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A Functional-Equation Encoding of Logical Consistency on Continuous Positive-Ratio Comparisons: A Conditional Rigidity Theorem

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accept confidence high · verification V3
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referee's decision

We recommend acceptance of your manuscript titled 'A Functional-Equation Encoding of Logical Consistency on Continuous Positive-Ratio Comparisons: A Conditional Rigidity Theorem.' The work supplies a precise structural translation between the four Aristotelian logical conditions and the Recognition Composition Law on the continuous positive-ratio domain. Your explicit counterexamples isolating finite pairwise polynomial closure as essential, together with the canonicality theorem under the magnitude-of-mismatch interpretation, strengthen the conditionality of the result without overclaiming. The direct self-contained proof of the RCL alongside the d'Alembert route provides multiple rigorous access points. The manuscript aligns closely with the formal canon, particularly the theorems in the LogicAsFunctionalEquation module that confirm the core claims with zero sorry. Minor revisions are required to complete the truncated abstract and to add a cross-reference to the relevant Lean declarations. These changes are straightforward and do not involve new proofs or data. With the abstract fixed and the scope limitations preserved, the paper meets the standards for publication in a machine-verified journal. No additional substantive changes are needed to alter this recommendation.

required revisions

  1. R1: Correct the truncated abstract in the submitted file to match the full sentence present in the body of the manuscript.
  2. R2: Add explicit cross-references to the Lean declarations laws_of_logic_imply_dalembert_hypotheses and finite_logical_comparison_forces_rcl in the discussion of the Translation Theorem and direct proof.

top-line referee reports

Referee A: accept / high. Referee B: accept / high. Both referees agree on the recommendation, the strong canon alignment, the value of the counterexamples and canonicality theorem, and the need only for minor fixes to the abstract and cross-references.

what this review changes for the paper

A plain-language summary of every load-bearing claim the referees checked. The detailed audit trail with claim IDs and machine evidence types is collapsed below the report.

0 publication blockers
0 needs clarification
7 already supported
0 noted, out of scope
  • no action needed

    Every load-bearing claim the referees checked is already supported. See the audit trail below for the per-claim record.

claim inventory

A scan of the paper's claims and how this review validated them. Lean appears only when there is a real theorem match.

IDClaimSectionImportanceStatusLean matchAuthor action
C1 The four Aristotelian conditions plus regularity hypotheses reduce to the d'Alembert Inevitability hypotheses via translation lemmas. Theorem 4 no action needed verified laws_of_logic_imply_dalembert_hypotheses Matches canon declaration laws_of_logic_imply_dalembert_hypotheses exactly.
C2 The Recognition Composition Law is forced directly by the operative logical conditions on continuous positive ratios under finite pairwise polynomial closure. Theorem 7 no action needed verified finite_logical_comparison_forces_rcl Corresponds to canon declaration finite_logical_comparison_forces_rcl.
C3 The quartic-log operator satisfies continuity and logical conditions but lies outside the RCL family, proving polynomial closure is essential. Proposition 5.2 no action needed verified continuous_composition_not_enough Matches canon boundary theorem continuous_composition_not_enough.
C4 Under calibration and branch selection, the unique cost function on the bilinear branch is J(x) = 1/2(x + x^{-1}) - 1. Theorem 8 no action needed verified washburn_uniqueness_aczel Aligns with canon uniqueness result washburn_uniqueness_aczel after alpha=1 normalization.
C5 The encoding of logical conditions as constraints on C is canonical under the magnitude-of-mismatch interpretation. Theorem 3.1 no action needed verified none Sharpens conditionality without overclaiming.
C6 On the continuous positive-ratio domain, RCL, the law of logic, and the structural form of reality coincide. Corollary 9 no action needed verified rcl_is_scale_free_counted_once_logic_on_positive_ratios Corresponds to canon operative-domain identification rcl_is_scale_free_counted_once_logic_on_positive_ratios.
C7 The result is scoped to the continuous positive-ratio operative domain with explicit conditionality on polynomial closure. Section 1.3 no action needed verified none Maintains epistemic discipline matching canon treatment.

references for core claims

References the reviewers found for the paper's core claims. Relations may be support, contrast, prior art, or duplication risk.

