Gap-protected spectral invariants on T^d/Z_2: dimensional rigidity and spectral determination at d = 4

Version 2 (July 2026). Full revision of this deposit: revised manuscript, supplementary material, and the seven-script verification suite. The v1 files are superseded. The paper studies the flat orbifold T^d/Z_2 and the question of how many Laplacian eigenmodes on the covering torus are needed to distinguish all 2^d singular points. Main results: (i) a spectral cutoff K*, defined by the rank of the dynamical Gram matrix on the fixed-point set, with K* = 5 at d = 4 and mode count N_4(5) = 137, decomposing as 137 = 1 + 8 + 128 into three sectors that are each topological invariants of the orbifold; (ii) gap protection: the count is stable under bounded perturbations small relative to the spectral gap; (iii) dimensional rigidity: the matching condition 2d = 2^(d-1), balancing first-shell lattice vectors against the fixed-point count, holds only at d = 4; (iv) a determination theorem for the theta function of Z^4 within an explicitly described family of quaternary forms; (v) diagonalization of the inter-fixed-point kernel by Krawtchouk polynomials on the Hamming scheme. The deposited manuscript extends the journal version by one closing section, which records a numerical observation conditional on three explicitly stated premises. Files: manuscript.pdf (main paper, extended edition), sm-appendix.pdf (supplementary material with complete proofs), seven Python scripts that reproduce every numerical claim (verify.py, verify_spectral_bounds.py, verify_krein.py, classify_crystallographic.py, verify_computational.py, verify_prime_family.py, verify_theta_determination.py), README.md, requirements.txt, LICENSE. Maintained code: https://github.com/kootru-repo/gap-protected-spectral-invariants