A Rigorous Mathematical Architecture for a 6D-to-3D Cut-and-Project and Incidence-Spectral Research Programme — FORM–Mosco Theorem Audit, Frozen-Target Refutation, and Homogenized Continuum Successor Programme — Rigorous Consolidated Core v2.8.0

Abstract

This record provides Rigorous Consolidated Core v2.8.0 of the Cut-and-Project / Incidence-Spectral Mathematical Core, with the Helix–Light–Vortex (HLV) framework retained as an active falsifiable testbed-theory instantiation under explicit claim boundaries. Version 2.8.0 integrates the completed CP-H1-FORM-MOSCO-001 theorem-audit chain through gate A6 and updates the continuum programme accordingly. The principal new result is negative and load bearing: A1 = PROVED A2 = PROVED A3 = PROVED A4 = PROVED A5 = PROVED A6 = REFUTED A7 = OPEN A8 = OPEN M1 = REFUTED M2 = OPEN Accordingly, the frozen CP-H1-FORM-MOSCO-001 continuum target E_infinity(u) = eta ∫_Omega |grad u|^2 dx cannot receive a theorem-level Mosco PASS. The frozen branch is therefore classified as: FAIL — FROZEN CONTINUUM TARGET REFUTED AT A6 / M1. This is an analytical theorem-level failure, not a failure of the previously locked finite computation. The authoritative finite CP-H1-FORM-MOSCO-001 v0.1.3 execution remains valid with implementation gates I0–I9 PASS. The finite diagnostic result is preserved exactly as originally reported. The theorem failure arises because an internal microscopic corrector of vanishing common-space amplitude can lower the asymptotic discrete energy below the frozen zero-corrector continuum target while the reconstructed states converge strongly in L2(Omega). Consequently, the geometry-derived coefficient eta = pi(64 - 23 sqrt(2))/48 remains mathematically valid as the common bulk pair-frequency coefficient and zero-corrector affine coefficient established in A4. It is not, however, the effective homogenized continuum coefficient for arbitrary bounded-energy sequences. This distinction is now explicit and permanent. The correct continuum successor must therefore solve an internal homogenization / cell problem rather than retrospectively retune eta inside the failed branch. The prospective successor architecture is based schematically on A_hom(xi) = inf_chi Σ_i ∫_{W_i} |xi · p_i + chi(y+q_i) - chi(y)|^2 dy. No value of the homogenized coefficient or tensor is assumed in v2.8.0. In particular, this record does not assume in advance whether the effective limit is isotropic, A_hom = c_hom I, or genuinely anisotropic. That question must be derived prospectively from the internal cell problem and symmetry structure. Published FORM–MOSCO theorem-audit chain ----------------------------------------- A1 theorem audit / closure: DOI: 10.5281/zenodo.22703339 A2 theorem audit / closure: DOI: 10.5281/zenodo.22709453 A3 theorem audit / closure: DOI: 10.5281/zenodo.22709633 A4 theorem audit / closure: DOI: 10.5281/zenodo.22709839 A5 theorem audit / closure: DOI: 10.5281/zenodo.22710024 A6 theorem audit / refutation: DOI: 10.5281/zenodo.22710226 The controlling CP-H1-FORM-MOSCO-001 provenance remains: Analytical architecture: DOI: 10.5281/zenodo.22657352 Pre-execution protocol: DOI: 10.5281/zenodo.22658157 Authoritative corrected implementation v0.1.3: DOI: 10.5281/zenodo.22664013 Locked finite diagnostic result: DOI: 10.5281/zenodo.22680099 Immediate public Mathematical Core predecessor: Rigorous Consolidated Core v2.7.1: DOI: 10.5281/zenodo.22685169 Residual preservation --------------------- Version 2.8.0 preserves all earlier positive, negative, open, invalid, and inconclusive results at their original scopes. In particular: - A1–A5 remain PROVED only within their published theorem scopes. - A4 remains a valid theorem concerning bulk pair-frequency control, boundary suppression, and the zero-corrector affine coefficient eta. - A5 remains a weak-gradient / directional-flux compactness result and is not retroactively promoted to strong L2 Rellich compactness. - A6 and M1 are permanently recorded as REFUTED for the frozen continuum target. - A7, A8, and M2 remain OPEN rather than being