Appendix R: Precision Audit Scorecard

This appendix provides the consolidated quantitative balance sheet for Geometric Consciousness Theory (GCT). It aggregates the results of the derivations performed across the three volumes, providing a high-resolution comparison between the structural eigenvalues of the 6D hardware and the empirical data measured by the international physics community (CODATA 2022 / PDG 2024).

The GCT Metric of Success: We evaluate the theory based on its ability to reduce the 26 free parameters of the Standard Model toward a small set of structural commitments: the geometric seed (the Golden Ratio $\phi$ and the icosahedral $H_3$ Coxeter group), the dimensional scale anchor ($m_e$), the disclosed A2 native-RGE endpoint anchor $\alpha_2^{-1}(M_{\rm GUT})\approx24$-$25$, the disclosed A3 low-energy $\alpha(0)$ anchor used only by measurement-anchored precision-comparison rows, and a finite set of Tier 3 integer anchors (notably $\phi^{-107}$ for the electron mass exponent and the harmonic exponents $N = 11, 17$ for the muon and tau) per the Parameter Ledger §0.1. The integer anchors, A2 boundary, and A3 precision-comparison input are empirical choices consistent with the framework, not unforced derivations — see Ch07 §7.2.2, App ZN, App H Open Problems O.14/O.5/O.19, and Parameter Ledger §0.1 for the load-bearing closure targets. Tier-discipline (this appendix's headline tier labels) reflects this layered disposition: Tier 2 mechanism + Tier 3 specific exponent/boundary + A3 where the corrected comparison row explicitly imports measured $\alpha(0)$.

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R.0 Scope of Parametric Closure — Law-Level vs. State-Level

GCT's parametric-closure claim is a claim about laws and mechanisms, not about the state of the universe at the moment of measurement. The distinction is structural and load-bearing for every entry in this scorecard:

The Selection Operator $F_{\rm sel}$ is irreducibly stochastic by construction (V1 Ch16 §16.3, Ch17 §17.4). The macroscopic sum of selection events that determines, e.g., the current dark-energy density, is therefore a measurement input to the theory, not a derivation output: a theory that predicted the exact magnitude of a state quantity arising from integrated selection would be inconsistent with its own selection-stochasticity (and with the free-will posture of Axioms 1–2). This is the same scope discipline that QED applies — the theory predicts the electromagnetic coupling structure once its law-level anchors are fixed, but it does not predict the photon occupation number of any specific cavity; the latter is measured.

In practice this means scorecard entries fall into three explicit categories:

1. Tier 2 mechanisms plus Tier 3/4 anchors, imports, residuals, and tensions across §R.1–R.8: closed-form geometric eigenvalues under the explicit structural anchors enumerated in the Parameter Ledger. The scorecard entry is the formula; the residual is the precision of the closure under the row's stated tier burden.
2. Law-level Tier 2 + integer-anchor Tier 3 predictions (electron mass exponent $-107$, harmonic exponents $N=11,17$): the mechanism is Tier 2, while the specific integer anchor is row-local and carries the tier stated in the row.
3. State-level claims (cosmic-scale integrated quantities, e.g., the magnitude of dark-energy acceleration, the total number of historical Polaron-bearing structures, the current matter-radiation ratio): treated as measurement inputs. GCT predicts the framework-level structure of these observables (the equation-of-state shape, the multi-channel composition, the asymptotic behavior, identifiable fingerprints of each contributing channel) but not their state-level magnitude at the present cosmological epoch. A state-level residual is therefore the numerical mismatch of a particular row against current data; a framework-level methodology claim is the structural rule or discriminator being tested.

Category-3 quantities are flagged explicitly throughout the manuscript (e.g., App H O.13 for biogenic dark energy: the channel contributes to total $\rho_\Lambda$ by an unknown fractional weight; the testable prediction is the fingerprint of the biogenic contribution within the total, not its magnitude in isolation). Frontmatter language ("law-level parametric closure") and the README headline reflect this scope; entries in §R.1–R.8 are law-level unless explicitly flagged as state-level.

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Note on Precision: GCT predictions represent 'Tree-Level' geometric eigenvalues. Higher-order radiative corrections (e.g., $\mathcal{O}(\alpha^2)$ phason loops) introduce a theoretical uncertainty floor of approximately 10-50 ppm. Discrepancies below this threshold (such as the 20 ppm Muon residual) require non-perturbative lattice simulations under the Protocol B follow-up program to resolve.

