Appendix G: Tier Classification Decision Tree

G.1 Classification Algorithm

To maintain the highest standards of scientific integrity, every claim within the GCT framework is subjected to a rigorous Epistemic Filter. This decision tree ensures that the reader can distinguish between logical certainties, geometric modeling, and phenomenological calibration.

The Decision Logic:

1. Does the claim follow necessarily from the Axioms (Presence and Intelligibility) and formal mathematics?

• YES: Proceed to Tier 1.

2. Is the claim forced by a combination of parsimony, empirical adequacy, and the Uniqueness Absolute, but where the requirement to impose those constraints is itself an architectural commitment not deductively forced by the Ontological Axioms alone?

• YES: Proceed to Tier 1/2 (Uniqueness-Justified Structural Postulate).

3. Does the claim depend on the specific Icosahedral Cut-and-Project ansatz ($E_8 \to 6D \to 3D$)?

• YES: Is the result a structural necessity of that specific geometry?
• YES: Proceed to Tier 2.

4. Does the claim require calibration to empirical data or rely on unproven hydrodynamic assumptions?

• YES: Proceed to Tier 3.

5. Is the claim an exploratory calculation where the dimensional structure is correct but the numerical coefficient is uncertain by more than one order of magnitude, or an ontological extension outside the core falsifiable theory?

• YES: Proceed to Tier 4 (Speculative / Exploratory). All Tier 4 claims carry explicit order-of-magnitude bounds and are segregated from the core falsifiable program.

G.2 Tier 1 Results Summary: Logical Necessities

These results are considered Exact. They are the fundamental constants of any intelligible reality. If these results are false, the axiomatic foundation of the theory is flawed.

• Wheeler–DeWitt Equation ($\hat{H}\Psi=0$): Derived necessarily from the requirement of a self-defining, closed-system Nullity.
• Dimensionality ($D=3$): Derived from the Knot-Theoretic Bound required for persistent identity and information storage.
• Gauge Masslessness ($m_\gamma = 0$): Derived from the requirement of Kinematic Agency (Phase Freedom) in a supersolid vacuum.
• Lattice Action Postulate ($\hat{\hbar}=1$): Discrete unit of action within the lattice; postulated as an irreducible structural assumption.

G.2.5 Tier 1/2 Results Summary: Uniqueness-Justified Structural Postulates

These results are uniquely forced by the conjunction of the Ontological Axioms and the Uniqueness Absolute, but the requirement to impose those constraints is itself an architectural commitment.

• Icosahedral Projection ($E_8 \to \mathbb{Z}^6 \to H_3$): The unique discrete projection satisfying parsimony and empirical adequacy under the Uniqueness Absolute (Lemma III, Appendix U §U.6.1–§U.6.2). Inherits the modular-reduction status of Theorem T-McKay (App U §U.7): the McKay-correspondence step ($2I \to E_8$) and the Elser–Sloane projection step ($E_8 \to \mathbb{Z}^6$) are closed at Tier 1; $\phi$ enters the projection geometry through the $H_4$ Cartan entry $-\phi$ and the root metric rather than as a Coxeter-element eigenvalue modulus (Lemma T-McK.4).
• Dimensional Embedding ($D=6$): Derived as the unique minimal periodic parent lattice supporting the icosahedral acceptance window (Theorem T-McKay, Appendix U §U.7). The classification is Tier 1/2 under the modular reduction of App U §U.7.1–§U.7.6. Four of the five required forcing lemmata are closed at Tier 1 in the published literature (T-McK.1a — Schur cover of $A_5$; T-McK.2 — McKay correspondence $2I \to E_8$; T-McK.3 — Elser–Sloane projection $E_8 \to \mathbb{Z}^6$; T-McK.4 — $\phi$ from the $H_4$ Cartan/root metric). The single remaining open step (Lemma T-McK.1b — Spinor stability via APS defect-index identification, App U §U.7.6.3) is reducible to a finite spectral-flow computation of the icosahedral $\eta$-invariant on the boundary $\partial \mathcal{M}_{RT}$ of the rhombic-triacontahedron acceptance window, with explicit references to Atiyah–Patodi–Singer 1975 + Connes 1994 + Connes–Moscovici 1995. Closure of U.7.6.3 would elevate the entire chain to Tier 1.

G.3 Tier 2 Results Summary: Geometric Consequences

These results are Structurally Robust within the Icosahedral ansatz. They represent the "Derived Scaling Laws" of the Operating System.

