A Rigorous Mathematical Architecture of the Helix–Light–Vortex Framework: Typed Operators, Abstract Incidence Dynamics, Golden Cut-and-Project Carriers, Locked Falsification Results, Gauge Hamiltonians, and Claim-Conditioned Validation — Rigorous Consolidated Core v2.1.5

Abstract

This record contains Rigorous Consolidated Core v2.1.5 of the Helix–Light–Vortex Framework (HLV). HLV is positioned in this revision as: Helix–Light–Vortex Framework (HLV) A Cut-and-Project and Incidence-Spectral Research Programme. The term “Framework” denotes the existing mathematical architecture and provenance of the programme. The active scientific direction is the HLV Cut-and-Project / Incidence-Spectral Research Programme. The present work is not presented as a validated fundamental physical theory. The consolidated core separates: - native 6D-to-3D cut-and-project carrier construction; - abstract incidence structure; - finite carrier fingerprints; - spectral and Hodge diagnostics; - explicitly postulated free dynamics; - continuum obligations; - gauge-sector mathematics; - theorem-level no-go boundaries; - and physical interpretation. Version 2.1.5 preserves the previous locked negative and bounded results while integrating the completed HLV-R-MECH-001 mechanism chain. The principal established structural constraints remain: 1. Native projected geometry HLV-LAYER-ORIGIN-007F / 007F-CERT establishes that the tested native projected tetrahedral assembly is not a strict global face-to-face simplicial realization. The local projected rank-three cell geometry remains mathematically valid, but the tested assembly cannot be promoted to a global native piecewise-flat or Regge manifold without a new validated global metric-complex construction. 2. Abstract incidence complex HLV-DG-001 independently certifies the retained parent-labelled structure as an exact finite oriented 0–3 chain complex with (N0, N1, N2, N3) = (1110, 5345, 6960, 2826), boundary ranks (1109, 4137, 2823), Betti vector (1, 99, 0, 3), and exact chain identities B1 B2 = 0, B2 B3 = 0. 3. Static and dynamic specificity HLV-DG-002 rejects HLV-specificity of the frozen cross-grade Hodge signature under the complete R/Q/W null ensemble. HLV-FA-DYN-001 rejects overall HLV-specific dynamic transport under its frozen R/Q/W ensemble. Its degree-preserving rewire family separates strongly, but the broader geometric controls defeat the complete specificity claim. HLV-DS-SPEC-001R separately rejects full native-carrier graph-spectral specificity under the frozen R/Q/W/IRR ensemble. Its locked machine verdict is: DSSPEC001R_FAIL_PARTIAL_SIGNATURE_OR_FAMILY_ONLY The degree-preserving R family passes both QSPEC and RRESP, while Q, W, and IRR fail the complete frozen criteria. This result established only a bounded graph-spectral structural residual and did not identify its mechanism. 4. HLV-R-MECH-001 mechanism localization Version 2.1.5 integrates the first prospectively frozen mechanism localization of that surviving graph-spectral R residual. Controlling protocol: Krūger, M. (2026). HLV-R-MECH-001: Prospective Triangle-Matched Mechanism Test of the Surviving Degree-Preserving Rewire Spectral Residual — Pre-Execution Protocol Freeze v0.1.0. Zenodo. DOI: 10.5281/zenodo.22166283 Authoritative corrected implementation: Krūger, M. (2026). HLV-R-MECH-001: Deterministic One-Click Engine for Triangle-Matched Rewire Mechanism Testing — Corrected Implementation Freeze v0.1.2 [Computer software]. Zenodo. DOI: 10.5281/zenodo.22170307 Locked confirmatory results: Krūger, M. (2026). HLV-R-MECH-001: Locked Confirmatory Results for Triangle-Matched Mechanism Testing of the Degree-Preserving Rewire Spectral Residual v0.1.0 [Computer software]. Zenodo. DOI: 10.5281/zenodo.22170620 Locked result ZIP SHA-256: 69f927faba83b203d7dffbf028de680e6a6a3e36818002bc0e1ab27d5c01797d The successful locked machine verdict is: RMECH001_PASS_TRIANGLE_MATCH_COLLAPSE_PATTERN The experiment used two fresh prospectively frozen control families. R_DEG preserves the exact labelled target degree sequence while allowing the global triangle count to vary. R_TRI preserves both the complete labelled degree sequence and the exact global target triangle count T = 6960. Both families were matched in rewiring depth. The fresh R_DEG baseline reproduces the earlier graph-spectral residual: QSPEC: PASS RRESP: PASS with robust margins approximately 5.7870 and 7.7221. Under exact triangle matching, R_TRI returns: QSPEC: FAIL RRESP: FAIL under the complete prospectively frozen multi-band gate. The target-control distance relative to R_DEG is reduced by approximately: 74.85% for QSPEC, and 69.39% for RRESP. This prospectively localizes triangle/face organization as a major contributor to the previously observed degree-preserving rewire spectral residual. The result has an exact low-order spectral basis. For a