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R.1 Lepton Mass Spectrum

The lepton hierarchy uses a Symmetry Octave Harmonics ladder on the $N=144$ dodecahedral core: the harmonic-ladder mechanism is Tier 2, while the specific exponents and coefficient normalizations remain Tier 3 pending the O.5/O.14/O.15/O.26 closure stack.

Tau tolerance gate: the App FM/falsifiability-registry tau row treats 50 ppm as a PASS/TENSION boundary, with a registered tri-state edge: 0-50 ppm PASS, >50-55 ppm AT-GATE TENSION, and >=55 ppm hard FAIL under the current convention. The current 50.85 ppm residual is TENSION pending O.26b and QLQCD-1L, because §R.2.1 discloses a 10-50 ppm unmodeled higher-loop theory floor.

* Statistical Significance: The broad internal look-elsewhere headline for the lepton exponent pair (N=11, N=17) is 2.6σ under the full multi-base / non-Lucas sweep reported in the firewall. The $C(18,2)=153$ depinning-pair prior is a narrower structural prior and gives the higher conditional figure; it is not the unconstrained headline. The $6N \pm 1$ symmetry-octave selection rule (V3 Ch08 §8.3.1), given the muon's $N = 11$, structurally selects $N = 17$ as the next octave ($11 + 6 = 17$); this is the operative structural restriction in the framework. The Galois Rationality Constraint (Postulate G, App E §E.5) restricts general mass exponents to $\mathbb{Z}[\phi]$ but does not reduce the search space to Lucas numbers specifically (17 is not in the Lucas sequence $\{2, 1, 3, 4, 7, 11, 18, 29, \ldots\}$), so a "Lucas-only" constrained significance is not supported by the framework's own structural postulates. Conditional figures such as the depinning-prior result must be cited with their conditioning prior named.

HVP baseline: WP2025 (Aliberti et al. 2025, arXiv:2505.21476) is treated here as the consolidated SM theory/HVP benchmark $a_\mu^{SM}=116592033(62)\times10^{-11}$, dominated by the high-HVP lattice synthesis (BMW, RBC/UKQCD, Mainz, and related inputs) with reduced dispersive-R-ratio weight relative to 2020-era WP. Fermilab E989/BNL supply the external experimental comparison, not an input to the SM benchmark. Engine: protocol_qed_audit.py and verify_independent/verify_g2_muon.py both consume WP2025 values.

[!CAUTION] HVP baseline and channel-disposition status. The GCT muon $g-2$ closure $\Delta a_\mu = (1/5)(\alpha_{\rm CODATA}/\pi)^3 \approx 250.65 \times 10^{-11}$ is scored against the consolidated SM White Paper baseline (Aliberti et al. arXiv:2505.21476): $a_\mu^{\rm SM} = (116592033 \pm 62) \times 10^{-11}$, combined against the Fermilab E989 final (2025) experimental value $a_\mu^{\rm exp, FNAL} = (116592070.5 \pm 14.8) \times 10^{-11}$ (127 ppb FNAL-combined precision; Aguillard et al. 2025, arXiv:2506.03069). The corresponding FNAL + BNL E821 combined world-average is $a_\mu^{\rm exp, WA} = (116592071.5 \pm 14.5) \times 10^{-11}$ (124 ppb world-average precision; Aguillard et al. 2025 arXiv:2506.03069 abstract, world-average summary). Comparing the WP2025 SM theory/HVP benchmark with the FNAL-combined and world-average experimental numbers gives the same $\sim$0.6σ consolidated residual: $\Delta a_\mu^{\rm gap} \approx 37.5 \times 10^{-11}$.

Under this consolidated baseline, the GCT $(\alpha/\pi)^3 / 5$ contribution is adverse evidence: it overshoots the SM+experiment closure by ~3.4σ when added to the WP2025 SM benchmark. The disposition is Tier 3 fitted/equal-weight vertex coefficient + A3 precision input on top of a geometric mechanism class; Tension under WP2025; no robust confirmation. The CMD-2/CMD-family R-ratio reanalysis context and lattice-HVP synthesis have reduced the world-average experiment–theory tension, so the row remains a numerological channel in tension unless a future HVP arbitration independently reopens a >3σ gap near the GCT offset.

Binary gate: future CMD-3 + KLOE + BaBar R-ratio reconciliation and refined lattice-QCD HVP (BMW + RBC

• ETMC) that opens a >3σ experiment–theory gap near the GCT offset restores the GCT correction to decisive-test status; continued ≤1σ consolidation leaves the closure as a WP2025 tension that does not decisively distinguish GCT from a higher-loop SM calculation.