artificially completed after M1 failure. - the locked finite FORM-MOSCO execution remains I0–I9 PASS. - the unfavorable finest-level restricted-Voronoi weight-stability diagnostic TV_{2^21→2^22}(j=6) = 0.02520942687988281 remains explicitly preserved. The historical 0.02 CP-H1-CONT-BRIDGE-001 comparator was not a frozen FORM-MOSCO theorem or I0–I9 verdict gate, so this numerical residual does not cause the A6/M1 theorem failure and does not invalidate the locked finite execution. It remains an independent unfavorable finite quadrature-stability residual. No-retuning rule ---------------- No replacement coefficient, modified edge shell, altered window, new physical domain, changed boundary condition, modified form, or post-exposure threshold is inserted into CP-H1-FORM-MOSCO-001. The failed branch is preserved as failed. Any corrected continuum theory must be introduced prospectively under a separately named successor. Forward programme ----------------- The immediate mathematical successor is a prospectively frozen homogenized continuum programme, provisionally designated: CP-H1-HOM-MOSCO-001. Its first obligations are: 1. define the admissible internal corrector space and the exact cell functional independently of new numerical outcomes; 2. establish existence, gauge/constant-mode structure, and coercivity of the cell problem; 3. derive the effective homogenized quadratic form or tensor without fitting; 4. determine whether native symmetry forces isotropy or permits anisotropy; 5. only after the analytical architecture is frozen, compute the homogenized object numerically if necessary; 6. formulate a new Mosco / generalized-resolvent convergence programme for that prospectively defined effective form. A separate non-verdict-changing numerical robustness programme may also revisit the restricted-Voronoi quadrature residual at larger deterministic budgets. Such work cannot alter the negative CP-H1-FORM-MOSCO-001 theorem verdict. Programme interpretation ------------------------- The mathematical core remains theory-neutral wherever the results do not depend on HLV ontology. HLV remains an active falsifiable testbed theory instantiated on the declared 6D-to-3D cut-and-project carrier. The A6/M1 failure removes the frozen uncorrected scalar continuum mechanism from the admissible hypothesis set. It does not by itself falsify the cut-and-project carrier, the abstract incidence mathematics, or every possible HLV realization. Conversely, no alternative HLV continuum realization is regarded as validated merely because the failed branch admits a mathematically motivated successor. Any successor must independently satisfy its own prospective theorem, numerical, and physical-validation obligations. Threshold and claim governance ------------------------------ Version 2.8.0 retains the explicit threshold-provenance classification: TP1 — theorem-derived / exact TP2 — physics-derived TP3 — uncertainty / error-budget TP4 — literature / standard TP5 — prospective sensitivity / discrimination TP6 — engineering / numerical tolerance Prospective freezing prevents post-result threshold movement but does not convert operational numerical cutoffs into physical constants. The programme continues to distinguish: C0 — definition / axiom / protocol C1 — proved mathematics C2 — locked finite computation C3 — conditional physical bridge C4 — independent physical validation. No C4 validation is claimed for a fundamental HLV substrate, golden specificity, unified field theory, spacetime, particle sector, gravity, or cosmology. Version 2.8.0 therefore records both genuine mathematical progress and a genuine theorem-level falsification, while defining the scientifically admissible route forward without retrospective repair. Keywords: cut-and-project geometry; incidence-spectral mathematics; HLV testbed theory; Mosco convergence; theorem audit; theorem refutation; homogenization; internal corrector; cell problem; quadratic forms; regular model sets; continuum limit; threshold provenance; falsification; residual preservation; reproducibility