• Icosahedral Selection Theorem: App U proves point-group uniqueness of icosahedral symmetry conditional on H1. The Golden Ratio ($\phi$) projection is the primary Diophantine candidate for the global entropy-packing optimum of $J(\Omega)$, not a closed slope-maximality theorem.
• Newton’s Gravitational Constant ($G$): Tier 2 thermodynamic/acoustic mechanism with Tier 4 Planck-link inheritance from O.14 and Tier 3 dimensional anchoring; the registered numerical value is a 2274 ppm postdiction.
• The Weinberg Angle ($\sin^2 \theta_W = \phi^{-3}$): Derived from the volume ratio of the $E_\perp$ and $E_\parallel$ subspaces.
• The Fine-Structure Constant ($\alpha$): Derived from the Golden Angle impedance mismatch and $N=144$ dual-surface potential screening.
• Lepton Mass Ratios ($\phi^{11}, \phi^{17}$): Derived from the fractal resonance gaps of the dodecahedral defect cage.
• Electron Mass Exponent ($\phi^{-107}$) [Tier 2 framework + Tier 3 integer anchor + Tier 4 K-theoretic physical-link conjecture pending O.14]: The K-theoretic gap-labeling framework (Bellissard + Connes-Moscovici + Pimsner-Voiculescu, Ch07 §7.2.2) is Tier 2 rigorous. The integer $|n| = 107 = 1^2 + 5^2 + 9^2 = \sum m_i^2(H_3)$ is canonically identified as the second Newton power-sum of the $H_3$ Coxeter-exponent multiset (engine protocol_o14_coxeter_exponent_squares.py; unique to $H_3$ among finite irreducible Coxeter groups at any rank) — Tier 2 group-theoretic. The structural-link chain from the $V$-side Coxeter invariant to the 6D trace-image projection $\tau_*(K_0(\mathcal{A}))$ remains Tier 4 conjectural, recorded as the residual of Open Problem O.14.
• Muon Drag Coefficient ($5\alpha$) [Tier 2 mechanism + Tier 3 coefficient normalization]: The $A_5$ channel multiplicity is algebraic; converting the five channels into the equal-weight pole-mass coefficient $+5\alpha$ is a physical-normalization rule pending O.5 self-energy closure.
• Electron Mass ($m_e$): Planck-link closure check anchored on measured $m_e$ (A1), with the physical-link chain pending O.14 ($m_e = M_{Planck} \cdot \phi^{-107} \cdot (1 - 5\alpha)$).
• Proton Mass Formula ($m_e \phi^{15.618}$): The geometric scaling law identifying the proton as a Baryonic Triad ($3 \times 5 = 15$) stabilized by electroweak projection pressure.
• Neutrino Mass Scale ($m_e \phi^{-36}$): Identified as the Second-Order Phason Coupling limit.
• Muon g-2 [Tier 3 fitted/equal-weight $1/5$ coefficient + A3 on top of a geometric mechanism class; Tension under WP2025; no robust confirmation]: The 3-loop phason vertex correction yields $\Delta a_\mu \approx 250.65 \times 10^{-11}$. The fivefold $I_h$ channel partition is the mechanism class; converting it to the equal-weight numerical $1/5$ vertex factor is a Tier 3 coefficient pending O.26/O.5, and the numerical comparison uses A3 measured low-energy $\alpha(0)$. Under the consolidated WP2025 lattice-HVP synthesis (Aliberti et al. 2025) and CMD-2/CMD-family R-ratio reanalysis context, the SM–experiment gap is $\sim 0.6\sigma$, and the GCT geometric correction sits $\sim 3.4\sigma$ above the consolidated SM total. See App V §P.5 HVP Condition.
• Biogenic Dark Energy — Lagrangian Structure ($\mathcal{L} \propto \dot{\mathcal{I}}$): [Tier 2] — The coupling of complexity growth to metric expansion is structurally derived from the phason vacuum action (V2 Ch14 §14B.1).
• Biogenic Dark Energy — Amplitude Prediction (first-principles $\rho_\Lambda$): [Tier 4 — Known Failure] — First-principles magnitude overshoots observed $\rho_\Lambda$ by $10^{220}$; explicitly quarantined (V2 Ch14 §14B.0).
• The Born Rule [Tier 2 manuscript theorem assertion — App D; non-executable in the engine]: The manuscript derives the result via Gleason's Theorem applied to the GCT lattice Hilbert space, but the engine records this as a manuscript-level theorem assertion rather than an implemented verifier: non-negativity, normalization, non-contextuality, and Gleason applicability are not independently checked by code.
• Light Quark Absolute Masses: Derived via Mixed-Harmonic Area Law ($\phi^2$) and exact topological surface friction ($12\alpha$, $5/4$).
• Gauge Group Structure ($SU(3) \times SU(2) \times U(1)$) and Gluon Count ($N=8$): Derived from the holonomy structure of the icosahedral cut-and-project ansatz; the eight gluons follow from the integer rank reduction of the ten three-fold axes of the rhombic triacontahedron acceptance window.
• Spin-Statistics Theorem: Derived from the topology of tethered ribbons in three-dimensional space (Appendix C §C.4). The result is Tier 2; the algebraic-topology step (WZW evaluation, normalization of the generator of $H^3(SU(2);\mathbb{Z})$) is Tier 1, and the identification of GCT's tethered defects with this WZW framework is Tier 2. Full Tier 1 elevation of the complete result is reduced to a single bounded analytic step: the APS spectral-flow computation of Lemma T-McK.1b (App U §U.7.6.3), which would close the defect-index identification ($n_{\mathrm{def}} = -107$ conjectured) by evaluating the icosahedral $\eta$-invariant on $\partial \mathcal{M}_{RT}$.
• Polaron Unity Proposition (V1 §11.12, Proposition 11.12.A, conditional on Open Problems O.18 + Y.6.3a/b): Topological argument is Tier 3 conditional for the canonical trefoil-knot case and for the general prime-knot extension via the modular reduction in Appendix Y. The finite-level ambient space is $M_n \cong S^1 \times \mathbb{R}^3$; the classical Wirtinger/product split is a 3-manifold argument and applies only after a fixed-slice reduction $H_{Y.1}$ that the Ch11 ansatz has not yet proved. The trefoil meridian trace computation in §Y.6.3a is a finite-matrix surrogate pending a unitary finite-dimensional faithful quotient of the meridian subalgebra; it does not close the ambient-type gap. Downstream uses inherit the Tier 3 conditional disposition pending the $H_{Y.1}$ sub-closure of O.18 plus the Anderson-Putnam-to-knot-complement extension, canonicity of $K \mapsto \mathcal{A}_K$, finite-quotient meridian trace construction, and KO-dim-6 sign verification (O.32).
• Tau Screening Coefficient ($-3.6\alpha$) [Tier 2 mechanism + Tier 2 integer-pair $(D=18, N=5)$ + Tier 3 combination rule $c = -D/N$ pending O.26b]: $D = 18$ is the load-bearing icosahedral anchor — the $H_3$ Shephard-Todd invariant-degree sum $\{2, 6, 10\}$, sum $= 18$, uniquely $H_3$ among rank-3 Coxeter groups (Anchor A per App H O.26; engine gct_tau_uniqueness.py). The 2D-RT-face-in-6D tangent-bundle decomposition $6 + 6 + 6 = 18$ enters as a 6D-ambient consistency cross-check (Anchor B), not as a second independent icosahedral anchor — every 6D-ambient theory with this triple-decomposition yields 18 regardless of icosahedral structure. $N = 5$ from RT pentagonal vertex enumeration (same anchor as muon $+5\alpha$). The combination rule $c = -D/N$ pairing the two integers as a ratio is a Tier 3 postulate pending Open Problem O.26b.
• Higgs VEV ($v \approx 246$ GeV) [Tier 2 mechanism + Tier 3 calibrated integer factor + Tier 3 numerical residual]: Derived via the Absolute Pipeline $v = m_\mu \times 1440 \times \phi$, where $1440 = N_{\rm cage} \times N_{C_3} = 144 \times 10$ is the canonical muon-defect-saturation pathway (V3 Ch05 §5.2.1). O.20 supplies enumeration-level closure: 1440 is non-unique in factorisation; three additional independent icosahedral factorisations $(120 \times 12)$, $(24 \times 60)$, $(15 \times 96)$ serve as dimensional cross-checks rather than competing derivations. The structural derivation is geometric; quantitative agreement with the experimental value is reported in Appendix R.

