed point and one cycle, the incoming map is constant onto the fixed point, and the outgoing map is constant on orbits with image inside the next fixed point, which forces it constant\. Sufficiency is direct, and \(J1\) follows from the planted witness\.

*Count and single orbit\.*The admissible data at stationiiare the choice of fixed point \(1\+pi1\+p\_\{i\}options\) and the cyclic order of the cycle \(\(pi−1\)\!\(p\_\{i\}\-1\)\!options\); gauge conjugation, acting at stationiionτi\\tau\_\{i\}and on the connecting maps byσi∘φi−1∘σi−1−1\\sigma\_\{i\}\\circ\\varphi\_\{i\-1\}\\circ\\sigma\_\{i\-1\}^\{\-1\}\(withσ0=id\\sigma\_\{0\}=\\mathrm\{id\}ataa\), is transitive on these data with stabilizer of order∏ipi\\prod\_\{i\}p\_\{i\}, and the orbit\-stabilizer count gives∏i\(1\+pi\)​\(pi−1\)\!\\prod\_\{i\}\(1\+p\_\{i\}\)\(p\_\{i\}\-1\)\!\.

*Kan pair and \(J2\)\.*Both comma index categories atbib\_\{i\}are singletons, because the only morphism from\{a,c\}\\\{a,c\\\}intobib\_\{i\}is the single incoming path and dually; henceLan⁡\(bi\)=Ran⁡\(bi\)=\[1\]\\mathrm\{Lan\}\(b\_\{i\}\)=\\mathrm\{Ran\}\(b\_\{i\}\)=\[1\], both strictified extensions equal the all\-singleton trivial extensionTT, andTTviolates exactly theK1i\\mathrm\{K1\}\_\{i\}\. \(J4\) holds in the strong form since everybib\_\{i\}receives a path fromaa\.

*Closed form\.*\|ExtN\|\|\\mathrm\{Ext\}\_\{N\}\|factors along the chain into a transfer recursion whose station weights count pairs\(τi,constant data\)\(\\tau\_\{i\},\\text\{constant data\}\)with\|Fix⁡\(τi\)\|\|\\mathrm\{Fix\}\(\\tau\_\{i\}\)\|\-weighted transitions; the recursion reproduces every enumerated count at seven configurations, including1,257,4091\{,\}257\{,\}409at\(3,\(2,2,2\)\)\(3,\(2,2,2\)\)withN=4N=4\.

*Primality\.*For compositepi=d⋅ep\_\{i\}=d\\cdot ewith1<d<pi1<d<p\_\{i\}, the station data with one fixed point and onedd\-cycle satisfies both constraints with\|F⁡\(bi\)\|=1\+d<1\+pi\|F\(b\_\{i\}\)\|=1\+d<1\+p\_\{i\}, so admissible profiles differ and \(J3\) fails; parts \(i\)–\(ii\) are therefore sharp\. ∎

## Appendix EProofs for Section[6](https://arxiv.org/html/2608.26187#S6)

*Characterization \(Proposition[6](https://arxiv.org/html/2608.26187#Thmproposition6)\)\.*A witness isomorphism restricted to𝖢\\mathsf\{C\}is a𝖢0\\mathsf\{C\}\_\{0\}\-rooted isomorphism onto the restriction ofGG, so the predicate selects exactly that gauge component; conversely a gauge transport ofGG’s restriction extends by identities on the held\-out objects, and conjugation preserves functoriality\. Gauge invariance and the containment under soundness follow from Proposition[2](https://arxiv.org/html/2608.26187#Thmproposition2); under \(J3\),Adm\\mathrm\{Adm\}is one component, so internal correctness pins the completion\.

