CP-H1-HOM-MOSCO-001: Prospective Homogenized Cell-Problem Continuum Architecture — Analytical Architecture / Pre-Theorem Hypothesis Freeze v0.1.0

Abstract

This record prospectively freezes the analytical architecture of CP-H1-HOM-MOSCO-001, the homogenized cell-problem successor to the theorem-level failed CP-H1-FORM-MOSCO-001 continuum target. It is a PRE-THEOREM / PRE-NUMERICAL architecture record. No HOM cell-problem solver has been executed or inspected for this freeze. No effective coefficient has been fitted. No isotropy assumption is imposed. No predecessor result is retrospectively modified. Predecessor status ------------------ The predecessor CP-H1-FORM-MOSCO-001 branch remains permanently classified as: FAIL — FROZEN CONTINUUM TARGET REFUTED AT A6 / M1. Its theorem ledger is preserved: A1 = PROVED A2 = PROVED A3 = PROVED A4 = PROVED A5 = PROVED A6 = REFUTED A7 = OPEN A8 = OPEN M1 = REFUTED M2 = OPEN. The decisive negative theorem-audit record is: Krüger, M. (2026). CP-H1-FORM-MOSCO-001: Analytical Theorem Audit and Refutation of Gate A6 — Internal Corrector Obstruction and Failure of the Frozen Mosco Liminf Target v0.1.0. Zenodo. DOI: 10.5281/zenodo.22710226 The predecessor finite execution remains valid with I0–I9 PASS. The A6/M1 failure is analytical and does not invalidate that finite computation. Scientific motivation --------------------- The predecessor continuum target was E_infinity(u) = eta ∫_Omega |grad u|^2 dx, with eta = pi(64 - 23 sqrt(2))/48. A4 established eta as the exact bulk pair-frequency / zero-corrector affine coefficient for the frozen spherical-window/native-shell geometry. A6 subsequently proved that microscopic internal-space correctors can lower the asymptotic energy below this zero-corrector value while their amplitude vanishes in the common-space L2 representation. Therefore eta cannot be promoted directly to the effective homogenized continuum coefficient. This successor does not repair or retune the failed predecessor target. It introduces a new prospectively frozen homogenization problem. Frozen native geometry ---------------------- The successor retains the exact 6D-to-3D cut-and-project geometry: p_i = P e_i q_i = Q e_i for i = 1,...,6, with PP^T = QQ^T = I_3 and PQ^T = 0. The internal window remains W = B_1(0), the physical comparison domain remains Omega = B_2(0), and the genuine-refinement scale remains a_j = 2^{-j/2}. The native positive edge shell remains {e_i}_{i=1}^6. For each direction, W_i = W ∩ (W - q_i). No window, edge family, physical domain, refinement law, or boundary condition is altered in order to repair the predecessor result. Corrector architecture ---------------------- The frozen zero-mean corrector space is X_0 = {chi in L2(W) : ∫_W chi(y) dy = 0}. The zero-mean condition fixes the additive-constant gauge. Define the six-channel Hilbert space V = direct sum over i=1,...,6 of L2(W_i). The internal difference operator is (D chi)_i(y) = chi(y + q_i) - chi(y), for y in W_i. The macroscopic-gradient injection is (B xi)_i(y) = xi · p_i. Primary frozen cell problem --------------------------- The central object of CP-H1-HOM-MOSCO-001 is prospectively frozen as A_hom(xi) = inf_{chi in X_0} Σ_{i=1}^6 ∫_{W_i} |xi · p_i + chi(y+q_i) - chi(y)|^2 dy. Equivalently, if R = closure(Ran D), then A_hom(xi) = ||Pi_{R-perp} B xi||^2. This defines a symmetric positive-semidefinite effective matrix A_eff through A_hom(xi) = xi^T A_eff xi, with the architecture-level bound 0 <= A_eff <= eta I_3. The predecessor A6 result additionally implies A_eff != eta I_3: for at least one nonzero macroscopic direction xi, the relaxed energy lies strictly below the zero-corrector value eta |xi|^2. No numerical value of A_eff is assigned in this freeze. Scalar-versus-tensor decision rule ---------------------------------- The architecture explicitly permits both outcomes. If exact native symmetry and representation theory prove that A_eff = c_hom I_3, then the effective continuum form is isotropic and the scalar coefficient c_hom may be reported as a derived result. If the exact symmetry permits a non-scalar tensor, the full tensor A_eff must be retained. Anisotropy may not be removed after exposure by averaging eigenvalues or replacing A_eff with a fitted scalar. Isotropy is therefore a theorem target, not an assumption. Prospective theorem programme ----------------------------- The successor receives a new theorem ledger. The immediate HOM-A analytical obligations are: H1 — Kernel / Poincare / closed-range structure Prove or refute that the internal difference operator has only the