G.4 Tier 3 Results Summary: Phenomenological Models

These represent the Frontier of the program. They are models where first-principles mechanical derivations are currently incomplete.

• Biogenic Dark Energy — Susceptibility Magnitude ($\chi \approx 10^{121}$): [Tier 3 — Imported from standard cosmology; equals the Bekenstein-Hawking entropy $S_{BH} = A_H/(4\ell_P^2) \approx 10^{122}$ of the observable universe] The scaling is grounded in standard cosmology but requires full non-linear closure of the GCT dark energy locking potential.
• CKM/PMNS Mixing Matrices & NCG $D_F$ Operator: Structurally linked to the geometric eigenvectors of the canonical 152-node $I_h$-closed $D_F$ adjacency matrix (engine: cage_builder.build_canonical_cage(size=152)), but exact numerical derivations from bare graph eigenvalues fail to align with observed values. Currently a structural ansatz.
• APS Boundary Pathway for $\alpha$ (App Z §Z.5–§Z.6): Conjectural topological identification of the 41.6 ppm residual with the value of an APS $\eta$-invariant on $\partial\mathcal{M}_{RT}$. Open program; the canonical Tier 2 derivation remains the bare baseline $\alpha^{-1} = 360\phi^{-2}$ refined by the $1/288$ bilayer correction.
• Phason Drag = Qualia Identification: Identifies "experience" with topological friction adopting Structural Russellian Monism. A consistent philosophical framework, but not uniquely derived from the axioms (an eliminativist interpretation holds equal mathematical weight).