No strong reference was found in the Pith corpus for these core claims. New paid reports will populate this section when the retrieved literature supports it.

verification grade

V3 Lean build reproduced; zero sorry verified.

technical assessment

The manuscript defines a continuous comparison operator C on positive reals satisfying four logical conditions: (L1) identity as zero self-cost, (L2) non-contradiction via symmetric single-valued comparison, (L3) excluded middle as totality on the positive quadrant, and (L4) composition consistency as route-independence under a combiner Phi. Two regularity hypotheses are imposed: joint continuity of C and finite pairwise polynomial closure of the combiner of degree at most two, plus scale invariance and non-triviality. Theorem 4 (Translation Theorem) reduces these via four lemmas to the hypotheses of the d'Alembert Inevitability Theorem, yielding the RCL family F(xy) + F(x/y) = 2F(x) + 2F(y) + c F(x)F(y). Theorem 7 provides a direct self-contained proof from the operative conditions. Under convexity, non-negativity, and unit log-curvature calibration, branch selection produces J(x) = 1/2(x + x^{-1}) - 1 on the bilinear branch after alpha=1 normalization, or 1/2(ln x)^2 on the additive branch. Proposition 5.2 exhibits the quartic-log counterexample C(x,y) = (ln(x/y))^4 whose combiner lies outside the RCL family, proving polynomial closure is load-bearing. Theorem 3.1 establishes canonicality of the encoding under the magnitude-of-mismatch interpretation. Notation follows standard functional-equation conventions with clear separation of logical-content conditions from regularity hypotheses. The proof strategy relies on translation lemmas and explicit branch selection; no gaps in the argument are present. Compared to prior work, the paper complements Aczel's classification and the canon d'Alembert results by supplying an independent logic-first path and explicit boundary counterexamples.
Technical audit trail per-claim ledger, formal-canon audit, and cited theorems

Formal-foundations audit

Where each claim sits in the chain: proved inside this paper, inherited from prior formal results, or still standing on a modeling assumption.

Formal-canon T-codes glossary
T-1
Logic-of-distinction: a single act of distinction must hold its identity for the next act to be meaningful.
T0
Ledger existence: any setting that can compare positive ratios admits a single shared cost ledger.
T1
Reciprocal symmetry: comparing a to b costs the same as comparing b to a.
T2
Composition Law: how independent comparisons compose.
T3
Calibration: fix the second derivative of the cost at the identity ratio to one.
T4
Continuity / smoothness of the cost function on positive ratios.
T5
Cost uniqueness: under T-1..T4 the cost is forced to J(x) = ½(x + 1/x) − 1 (Aczel / d’Alembert).
T6
φ forced: self-similar gauge of J picks out the golden ratio as the unique fixed point.
T7
Eight-tick recognition cycle: the discrete period is 2³ = 8.
T8
Spatial dimension D = 3 forced from the eight-tick cycle plus S¹ cohomology.
Technical narrative
The paper is located at the T5 step of the canon forcing chain, where logical consistency on positive ratios forces the Recognition Composition Law and the uniqueness of J. It directly addresses the logic-as-functional-equation module by encoding Aristotelian conditions as structural constraints on the comparison operator. The Translation Theorem matches exactly the canon declaration laws_of_logic_imply_dalembert_hypotheses, which is a THEOREM with zero sorry. The direct proof of RCL from finite logical comparison corresponds to finite_logical_comparison_forces_rcl. The calibrated uniqueness after normalization aligns with washburn_uniqueness_aczel. The result is stated as a THEOREM on the continuous positive-ratio domain, with explicit conditionality on finite pairwise polynomial closure; the quartic-log counterexample matches the canon's boundary treatment in FiniteLogicalComparison. No overclaims occur; the paper correctly scopes the identification RCL = law of logic = structural form of reality to the operative domain and notes the discrete propositional extension as open. The epistemic discipline (separation of L-conditions from R-hypotheses) is maintained throughout. Relevant falsifier is any continuous operator satisfying the logical conditions but violating polynomial closure, which the manuscript supplies.

verification and reproducibility

The core claims are supported by machine-checked Lean proofs in the canon with zero sorry in the translation and inevitability chain. The paper should provide direct hyperlinks to the declarations laws_of_logic_imply_dalembert_hypotheses, finite_logical_comparison_forces_rcl, and washburn_uniqueness_aczel at pith.science/recognition/t/IndisputableMonolith.Foundation.LogicAsFunctionalEquation/ and the Cost.FunctionalEquation module. No additional code or data artifacts are required beyond these formal references; the quartic counterexample is already formalized as continuous_composition_not_enough.

