CP-H1-SRC-XSC-001: Cross-Scale Stability Architecture for the Fixed-15D Exterior-Moment Source Sector — Analytical Architecture / Pre-Source-Exposure Freeze v0.1.0

Abstract

This record freezes the analytical architecture for CP-H1-SRC-XSC-001, the next load-bearing source-sector programme of the theory-neutral 6D-to-3D Cut-and-Project / Incidence-Spectral research programme. CP-H1-SRC-XSC-001 follows the completed CP-H1-CONT-BRIDGE-001 common-space comparison programme. The authoritative locked bridge result is: CP-H1-CONT-BRIDGE-001: Locked Common-Space Technical Audit Results and Post-Exposure Corrected-Protocol Confirmation v1.0.0 DOI 10.5281/zenodo.22299784 That programme established finite deterministic numerical realizability and algebraic conformance of an HFF-independent restricted-Voronoi common-space bridge across all five frozen genuine-refinement levels. It did not establish source convergence, Mosco convergence, strong-resolvent convergence, a continuum operator, or continuum physics. CP-H1-SRC-XSC-001 now addresses a narrower mathematical question: whether the carrier-derived exterior-moment source sector exhibits cross-scale stability when represented in its common fixed 15-dimensional parent-bivector channel space. The frozen refinement levels remain a_j = 2^(-j/2), j = 0,...,4. For this programme the source is constructed on the rank-3/Freudenthal incidence complex K_core,j, not on the larger raw accepted Voronoi carrier V_core,j. This distinction is explicit in order to prevent the object-semantic mismatch identified and corrected during CP-H1-CONT-BRIDGE-001. The source channel is the fixed exterior space Lambda^2 R^6 ≅ R^15. For an oriented edge e=(v,w), with exact parent labels n_v,n_w and level-dependent parent-coordinate centroid nbar_j, the centered exterior-moment source is Omega_{j,e} = (n_v-nbar_j) wedge (n_w-nbar_j). Using the counting cochain metric, define the degree-one Hodge Laplacian Delta_{1,j} = B_{1,j}^T B_{1,j} + B_{2,j} B_{2,j}^T and the orthogonal harmonic projector P_{H,j}: C^1(K_j) -> ker Delta_{1,j}. If A_j is the edge-by-15 source matrix, the level-j source Gram operator is G_j = A_j^T P_{H,j} A_j in R^(15 x 15). The fixed 15-dimensional channel space makes direct cross-scale comparison possible even though the dimensions of the discrete harmonic spaces may vary with refinement level. No physical source amplitude or retrospective source renormalization is introduced in this architecture. In particular, no factor a_j^(-q), density compensation, coupling constant, flux normalization, or other scale factor may be selected after inspecting cross-scale behaviour. A scale-free secondary object is Ghat_j = G_j / tr(G_j), provided tr(G_j)>0. If the leading eigenvalue of G_j is simple and satisfies the prospectively defined spectral-gap admissibility rule to be frozen in the subsequent numerical protocol, the associated sign-independent rank-one projector is P_j = u_j u_j^T. Cross-scale directional comparison is then based on the projector distance d_jk = (1/sqrt(2)) ||P_j-P_k||_F = sqrt(1-(u_j^T u_k)^2), which is invariant under u_j -> -u_j. The architecture also defines the scale-free Gram distance g_jk = (1/sqrt(2)) ||Ghat_j-Ghat_k||_F, the normalized leading spectral gap gamma_j = (lambda_1,j-lambda_2,j)/lambda_1,j, and the leading spectral concentration eta_j = lambda_1,j / tr(G_j). All ten unordered level pairs are to be compared. No favourable subset of refinement levels may be selected after source exposure. This record intentionally does not specify numerical PASS/FAIL thresholds for projector stability, Gram stability or spectral-gap admissibility. Those quantities must be fixed prospectively in a separate CP-H1-SRC-XSC-001 Pre-Execution Numerical Protocol Freeze before any real G_j, source eigenvalue, source eigenvector, source projector or cross-scale distance is computed. If the leading eigenspace becomes degenerate or fails the prospectively frozen gap condition, the programme may not retrospectively change to a more favourable rank-r projector, selector, normalization or subset of levels. Such an outcome must be handled according to the later frozen failure semantics. The exposure firewall prohibits inspection of real cross-scale source Gram matrices, eigenvalues, eigenvectors, eigenprojectors, projector distances, Gram distances, beta-number trends or harmonic-basis trends before the numerical protocol and implementation are publicly frozen. This architecture also does not address Dirichlet-form normalization, Mosco convergence, strong- or generalized-resolvent convergence, continuum Hodge operators, physical source amplitudes, flux quantization, gauge coupling, particle physics, gravity, cosmology or experimental validation. A future PASS of CP-H1-SRC-XSC-001 could establish only finite cross-scale stability of the declared fixed-15D exterior-moment source sector over the five frozen refinement levels under prospectively fixed criteria. Five finite levels cannot by themselves establish an asymptotic continuum theorem. The fixed successor order is: 1. CP-H1-SRC-XSC-001 Pre-Execution Numerical Protocol Freeze. 2. CP-H1-SRC-XSC-001 deterministic Implementation Freeze. 3. Locked five-level source-sector execution and results. 4. Only after completion of this source programme: CP-H1-FORM-MOSCO-001. 5. Physical interpretation only after mathematical continuum closure. Principal provenance: CP-H1-SRC-001: DOI 10.5281/zenodo.22180728 CP-H1-SRC-UNI-001: DOI 10.5281/zenodo.22206661 CP-H1-ORI-SCALE-001: DOI 10.5281/zenodo.22207132 CP-H1-INT-EXACT-001: DOI 10.5281/zenodo.22209374 CP-H1-REF-001: DOI 10.5281/zenodo.22211960 CP-H1-REF-FEAS-001 Locked Results: DOI 10.5281/zenodo.22257899 CP-H1-CONT-BRIDGE-001 Locked Results: DOI 10.5281/zenodo.22299784 Mathematical Core v2.3.0: DOI 10.5281/zenodo.22258221