*Separation \(Proposition[7](https://arxiv.org/html/2608.26187#Thmproposition7)\)\.*Relax the seed’s size bound to44; the same forcing as in Appendix[D](https://arxiv.org/html/2608.26187#A4)now allows exactly two structures, the tight one \(Y3Y\_\{3\}, three tables\) and the tight one plus a spectator fixed point outside the incoming image \(Y4Y\_\{4\}, twelve tables\), soAdm=Y3⊔Y4\\mathrm\{Adm\}=Y\_\{3\}\\sqcup Y\_\{4\}with\|Adm\|=15\|\\mathrm\{Adm\}\|=15and both components admissible\. The world𝖢′\\mathsf\{C\}^\{\\prime\}adds two held\-out objectsu,vu,vwith probesμ:u→b\\mu\\colon u\\to b,ρ:b→v\\rho\\colon b\\to vabsorbingtt, and the ground truthGGrealizes theY4Y\_\{4\}structure withG⁡\(v\)=\[2\]G\(v\)=\[2\]and records that the two composite probes intovvtake distinct values\. A completion inY3Y\_\{3\}has a singleτ\\tau\-fixed point, and consistency of the recorded probe values is equivalent toτ\\tauhaving two fixed points, so no member ofY3Y\_\{3\}extends to the recorded data, while every member ofY4Y\_\{4\}extends and is isomorphic toGGrooted at𝖢0\\mathsf\{C\}\_\{0\}\. Category axioms, admissibility, the counts, and the refutation equivalence are machine\-verified\.

*Well\-definedness of incorporation \(Theorem[2](https://arxiv.org/html/2608.26187#Thmtheorem2)\)\.*The constraint language is invariant under conjugation by arbitrary automorphism families, since words transform by conjugation, limiting cones transport along isomorphisms of diagrams, sizes are preserved, and identity atoms reflect\. A gauge relating two commitments extends by identities to a data isomorphism of the successor instances; transport by conjugation is a bijection commuting with the successor gauge action, preserves admissibility by the invariance just stated, and preserves the canonical class becauseLan∗\\mathrm\{Lan\}^\{\*\}andRan∗\\mathrm\{Ran\}^\{\*\}are iso\-natural, the chosen colimits being unique up to the canonical comparison\. All truth values, chance, and component profiles therefore agree\.

*Monotonicity \(Lemma[1](https://arxiv.org/html/2608.26187#Thmlemma1)\)\.*A functor admissible through stagek\+1k\+1restricts on the stage\-kkcategory to the stage\-kkcommitment, which is admissible; descending the ladder gives the decreasing sequence, and its last term is the last stage’s admissible set by strictness of the restrictions\.

*Stabilization and entrenchment \(Theorem[3](https://arxiv.org/html/2608.26187#Thmtheorem3)\)\.*The retrospectively correct sets are antitone inside the finiteAdm⁡\(Sk\)\\mathrm\{Adm\}\(S\_\{k\}\)and stabilize\. For the entrenchment scheme, stage one is the relaxed seed of Proposition[7](https://arxiv.org/html/2608.26187#Thmproposition7); stage two attaches one objectddwith a single generatorr:d→br\\colon d\\to babsorbingtt, the marked\-cone constraint that\(d,r,r\)\(d;r,r\)is a limit of the parallel pair\(t,idb\)\(t,\\mathrm\{id\}\_\{b\}\), and the size bound11\. Over a commitmentXX, the limit constraint forces the value atddto biject onto the fixed set ofX⁡\(t\)X\(t\); forX∈Y3X\\in Y\_\{3\}the fixed set is a single point, giving exactly one admissible extension, which coincides with the right Kan extension of the incorporated data, and forX∈Y4X\\in Y\_\{4\}the fixed set has two points, contradicting the size bound, so no extension exists\. The verdict is representative\-independent by Theorem[2](https://arxiv.org/html/2608.26187#Thmtheorem2)\. All stage counts and both verdicts are machine\-verified against the generator\. ∎

## Appendix FExperimental details

#### Plain\-language pipeline\.

One instance travels the following path\. The generator writes the composition tables, the observed functor, and the constraint list\. The certifier either enumerates every extension within the declared bound and checks the four conditions directly, or, on the theorem\-certified tier, verifies the hypotheses of Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)and reads off the counts\. The renderer wraps the tables in a nonce cover story, the grader parses an answer back into a candidate extension and reports one of five outcomes, and the harness logs every call together with its finish reason\. Every step is replayable from the released code\.