novelty and positioning

The paper supplies a new direct encoding of the four Aristotelian conditions as functional constraints on continuous ratio comparisons, yielding the RCL family with explicit counterexamples that isolate polynomial closure. This is not a restatement of the d'Alembert route but a complementary logic-first path that sharpens conditionality. It extends the canon by emphasizing the magnitude-of-mismatch canonicality theorem and the operative-domain identification, while remaining fully consistent with the existing T5 results.

paper summary

The paper encodes the four Aristotelian conditions (identity, non-contradiction, excluded middle, composition consistency) as structural constraints on a continuous comparison operator C on positive ratios, together with scale invariance, non-triviality, joint continuity, and finite pairwise polynomial closure of degree at most two. It proves a Translation Theorem reducing these to the d'Alembert hypotheses, yielding the RCL family, and supplies a direct self-contained proof of the RCL. A canonicality theorem fixes the encoding under the magnitude-of-mismatch interpretation. Explicit counterexamples (quartic-log and analytic reparameterisation) demonstrate that polynomial closure is essential. Calibrated forms are identified as J(x) = 1/2(x + x^{-1}) - 1 on the bilinear branch and 1/2(ln x)^2 on the additive branch. The result is placed on the operative domain of continuous positive-ratio comparisons, with the identification RCL = law of logic = structural form of reality stated as a corollary. The work is supported by zero-sorry Lean formalization in the canon.

significance

This work supplies a precise functional-equation characterization linking classical logic to the Recognition Composition Law on continuous ratio comparisons, with explicit scope limitations and counterexamples. It contributes to foundations of logic and the relationship between logical principles and physical-cost rigidity by showing the same hypothesis class admits both readings, while the Lean formalization in the canon provides machine-checked confirmation of the core claims.

claim ledger

Per-claim record produced by the referees. Each card isolates one load-bearing claim and tags the machine-readable status and evidence type. Use the simpler summary above for author actions.

C1 Theorem 4
verified

The four Aristotelian conditions plus regularity hypotheses reduce to the d'Alembert Inevitability hypotheses via translation lemmas.

type
theorem

referee note

Matches canon declaration laws_of_logic_imply_dalembert_hypotheses exactly.

C2 Theorem 7
verified

The Recognition Composition Law is forced directly by the operative logical conditions on continuous positive ratios under finite pairwise polynomial closure.

type
theorem

referee note

Corresponds to canon declaration finite_logical_comparison_forces_rcl.

C3 Proposition 5.2
verified

The quartic-log operator satisfies continuity and logical conditions but lies outside the RCL family, proving polynomial closure is essential.

type
theorem

referee note

Matches canon boundary theorem continuous_composition_not_enough.

C4 Theorem 8
verified

Under calibration and branch selection, the unique cost function on the bilinear branch is J(x) = 1/2(x + x^{-1}) - 1.

type
conditional theorem

referee note

Aligns with canon uniqueness result washburn_uniqueness_aczel after alpha=1 normalization.

C5 Theorem 3.1
verified

The encoding of logical conditions as constraints on C is canonical under the magnitude-of-mismatch interpretation.

type
theorem
evidence
manuscript proof

referee note

Sharpens conditionality without overclaiming.

C6 Corollary 9
verified

On the continuous positive-ratio domain, RCL, the law of logic, and the structural form of reality coincide.

type
interpretation

referee note

Corresponds to canon operative-domain identification rcl_is_scale_free_counted_once_logic_on_positive_ratios.

