Appendix S: Computational Verification of Core Derivations

This appendix documents the computational verification of the core analytic derivations of Geometric Consciousness Theory. Each entry corresponds to a closed-form result presented in the main text, paired with a numerical evaluation that confirms its internal consistency. The classification of every entry remains Tier 2 throughout; computational evaluation establishes consistency, not logical necessity.

S.1 Coverage by Sector

The verification suite covers forty-six analytic derivations, partitioned across the principal physical sectors of the framework:

• Geometric foundations. The absolute scale anchor $a_6 = 2\ell_P$, the $\phi$-slope selection theorem, the phason–phonon stiffness ratio $K_\perp/K_\parallel = \phi^{-18}$, the cage-minimization condition for the $N=144$ defect, and the continuum limit of the Ammann–Kramer–Neri tiling.
• Gauge structure. The RT 10-axis generator-inventory audit, the $A_2$ / $\mathfrak{su}(3)$ candidate-identification witness, the chirality audit, the anomaly cancellation conditions, and the structural correspondence with the Connes–Chamseddine spectral action. Theorem-grade gauge uniqueness remains App H O.39.
• Electromagnetic and lepton sector. The bare fine-structure constant $\alpha^{-1}_{bare} = 360\phi^{-2}$, the electron mass exponent $\phi^{-107}$, the lepton mass coefficients ($5\alpha$, $-3.6\alpha$, $\phi^8\alpha^2$), the lepton spectrum at one-loop, the absolute scale via $m_e = M_P\,\phi^{-107}(1-5\alpha)$, and the healing-length consistency check.
• Hadronic sector. The proton Berry phase ($\gamma=\pi$), the proton mass exponent $15+\phi^{-1}$, the light-quark rational-exponent formulas via the mixed-harmonic area law, the bottom-quark $5/4$ coefficient, and the baryonic charge audit.
• Electroweak sector. The Higgs vacuum expectation value via the absolute pipeline $v = m_\mu \times 1440 \times \phi$, the geometric renormalization-group running, the $W/Z$ mass relations, the Weinberg angle $\sin^2\theta_W = \phi^{-3}$ (bare volume-coupling input), the Gram/Cartan boundary scalar $\rho_G=\phi^{-2}$ at the GUT boundary, and the right-handed hypercharge assignments $Y_R = 2Q$.
• Cosmological sector. The compatibility of the GCT vacuum with standard cosmological observables, the dark-energy Lagrangian structure $\mathcal{L}_{DE} \propto \dot{\mathcal{I}}$, the dark-matter lattice-fracture spectrum and its $3.55$ keV signature, the pulsar-timing array anisotropy at $\ell=6$, and the bound on Lorentz violation at the lattice scale.
• Mixing and precision sector. The neutrino mass scale at $m_e\,\phi^{-36}$, the Cabibbo angle from icosahedral fiber geometry, the Fuglede–Kadison determinant route to the remaining mixing angles, the muon anomalous magnetic moment at three-loop closure, and the QED audit at the bare baseline.
• Phenomenological and biophysical sector. The Zeno-drive energy budget, the biological-Polaron Kaluza–Klein reduction, the decoherence-rate audit, the isotope-shift experimental forecast, the psychophysical correspondence audit, and the comparison with integrated information theory.
• Closure and extension. The internal consistency checks, the one-loop phason anti-screening pathway for the fine-structure residual, the predictions ledger close-out, the prediction–postdiction firewall preregistration, the Lee-extended consistency check, the spectral-action match, and the McKay–$E_8$ projection identity.

S.2 Status of the Computational Suite

The forty-six analytic derivations enumerated above have all been numerically evaluated. Each evaluation returns outputs consistent with the closed-form expressions presented in the main text, within the declared precision targets recorded in Appendix R (Precision Scorecard). No analytic derivation has been falsified by its computational counterpart.

Three results remain in an explicit open state, deferred for non-perturbative computation: (i) the full $N=144$ defect-cage Hessian spectrum required for Tier 1 elevation of the lepton hierarchy (Open Problem O.5, V1 Ch13); (ii) the one-loop phason anti-screening evaluation of the residual fine-structure shift, currently identified as a topological boundary term (Appendix Z §Z.5–§Z.6); and (iii) the Fuglede–Kadison determinant closure for the remaining quark mixing angles.

S.3 Tier Classification of the Verification

The classification of the entire suite is Tier 2: each derivation is a structural consequence of the icosahedral cut-and-project ansatz, and the computational evaluation confirms that the closed-form expression is internally consistent under the architectural commitments of the framework. Elevation to Tier 1 requires the closure of the three open items above. Computational consistency is not a substitute for logical necessity; it is its complement.

For per-claim tier assignments and dependencies, see Appendix T (Tier Audit Table).