simple graph Laplacian L = D - A, the identities Tr(L) = sum_i d_i, Tr(L^2) = sum_i d_i^2 + sum_i d_i, and Tr(L^3) = sum_i d_i^3 + 3 sum_i d_i^2 - 6T hold. Consequently, preserving the complete degree sequence and exact triangle count fixes the target values of the first three raw Laplacian spectral moments exactly. This mechanism result does not prove that global triangle count is the sole cause of the residual. In R_TRI, the target distances remain above the corresponding maximum leave-one-out distances and retain robust margins above 1.5, but only one of three frozen spectral bands passes in each signature. Predeclared secondary diagnostics also retain structural differences in: - local per-vertex triangle distribution; - clustering; - four-cycle counts; - assortativity; - algebraic connectivity; - and other higher-order local structure. The natural successor is therefore a separately prospectively frozen local-triangle-profile and short-cycle mechanism test. 5. Carrier fingerprint status The finite orientation-sensitive carrier fingerprint remains a bounded C2 result. Stage 6B and Stages 12–16 support an internally replicated orientation-sensitive finite carrier fingerprint under the stated frozen nulls. Stage 17 blocks the stronger fixed-window R = 2,3,4 scaling claim. Stage 18 remains diagnostic and does not overwrite that result. No injectivity theorem currently maps the finite fingerprint uniquely back to a microscopic carrier or physical spacetime. 6. Minimal free dynamics HLV-FA-0 remains explicitly axiomatic rather than derived. It postulates the cochain Hilbert space H_FA = direct sum from p=0 to 3 of C^p(K_abs; C), the minimal incidence-linear self-adjoint Hodge–Dirac generator D_K = d + delta, and one symbolic positive energy scale E_H. The numerical value of E_H is not predicted. The DG-001 Betti vector implies 103 exact Hodge–Dirac zero modes on the finite target. No particle masses, gauge interactions, gravity, dark-sector portal, or absolute physical energy scale follows from FA-0 alone. 7. Scalar-mode and gauge boundaries The core retains theorem-level no-go and covariance results showing, among other things, that: - a positive carrier Laplacian cannot generate a homogeneous negative quadratic direction from a nonnegative local mass; - a centered deformation F(L_G) with F(0)=0 leaves the constant-mode quadratic coefficient unchanged; - a bare tensor Laplacian L_G tensor I_r is not locally U(r)-frame covariant without independently supplied link transporters; - a fixed wrong-sign coefficient on a genuine refinement generator with diverging ultraviolet edge produces an unbounded negative spectral minimum. The finite compact-group gauge Hamiltonian remains mathematically well-defined with a positive finite-complex spectral gap, but this does not establish a continuum Yang–Mills mass gap. 8. Continuum boundary A single fixed bounded-degree, bounded-weight carrier has bounded Laplacian spectrum and therefore no intrinsic ultraviolet limit. A genuine continuum programme requires a changing refinement family, explicit scaling, identification maps, and an appropriate convergence theorem such as Mosco or generalized strong-resolvent convergence. No such native HLV continuum theorem is established in the present core. Scientific status after v2.1.5 The HLV Framework contains a reproducible mathematical cut-and-project construction, a certified finite abstract incidence complex, several rigorous operator sectors, theorem-level no-go results, finite carrier diagnostics, preserved negative specificity results, and a prospectively confirmed graph-spectral mechanism localization. The new HLV-R-MECH-001 result substantially clarifies the origin of the earlier degree-preserving R residual: triangle/face organization is a major contributor. It does not rescue the previously failed full carrier-specificity claims and does not establish that the native golden 6D-to-3D carrier is a fundamental physical substrate. The present evidence does not establish: - physical selection of the golden ratio; - a unique microscopic 6D-to-3D geometry; - Lorentzian spacetime; - a native Regge manifold; - Standard-Model recovery; - a Higgs mechanism; - particle masses; - an absolute HLV energy scale; - continuum Yang–Mills; - gravity; - dark matter; - dark energy; - cosmology; - or experimental validation. The active programme is therefore intentionally narrower: to determine which structural, incidence, spectral, and refinement properties of cut-and-project and related discrete systems survive increasingly strong matched alternatives, and to distinguish general mathematical mechanisms from genuinely carrier-specific effects before any physical interpretation is attempted. This revision strengthens mechanism identification and falsification discipline while leaving the fundamental physical claim level unchanged.