Status: Tier 2 law-level mechanism + Tier 3 state-level residuals. The lepton harmonic-ladder mechanism (resonant overtones of the 6D hardware at $N = 11, 17$) is Tier 2 structural; the absolute precision figures are state-level residuals against pole-mass targets: the electron sits at 1006 ppm against CODATA 2022 (conditional on Open Problem O.14 closure of the $n = -107$ K-theoretic gap-label link), the muon at ~21 ppm and tau at ~51 ppm (both against PDG 2024 anchors; conditional on the §R.2.1 Loop-Order Discipline assumption that the unmodeled GCT higher-loop sector matches the SM 2-loop EW + 3-loop QED corrections absorbed into the pole-mass targets).

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R.2 Fundamental Coupling Constants

These constants define the interaction impedances of the force sectors. Residuals here are identified as signatures of RGE Flow (radiative corrections) from the vacuum crystallization scale.

[!IMPORTANT] Higgs sector handle accounting [Tier 2 mechanism + Tier 3 calibrated handles]: the VEV derivation $v = m_e \phi^{11}(1 + 5\alpha + \phi^{8}\alpha^{2}) \cdot 1440\phi$ depends on the muon-mass formula, the $N=11$ harmonic anchor, and the $1440$ saturation factor (one of the documented icosahedral factorizations per O.20). The packing-efficiency identity $\lambda_{\rm bare}=1/8$ and the spectral-action cutoff $\Lambda_{\rm Higgs}$ belong to the separate Higgs-mass route, not to the VEV verifier. The headline precision claims therefore separate into: VEV at 181 ppm as a Tier 2 mechanism + Tier 3 VEV-handle residual; Higgs mass at 1.6% as the independent bare-mass route pending the one-loop/spectral-action closure.

**** Weinberg Boundary Theorem: $\phi^{-2}$ is the exact Tier 1 Gram/Cartan boundary scalar $\rho_G$ at the GUT scale; the standard normalized Cartan share is $\rho_G/(1+\rho_G)=0.27639$, while $\phi^{-3}$ is the volume-coupling Weinberg-angle input (App U §U.9) entering the geometric RGE flow.*

Status: Tier 2 law-level mechanism + Tier 3 state-level residuals. The icosahedral projection assigns coupling strengths as projection weights of the unit cell axes (Tier 2 law-level structure); the specific numerical residuals against current empirical anchors are Tier 3 state-level closure targets. Notably: $\alpha^{-1}$ sits at 3442 ppm bare residual (closure conditional on QLQCD-1L per O.5); $\alpha_s^{-1}(M_Z)$ has the largest gauge-coupling gap, with the registered bare handle $10\phi^2=26.18$ versus the PDG-scale value $\approx8.44$ (67.6% bare-to-$M_Z$ residual pending O.42 / QLQCD-2); Higgs mass carries a +1.6% bare gap (closure conditional on O.5 1-loop); muon g-2 carries a "Tension under WP2025; falsification conditional on long-term HVP-synthesis arbitration" disposition under the WP2025 lattice HVP synthesis; PMNS $\theta_{23}$ carries a >4σ tension under standard interpretation (see §R.4). The law-level mechanism is preserved, while the headline status records that several state-level numerical residuals depend on Open Problem O.5 closure.

R.2.1 Loop-Order Discipline — Radiative-Correction Provenance

The Tier 2 GCT formulas in §R.1–R.2 (lepton masses, $\alpha^{-1}_{\text{tree}}$, Cabibbo angle, sin²θ_W, etc.) are labeled "tree-level" or "tree + 1-loop" according to the loop order actually present in the displayed expression. The ppm-level residuals quoted for the muon and tau are scored against CODATA / PDG pole masses, which already include standard-model 2-loop electroweak + 3-loop QED radiative corrections at the precision relevant here. The residual table therefore separates formula residuals, anchor sensitivity, and unmodeled loop bands. Lepton precision rows are Tier 2 mechanism + Tier 3 pole-mass residual under the SM-equivalent higher-loop assumption.

Implication for Tier 2 status: the ppm-precision claims are valid only with the stated loop-order discipline. They are claims that the GCT geometric form plus displayed correction matches the pole-mass target to the quoted residual, with the residual representing GCT truncation, unmodeled GCT higher-loop terms, and experimental anchor uncertainty. QLQCD-1L closure (App H O.5) is the path that would replace the SM-equivalent higher-loop assumption with a GCT-internal loop derivation.