#### Sensitivity to the size bound\.

The declared bound enters the protocol only through the denominator of the chance level, and loosening it enlarges the answer space\. Atm=2m\{=\}2,p=\(2,2\)p\{=\}\(2,2\)the count grows from4,3874\{,\}387atN=4N\{=\}4to117,867117\{,\}867atN=5N\{=\}5and chance falls from2\.1×10−32\.1\\times 10^\{\-3\}to7\.6×10−57\.6\\times 10^\{\-5\}; atm=3m\{=\}3,p=\(2,3,3\)p\{=\}\(2,3,3\)chance falls from8\.1×10−78\.1\\times 10^\{\-7\}to1\.8×10−91\.8\\times 10^\{\-9\}\. The declared bound is not the tightest admissible one\. AnyN≥maxi⁡\(1\+pi\)N\\geq\\max\_\{i\}\(1\+p\_\{i\}\)is admissible, and the generator’s defaultN=maxi⁡\(1\+pi\)\+1N=\\max\_\{i\}\(1\+p\_\{i\}\)\+1sits one above that minimum, so the reported chance levels are tied to the declared bound and would be larger at the minimum, by factors from2\.82\.8atm=1m=1to280280at\(3,\(2,3,3\)\)\(3,\(2,3,3\)\), where chance rises from8\.1×10−78\.1\\times 10^\{\-7\}to2\.3×10−42\.3\\times 10^\{\-4\}\. The released instance\(2,\(3,2\)\)\(2,\(3,2\)\)already uses the minimum,N=4N=4\. Every above\-chance conclusion survives either convention, since the largest minimal\-bound chance is0\.440\.44at\(1,\(3\)\)\(1,\(3\)\)against accuracies of0\.830\.83and above\. Padding within the bound is also excluded in the family, since the size constraintK2i\\mathrm\{K2\}\_\{i\}admits at most1\+pi1\+p\_\{i\}elements and the admissible completion already uses all of them, so no spectator element can be added without violating a constraint\.

#### Models and access\.

GPT\-5\.6 Luna Pro, Claude Sonnet 5, Gemini 3\.1 Pro, and DeepSeek V4 Pro, accessed through a single aggregator API between August 20 and 21, 2026\. The strictest tier of one provider refused every rendering of the task, including a pure\-mathematics statement, with a content\-filter finish reason; the mid tier answered all renderings, and the reported data point for that family is therefore mid\-tier \(Appendix[C](https://arxiv.org/html/2608.26187#A3)\)\.

#### Sampling\.

Per instance and model under the primary rendering, six jump, four calibration, and two control samples, the first greedy and the rest at temperature0\.70\.7\. The wording arm adds four jump samples per model for two instances under a second cover story\. In total464464graded answers enter every count in the paper\. A seed pilot of165165answers, run under an earlier harness with a6,0006\{,\}000\-token budget and no finish reasons, is released but excluded from all counts; on the seed rendering GPT\-5\.6 solved55of1111jump samples with six rule violations, which motivated the wording arm\. Output budgets in the main run are12,00012\{,\}000tokens; finish reasons are recorded; answers with finish reason length and empty answers are reported separately and excluded from conditional rates\.

#### Grading\.

Answers are parsed into per\-station tables and classified as admissible, Kan default, valid\-but\-inadmissible, rule\-violating, or invalid, using the structural characterization of Theorem[1](https://arxiv.org/html/2608.26187#Thmtheorem1)\(i\); the grader agrees with certified membership on all138,691138\{,\}691enumerated extensions of the six releasedm≤2m\\leq 2instances under both renderings, a check that is part of the released test suite, and it passes the adversarial renaming and junk\-answer tests of the released suite\.

#### Aggregates\.

Calibration answers that are neither truncated nor empty are the Kan default in129129of132132cases \(0\.980\.98;0\.900\.90of all144144\)\. Controls are passed in6969of7272cases, the three failures being DeepSeek answers atm=3m=3that were truncated or empty\. The Kan\-default rate is00in all248248constrained jump samples\. The2222jump failures are1515truncations,33empty answers,33rule violations, and11valid\-but\-inadmissible completion; atm=3m=3the split is66,33,11, and11\. Total evaluation cost was below fifteen dollars, and a full four\-model evaluation of one instance completes in under half an hour\.