additive-constant kernel and establish the coercive / closed-range estimate needed for the cell problem. H2 — Cell minimizer existence and uniqueness Prove existence and uniqueness modulo constants of cell-problem minimizers, or classify the formulation honestly as relaxed-only if exact minimizers do not exist. H3 — Regular-core density Prove that smooth zero-mean correctors approximate the relaxed L2 cell energy and therefore support later discrete recovery constructions. Only after H1–H3 are resolved may the programme proceed to: H4 — exact symmetry / scalar-versus-tensor classification; H5 — strict ellipticity / positive definiteness of A_eff; followed by new homogenized liminf, recovery and boundary-identification obligations. A theorem-level HOM-MOSCO PASS requires the complete prospectively defined theorem chain, including new liminf and recovery proofs for the relaxed homogenized target. No finite solver trend may substitute for those proofs. Numerical firewall ------------------ No numerical cell-problem calculation belongs to this freeze. Only after the analytical objects and theorem obligations have been fixed may a deterministic numerical protocol be frozen. Any later numerical programme must prospectively specify: - approximation space; - gauge enforcement; - quadrature rule and error budget; - solver tolerances; - convergence diagnostics; - tensor extraction; - symmetry checks; - independent discretization or robustness controls; - deterministic seeds where applicable; - machine-readable provenance; - explicit PASS / FAIL / INCONCLUSIVE semantics. Numerical output may estimate a mathematically defined A_eff. It may not redefine the mathematical target after exposure. Anti-retuning / anti-rescue rule -------------------------------- CP-H1-HOM-MOSCO-001 is a new prospective branch. It may not be used to rewrite CP-H1-FORM-MOSCO-001. In particular, the following are prohibited as retrospective repairs of the predecessor: - replacing eta by a fitted value; - inserting a newly computed A_eff into the old target; - modifying the acceptance window; - changing the native edge shell; - changing the physical domain; - changing the boundary condition; - changing the refinement law; - introducing a post-exposure numerical threshold; - relabelling A6/M1 as OPEN or PASS. The predecessor FAIL remains permanent. Claim boundary -------------- This architecture does not establish: - existence of an exact cell minimizer; - a Poincare inequality for the corrector operator; - closed range of D; - isotropy of A_eff; - a numerical value of A_eff; - strict positive definiteness of A_eff; - homogenized Mosco convergence; - generalized or strong-resolvent convergence; - continuum Hodge theory; - physical spacetime; - gauge dynamics; - particle physics; - gravity; - cosmology; - empirical validation of HLV. It defines the prospective mathematical problem required to test those later continuum claims rigorously. Programme position ------------------ The mathematical programme remains theory-neutral where its statements depend only on the declared cut-and-project and incidence-spectral structures. HLV remains an active falsifiable testbed-theory instantiation under explicit claim boundaries. The predecessor A6/M1 failure removes the frozen uncorrected scalar continuum mechanism from the admissible hypothesis set. This successor tests a mathematically motivated homogenized alternative without claiming in advance that it succeeds. Next operation after publication -------------------------------- Do not run a cell-problem solver yet. The next scientific operation is the HOM-A theorem audit: H1 — kernel / Poincare / closed-range theorem; H2 — existence and uniqueness modulo constants of minimizers; H3 — smooth-corrector density in the relaxed cell energy. Only after those analytical obligations are resolved should H4 symmetry, H5 ellipticity, and a prospectively frozen deterministic numerical cell-problem protocol be attempted. Evidence status --------------- This record is an analytical architecture / hypothesis freeze. It contains definitions, exact inherited geometry, architecture-level Hilbert-space consequences, theorem obligations and failure semantics. It contains no new HOM numerical target result and no HOM-MOSCO convergence claim. Keywords: cut-and-project geometry; homogenization; cell problem; Mosco convergence; internal corrector; quasiperiodic graph; regular model set; effective tensor; quadratic forms; continuum limit; theorem audit; hypothesis freeze; residual preservation; falsification; HLV testbed theory; reproducibility