R.2.2 Hypercharge Assignment and Anomaly Cancellation

The L/R-asymmetric Standard Model hypercharges follow from the icosahedral identification of right-handed singlets with $E_\perp$-sterile states (V3 Ch04): for right-handed sterile singlets ($I_3 = 0$) the Gell-Mann–Nishijima relation $Q = T_3 + Y/2$ reduces to $Y_R = 2Q$. The full one-generation hypercharge assignment in the $Q = T_3 + Y/2$ convention is therefore:

The four Standard Model gauge-anomaly cancellation conditions on this one-generation assignment evaluate to zero algebraically:

$$\begin{aligned} [SU(3)]^2 \cdot U(1)_Y :\quad & 2\,Y_Q - Y_u - Y_d = 2\cdot\tfrac{1}{3} - \tfrac{4}{3} - (-\tfrac{2}{3}) = 0 \\ [SU(2)]^2 \cdot U(1)_Y :\quad & 3\,Y_Q + Y_L = 3\cdot\tfrac{1}{3} + (-1) = 0 \\ \text{grav}^2 \cdot U(1)_Y :\quad & 6\,Y_Q + 2\,Y_L - 3\,Y_u - 3\,Y_d - Y_e = 2 - 2 - 4 + 2 + 2 = 0 \\ [U(1)_Y]^3 :\quad & 6\,Y_Q^3 + 2\,Y_L^3 - 3\,Y_u^3 - 3\,Y_d^3 - Y_e^3 = 0 \quad \text{(machine-precision residual)} \end{aligned}$$

The colour multiplicity 3 enters quark contributions; the doublet multiplicity 2 enters the gravitational and $U(1)^3$ traces via LH-doublet components. Coefficient signs (+ for LH, − for RH via chirality flip) reflect the anomaly-trace convention. Engine verification computes all four traces and the right-handed-singlet identity in a single L_∞ residual: GCT_Physics_Engine/verify_independent/verify_hypercharges.py reports machine-precision PASS (max residual $\sim 10^{-15}$ from $[U(1)]^3$ float roundoff).

Tier discipline (two distinct claims must be separated). The vanishing of the four anomaly traces given the standard one-generation L/R-asymmetric hypercharge assignment is a textbook SM identity (Bilal 2008 arXiv:0802.0634 §7.3; standard QFT references); calling that arithmetic Tier 2 GCT-specific would be inflation. Two distinct claims live in this subsection:

1. The four-trace cancellation given the standard hypercharge assignment is Tier 1 textbook algebraic identity — it does not depend on GCT-specific content. We list it here for completeness, to demonstrate that the verifier verify_independent/verify_hypercharges.py reproduces the textbook result to machine precision (max residual $\sim 10^{-15}$), not to claim GCT-specific credit for it.

2. The L/R-asymmetric hypercharge assignment itself — that right-handed singlets are $E_\perp$-sterile and that $Y_R = 2Q$ follows from the icosahedral $E_\parallel/E_\perp$ split (V3 Ch04) — is a Tier 2 GCT-specific structural claim. The load-bearing GCT content of this section is the derivation of the multiplet structure from the icosahedral architecture; the subsequent anomaly arithmetic is included for completeness against the standard SM consistency check.

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R.3 Hadron Sector

The Proton mass represents the global energy minimum of the icosahedral projection, derived from the winding sum of the Baryonic Triad.

Proton mass — three values guide:

1. Formula prediction (the GCT value): $m_p = m_e \cdot \phi^{15+\phi^{-1}} \approx 938.417$ MeV. This is the tree-level output of the baryonic-triad mechanism evaluated against the CODATA $m_e$. The residual against the CODATA $m_p$ is $\approx 155$ ppm. Disposition: Tier 2 mechanism + Tier 3 sheet/exponent handle pending AKN-action closure, not hadron-mass closure from the current postulate set.
2. CODATA observation: $m_p = 938.272$ MeV. The empirical anchor against which the prediction is scored.
3. Ratio formula: $m_p/m_e = \phi^{15+\phi^{-1}} \approx 1836.437$ vs CODATA observed $1836.153$. Same $\sim0.015\%$ residual, expressed as a dimensionless ratio.

The $\sim0.015\%$ residual is the current tree-level precision of the $\phi$-power formula under the layered Tier 2 mechanism + Tier 3 sheet/exponent disposition; closure to higher precision and promotion of the sheet-doubling handle are QLQCD-1L / AKN-action targets (App Z). Ppm-level display is not used as evidential precision for this offset-fitted handle.

Quark masses are renormalized QCD parameters, not invariant rest masses. The comparison convention is therefore explicit in the table: $u,d,s$ use $\overline{\mathrm{MS}}$ at 2 GeV; $c,b$ use $\overline{\mathrm{MS}}$ at $\mu=m_q$; $t$ uses a pole-mass convention with the pole/MS-bar ambiguity retained as a Tier 3 residual.