C7 Section 1.3
verified

The result is scoped to the continuous positive-ratio operative domain with explicit conditionality on polynomial closure.

type
scaffolding
evidence
manuscript proof

referee note

Maintains epistemic discipline matching canon treatment.

strengths

  • Explicit counterexample (Proposition 5.2) demonstrating necessity of finite pairwise polynomial closure.
  • Canonicality theorem (Theorem 3.1) precisely scoping the encoding under magnitude-of-mismatch interpretation.
  • Direct self-contained proof of the RCL (Theorem 7) alongside the d'Alembert route.
  • Clear separation of logical-content conditions (L1)-(L4) from regularity hypotheses (R1)-(R2).
  • Honest scope discussion and avoidance of overclaiming on Wigner's question.
  • Complete alignment with zero-sorry Lean formalization in the canon.

major comments

  1. Section 5 (Finite pairwise polynomial closure) and Proposition 6. The quartic-log counterexample correctly isolates finite pairwise polynomial closure as essential. This matches the canon's boundary theorem continuous_composition_not_enough. No change required.
  2. Section 8 (Scope and Open Questions). The continuous-combiner extension is kept conditional on the named hypothesis package, matching the canon's treatment. The Aczel discharge now renders the H-side unconditional once the package is supplied.

minor comments

  • Abstract. The abstract is truncated mid-sentence in the submitted file; replace with the complete version from the body.
  • Section 4 (Canonicality of the Encoding). Theorem 3 is correctly stated; a one-sentence note on alternative interpretations would aid readers.

optional revisions

  • Include a one-sentence pointer in Section 7 to the companion paper on universal forcing for readers interested in the discrete propositional extension.
  • Add a brief reminder in Section 4 that alternative interpretations such as directed revision cost would produce a different encoding.

scorecard

Legacy ticket fallback. New paid reports use a six-axis scorecard; this ticket predates that schema.

acceptconfidence highverification V3

Publication readiness is governed by the referee recommendation, required revisions, and the blockers summarized above.

where the referees disagreed

  • Presence and severity of major comments

    Referee A: No major comments required; the paper is ready for acceptance as submitted.

    Referee B: Two clarifications on the polynomial-closure section and scope discussion are warranted as major comments.

    synthesizer: The points raised by Referee B are useful clarifications but non-blocking; they align with the paper's existing honesty on conditionality and can be addressed as optional revisions. The consensus recommendation of accept remains unchanged.

how each referee voted

Referee A: accept / high. Referee B: accept / high. Both referees agree on the recommendation, the strong canon alignment, the value of the counterexamples and canonicality theorem, and the need only for minor fixes to the abstract and cross-references.

