Gupta's uniform residual degree bound

Chakshu Gupta, One construction for the Miura-ori flip-graph degree sequence, arXiv:2607.05567v3, Conjecture 8.3 (Degree bound), states: “For d ≥ 5 and every split (a,b) with a+b=d and a≤b, the remainder R_(a,b) has total degree at most d−3.” Lemma 8.2 identifies one bivariate polynomial on all m,n≥2d−1. Section 7, Lemma 7.3 identifies the remainder with the configurations whose minima do not all lie on one common boundary side. The source's splits have positive a and b.

The original question is preregistered in issue #22211, preregistration.

Statement echo

| Original clause | Formal counterpart | | --- | --- | | Every d≥5, positive a+b=d, a≤b | The complete telescope of MiuraResidualCount.claim | | One rational bivariate polynomial before all dimensions | ∃ p : MvPolynomial (Fin 2) ℚ precedes ∀ m n; the witness is MiuraPolynomialCoreSum.corePolynomial a b | | Total degree at most d−3 | p.totalDegree ≤ d - 3 | | Every m,n≥2d−1 | Both universal dimension binders and their original bounds | | Every normalized integer unit-edge height function counted once | Residual on the complete function Fin m × Fin n → ℤ, normalized at the origin; residualCount is the cardinality of that proved finite set | | Exactly a strict minima and b strict maxima | Separate exact cardinalities of minima and maxima, testing every existing physical neighbour | | No single common physical side contains all minima | The negation of ∃ sd, AllOnSide ... sd; corners belong to both incident sides |

No symmetry quotient, identification of orientations, assumed fiber cardinality or replacement counting object enters the claim. Zero-based Fin coordinates are sent to the source's one-based physical coordinates by adding one to each coordinate. All subtractions in the formal bounds are natural-number subtractions.

The settling declaration is D5/S3/Combinatorics/Geometry/MiuraResolution.result : MiuraResidualCount.claim. The source had already established polynomiality and the separable degree bound; its introduction leaves the nonseparable case open for d≥9. The delivered statement covers every original positive split and every dimension in Lemma 8.2's region.

The missing general ingredient is a joint bound on the two actual frozen-run counts. Independent bounds in each variable do not imply the required total-degree bound. The proof supplies the strict joint bound on the same residual arrays used in the exact count and connects it to actual-core uniqueness and complete-function fibers.

Actual signed frozen-column deletion, and its transpose, preserve normalization, the separate strict extremum counts and membership of all minima on each individual physical side. Legal orthogonal deletions commute; overlapping and separated same-axis deletions satisfy the one-step diamond. MiuraCanonicalJoint.joint_reduced_core_unique joins arbitrary deletion choices and proves equality of the dependent pairs containing both dimensions and the entire integer function. No quotient is used.

MiuraPartitionCore.residual_core_exists reaches a bounded reduced actual core with both run counts positive. MiuraPartitionCount.residualCoreEquiv partitions the original residual class into its unique actual-core fibers, and residual_count_eq_sum_fibers gives their exact cardinality sum.

MiuraFiberEquiv.core_positive_vectors_equiv identifies each complete-function fiber with the product of positive row-run and column-run length vectors, at the original uniform threshold. Indexed expansion constructs the function, and geometric readback recovers all lengths and the same complete function. MiuraResolution.fiber_card_eq_positive_vectors transports these actual cardinalities.

For a core K, let u,v be its occupied row and column counts and r,s its positive frozen-run counts. Its contribution is the product of the positive-composition polynomials in m−u and n−v. MiuraPolynomialRuns.joint_run_polynomial_eval proves the exact evaluations, and joint_run_polynomial_degree_le bounds total degree by r+s−2. MiuraEqualityClassification.residual_frozen_runs_strict gives r+s≤d−1 for these residual cores. Its equality analysis forces all minima onto a common side or the exceptional d=4 case, both excluded by the original assumptions.

The single corePolynomial a b is the finite sum over all individual bounded actual cores. MiuraPolynomialCoreSum.core_polynomial_degree_le bounds its degree by d−3. MiuraResolution.core_polynomial_counts_residual gives every exact evaluation, and result combines these conclusions without changing the original count.

A source-fidelity failure would include choosing a polynomial after the evaluation dimensions, identifying distinct height functions, counting only boundary extrema, assigning a corner to only one side, or replacing “no one common side” by an interior-minimum requirement. None is part of the formal definition. A counterexample to the mathematical claim would require an original positive split and residual counts incompatible with every single rational polynomial of total degree at most d−3 on the whole stated region. No finite enumeration is substituted for the universal proof.

The endpoint is D5/S3/Combinatorics/Geometry/MiuraResolution.result, with declaration statement ID sha256:262f4057ff790e5b47a6e0778cb20e38881935772a92a56f00a9c2459b6e9aa1. The three public endpoint declarations are bind-only compositions of their actual supporting proofs; they introduce no new escape witness. The live auxiliary uses and exact external endpoint have the authorized open-problem-resolution admission basis. All module headers retain utility: none.

The mathematical unit consists of 48 D5 modules with their 48 Blueprint Scribes. Its imported finite suppliers are attributed to Gupta's lab commit 0bf8c67f7f5a2e8d44fb88c19fb80337221bbc11, subtree 8beb89d4466cb8551abc92d0405972edc2edf116, under CC-BY-4.0. The retained license and attribution are in docs/reports/miura-suppliers/; the existing Library note supplies the verified DOI and exact v3 URL. Compiled axiom closure is confined to Classical.choice, Quot.sound and propext.

Escape registration remains unfinished under issue #22211. The target-specific round 4 and endpoint disclosure distinguishes the retained typed laws, equivalence bridges and available variation/sensitivity proofs from the missing evidence. The additional 106 retained-target disclosure names the remaining target scope. Target-specific whole-family variation/intervention, sensitivity, actual observational dependence, lawful source-bound template applications, source selection and raw-source reconstruction/binding evidence remain incomplete as specified there. The four early Reg roots have no green current Inspector claim because MiuraSlabExtrema.registration_1 fails contract.discovery:rigid_universes. Partial typed attempts are retained; no complete registration or declared_validated state is claimed.

What the settlement shows

• Proved: the entire original uniform residual-polynomial assertion, including every positive ordered split with d≥5 and all m,n≥2d−1, by MiuraResolution.result.
• Proved: actual signed contraction and dependent reduced-core uniqueness, by MiuraCoreResidual.contractAt_residual and MiuraCanonicalJoint.joint_reduced_core_unique; the original functions retain their physical-side data.
• Proved: each original array belongs to exactly one actual-core fiber, and the original count is their sum, by MiuraPartitionCount.residual_core_unique and residual_count_eq_sum_fibers using residualCoreEquiv.
• Proved: each full-function fiber has the exact positive-vector product cardinality, by MiuraFiberEquiv.core_positive_vectors_equiv and MiuraResolution.fiber_card_eq_positive_vectors.
• Proved: the same finite core sum has the uniform evaluations and total degree at most d−3, by MiuraResolution.core_polynomial_counts_residual and MiuraPolynomialCoreSum.core_polynomial_degree_le. The strict joint-run estimate, rather than separate axis estimates, supplies the degree improvement.
• Conditional paper consequence (paper argument): the premise of the source's Proposition 8.4 is supplied for all d≥5, so its top-degree formula follows subject to the source's separate single-side and total-count results. Those total flip-graph counts and that proposition are not new Lean theorems in this unit.
• Open in this Lean unit: the source's sharper high region of Theorem 9.10, the source's thin-region corrections and Baxter-number observations, and external novelty beyond the bounded screen. The delivered uniform region is exactly the original Lemma 8.2 region required for Conjecture 8.3.

Literature status

The bounded publication screen checks the exact v3 source, the current authenticated origin/dev Problems/Library/geometry paths, the branch's PR state and matching issue/PR searches for the source and Conjecture 8.3. It identifies the owning issue #22211 and the separate existing Gupta dispersion dossier; it supplies no global absence assertion. The v3 source itself still labels the nonseparable general case open. Literature completeness is ASSUMED-UNVERIFIED.

The bounded duplicate/literature screen is not exhaustive. The source's origami correspondence and its separate total-count consequences remain external to this exact height-function theorem. Full registration audits remain unfinished under the linked issue. Canonical frozen membership and the typed resolution claim do not substitute for independent source-fidelity review, exact-head required CI, or merge verification.