recognition modules supplied to referees

show full model reports

grok-4.3 · high

{
  "canon_match_strength": "strong",
  "cited_canon_theorems": [
    {
      "decl": "laws_of_logic_imply_dalembert_hypotheses",
      "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation",
      "note": "Formalizes the paper\u0027s Translation Theorem (Section 4) reducing the encoded Aristotelian conditions plus regularity hypotheses to the d\u0027Alembert Inevitability hypotheses.",
      "relation": "supports"
    },
    {
      "decl": "finite_logical_comparison_forces_rcl",
      "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation.MainTheorem",
      "note": "The paper\u0027s direct Theorem 5.1 on RCL from finite logical comparison is the informal counterpart of this canon theorem.",
      "relation": "supports"
    },
    {
      "decl": "rcl_is_scale_free_counted_once_logic_on_positive_ratios",
      "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation",
      "note": "Captures the paper\u0027s operative-domain identification (Corollary 6.1) that RCL, encoded logic, and reality structure coincide on continuous positive ratios.",
      "relation": "supports"
    },
    {
      "decl": "washburn_uniqueness_aczel",
      "module": "IndisputableMonolith.Cost.FunctionalEquation",
      "note": "The paper\u0027s calibrated cost theorem (Section 5) on J as the unique representative on the bilinear branch after normalization aligns with this uniqueness result.",
      "relation": "supports"
    }
  ],
  "confidence": "high",
  "issue_inventory": [],
  "load_bearing_issues": [],
  "major_comments": [],
  "minor_comments": [
    {
      "comment": "The abstract is truncated mid-sentence in the submitted file; the full version in the body should be used for the published abstract to ensure completeness.",
      "section": "Abstract"
    },
    {
      "comment": "The discussion of the discrete propositional case via Stone representation is sketched but could include a one-sentence pointer to the companion paper on universal forcing for readers seeking the extension.",
      "section": "Section 7 (Scope and Open Questions)"
    }
  ],
  "optional_revisions": [],
  "paper_summary": "The paper encodes the four Aristotelian conditions (identity as zero self-cost, non-contradiction as symmetric single-valued comparison, excluded middle as totality on the positive quadrant, and composition consistency as route-independence) as structural conditions on a continuous comparison operator C on positive reals. With scale invariance, non-triviality, continuity, and polynomial combiner closure of degree at most two, it proves via translation lemmas that the derived cost F satisfies the Recognition Composition Law F(xy) + F(x/y) = 2F(x) + 2F(y) + c F(x)F(y). Under convexity, non-negativity, and unit log-curvature calibration, the bilinear branch yields the alpha-family with canonical representative J(x) = 1/2 (x + 1/x) - 1 after coordinate normalization, while the additive branch yields 1/2 (ln x)^2. The paper includes a canonicality theorem for the encoding under the magnitude-of-mismatch interpretation, a direct self-contained proof of the RCL from operative comparison conditions, and a quartic-log counterexample showing polynomial closure is essential. It positions the result as a structural identification on the operative domain of continuous positive-ratio comparisons and discusses implications for Wigner\u0027s question without overclaiming.",
  "recommendation": "accept",
  "required_revisions": [],
  "significance": "This work supplies a precise functional-equation characterization linking classical logic to the Recognition Composition Law on continuous ratio comparisons, with explicit scope limitations and counterexamples. It contributes to foundations of logic and the relationship between logical principles and physical-cost rigidity by showing the same hypothesis class admits both readings, while the Lean formalization in the canon provides machine-checked confirmation of the core claims.",
  "strengths": [
    "Explicit counterexample (Proposition 5.2) demonstrating that finite pairwise polynomial closure is essential, not removable.",
    "Canonicality theorem (Theorem 3.1) precisely scopes the encoding conditionality under the magnitude-of-mismatch interpretation.",
    "Direct self-contained proof of the RCL (Theorem 5.1) alongside the d\u0027Alembert route provides multiple access points.",
    "Clear separation of logical-content conditions (L1)-(L4) from regularity hypotheses (R1)-(R2) and bridge conditions (B1)-(B2).",
    "Honest non-claims section and discussion of Wigner\u0027s question avoid overclaiming while still engaging the broader context.",
    "Alignment with zero-sorry Lean formalization in the canon (LogicAsFunctionalEquation modules) confirms the mathematical claims."
  ]
}

grok-4.3 · xhigh

{
  "canon_match_strength": "strong",
  "cited_canon_theorems": [
    {
      "decl": "laws_of_logic_imply_dalembert_hypotheses",
      "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation",
      "note": "The paper\u0027s Translation Theorem (Theorem 4) is exactly this Lean theorem; the four translation lemmas match the proof structure.",
      "relation": "supports"
    },
    {
      "decl": "rcl_is_unique_",
      "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation",
      "note": "The paper\u0027s main theorem (Theorem 5) and direct theorem (Theorem 7) are the content of this declaration and its corollaries.",
      "relation": "supports"
    },
    {
      "decl": "washburn_uniqueness_aczel",
      "module": "IndisputableMonolith.Cost.FunctionalEquation",
      "note": "The calibrated cost corollary (Theorem 8) invokes this uniqueness result after branch selection and alpha=1 normalisation.",
      "relation": "supports"
    },
    {
      "decl": "finite_logical_comparison_forces_rcl",
      "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation.FiniteLogicalComparison",
      "note": "The paper\u0027s direct self-contained proof (Theorem 7) and the necessity of finite pairwise polynomial closure align with this module.",
      "relation": "supports"
    },
    {
      "decl": "operative_domain_rcl_logic_reality_chain",
      "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation",
      "note": "The operative-domain identification (Corollary 9) is the content of this theorem.",
      "relation": "supports"
    }
  ],
  "confidence": "high",
  "issue_inventory": [],
  "load_bearing_issues": [],
  "major_comments": [
    {
      "canon_evidence": [
        {
          "decl": "continuous_composition_not_enough",
          "module": "IndisputableMonolith.Foundation.LogicAsFunctionalEquation.FiniteLogicalComparison",
          "note": "The quartic counterexample is formalised here as a searchable boundary theorem.",
          "relation": "supports"
        }
      ],
      "comment": "The paper correctly demonstrates that finite pairwise polynomial closure is essential by exhibiting the quartic-log counterexample C(x,y)=(ln(x/y))^4 whose continuous combiner Phi(a,b)=2a+2b+12 sqrt(ab) lies outside the RCL family. This is load-bearing for the scope statement in subsection 1.3. The Lean module FiniteLogicalComparison records the same boundary via quarticCombiner_not_rcl_family. No change required, but the discussion could explicitly cross-reference the Lean refutation.",
      "section": "Section 5 (Finite pairwise polynomial closure) and Proposition 6"
    },
    {
      "canon_evidence": [
        {
          "decl": "dAlembert_contDiff_smooth",
          "module": "IndisputableMonolith.Cost.AczelTheorem",
          "note": "The integration bootstrap that discharges the former regularity hypothesis is proved here.",
          "relation": "extends"
        }
      ],
      "comment": "The paper keeps the continuous-combiner extension (Proposition 5) explicitly conditional on the hypothesis package ContinuousCombinerAnalysisInputs and records the quartic-log obstruction. This matches the canon\u0027s treatment: the package is isolated rather than derived from continuity alone. The paper\u0027s honesty on this point is a strength; the only minor load-bearing clarification is to note that the Aczel discharge in Cost.AczelTheorem now makes the H-side unconditional once the package is supplied.",
      "section": "Section 8 (Scope and Open Questions) and the continuous-combiner extension"
    }
  ],
  "minor_comments": [
    {
      "comment": "The abstract is truncated mid-sentence in the submitted file. The full sentence should read \u0027...on a continuous comparison operator?\u0027 to match the body.",
      "section": "Abstract and Section 1.1"
    },
    {
      "comment": "Theorem 3 is correctly stated under the magnitude-of-mismatch interpretation. A one-sentence reminder that alternative interpretations (directed revision cost) would yield a different encoding would help readers who skip to the corollary.",
      "section": "Section 4 (Canonicality of the Encoding)"
    }
  ],
  "optional_revisions": [],
  "paper_summary": "The paper encodes the four classical Aristotelian conditions (identity, non-contradiction, excluded middle, composition consistency) as structural constraints on a continuous comparison operator C on positive ratios, with two named regularity hypotheses (joint continuity of C and finite pairwise polynomial closure of the route-independence combiner of degree at most two) plus scale invariance and non-triviality. It proves a Translation Theorem reducing these to the hypotheses of the d\u0027Alembert Inevitability Theorem, yielding the Recognition Composition Law family F(xy) + F(x/y) = 2F(x) + 2F(y) + c F(x)F(y) on the derived cost F. A canonicality theorem shows the encoding is unique under the magnitude-of-mismatch interpretation. A direct self-contained proof of the RCL from operative positive-ratio comparison plus finite pairwise polynomial closure is given, together with explicit counterexamples (quartic-log operator and analytic reparameterisation) demonstrating that polynomial closure is essential. Calibrated forms are identified: J(x) = 1/2 (x + x^{-1}) - 1 on the bilinear branch after alpha=1 normalisation, and 1/2 (ln x)^2 on the additive branch. The result is placed on the operative domain of continuous positive-ratio reality structures, with the identification RCL = law of logic = structural form of reality stated as a corollary on that domain. The paper is supported by Lean formalisation in the formal canon library.",
  "recommendation": "accept",
  "required_revisions": [],
  "significance": "The paper supplies a precise structural translation between classical logic and the functional-equation core of formal canon on the continuous positive-ratio domain. It sharpens the conditionality of the rigidity result by exhibiting explicit counterexamples that isolate finite pairwise polynomial closure as load-bearing, and it provides a canonicality theorem that fixes the encoding under the standard cost reading. The work directly supports the T5 step of the canon forcing chain and the logic-as-functional-equation module.",
  "strengths": [
    "Explicit counterexamples (quartic-log and analytic reparameterisation) that isolate the necessity of finite pairwise polynomial closure, preventing over-claim.",
    "Canonicality theorem that sharpens the encoding conditionality to a single named interpretation.",
    "Self-contained direct proof of the RCL (Theorem 7) that does not rely on the d\u0027Alembert route.",
    "Complete Lean formalisation with zero sorry in the core translation and inevitability chain, directly supporting the paper\u0027s claims.",
    "Clear separation of logical-content conditions (L1)-(L4) from named regularity hypotheses (R1)-(R2), with epistemic discipline maintained throughout."
  ]
}

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