* Calculated using the absolute GCT Higgs VEV derivation of 246.18 GeV. The 5/4 coefficient is a Tier 2 representation-theory coefficient, but the absolute bottom-quark prediction imports the Tier 3 $m_c$ input and is therefore Tier 3 as a state-level residual conditional on O.5.

Status: Tier 2 law-level mechanism (Fuglede-Kadison-determinant chain rooted natively in the Planck scale) + Tier 3 state-level residuals. The hadron mass chain follows the FK-determinant law-level structure. The down-quark row uses the primary $\phi^{\phi}$ infinite-volume-limit closed form and is POSTDICTION-CONSISTENT at $+0.33\%$ inside the registered shell-resonance band, conditional on O.5 for rigorous convergence and algebraic-field identification.

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R.4 Mixing Geometry Matrices

[Tier 3 candidate identification]: CKM and PMNS matrices are CANDIDATELY identified with the topological tunneling and axis sliding operations; closure pending O.5/O.7.

† PMNS $\theta_{23}$ disposition status — octant-ambiguity framing. The bare GCT prediction is $\theta_{23} = 45°$, which sits at the maximal-mixing boundary between the two octant solutions of the global oscillation fit. The current global posterior on $\theta_{23}$ is bimodal under the standard interpretation: NuFit-6.0 (Esteban, Gonzalez-Garcia, Maltoni, Martinez-Soler, Pinheiro & Schwetz 2024, arXiv:2410.05380; JHEP 12, 2024, 216) explicitly states "the third mixing angle $\theta_{23}$ still suffers from the octant ambiguity, with no clear indication of whether it is larger or smaller than $45°$". The 49.5° ± 1.1° (NOvA/T2K) value cited above corresponds to the upper-octant best-fit solution; the lower-octant best-fit sits closer to $42°$–$45°$, placing the bare GCT prediction at $\sim 1$–$2\sigma$ from that solution. The MSW matter effect for NOvA (L=810 km) shifts $\theta_{23}$ by only $\sim 0.00004°$ — five orders of magnitude too small to close the gap to the upper-octant solution. The operative disposition is therefore: bare GCT prediction $\theta_{23} = 45°$ is in $> 4\sigma$ tension with the upper-octant solution, and within $\sim 1$–$2\sigma$ of the lower-octant solution; the global posterior is bimodal pending T2K-II / DUNE arbitration of the octant question. The bare prediction is held at Tier 3 pending either octant resolution or a geometric derivation of the deviation from $45°$. The residual is bundled with the QLQCD-1L research debt (App H §H.5 Open Problem O.5). If next-generation T2K-II / DUNE measurements (a) resolve the octant in favour of the upper solution and (b) confirm the upper-octant best-fit value at $> 5\sigma$ without a closing GCT-side derivation, the bare prediction must be revised; resolution in favour of the lower octant would tighten the GCT prediction to a $\sim 1$–$2\sigma$ tension with the data.

‡ 3×3 Unitarity check: The individual mixing-angle predictions above are not independent — they are constrained by the full $3 \times 3$ unitarity of the CKM and PMNS matrices ($V^\dagger V = 1$, nine real constraints from three row-norms, three column-norms, and three off-diagonal sum rules). The CKM and PMNS diagnostic rows check the constructed matrices after applying the standard unitary parameterization: unitarity is exact by construction to machine precision. The real CKM support/constraint sits in the mixing-angle comparison rows, where the $\sim 1\%$ mismatches against PDG global-fit central values remain pending O.7; they are not a unitarity defect. The real PMNS tension is the $\theta_{23}$ angle row above; a determinant or row-norm check of a standard 3×3 parameterization cannot independently validate or falsify that angle. Verifier: verify_independent/verify_mixing_unitarity.py and verify_independent/verify_pmns.py.

Status: Tier 2 mechanism + Tier 3 transfer-correction residual; PDG $s_{12}$ miss for the corrected Cabibbo formula $s_{12} = \phi^{-3}\cdot(1-5\alpha)$ is 3.6σ (residual 0.00245 / PDG uncertainty 0.00068); for the bare formula $s_{12} = \phi^{-3}$ the miss is 16.2σ (residual 0.011 / 0.00068). CKM precision closure pending. PMNS $\theta_{23}$ disposition is octant-ambiguity-dependent per the §R.4 footnote. CKM 3×3 unitarity is exact by construction under the standard unitary parameterization; the precision issue is the mixing-angle comparison rows, not a unitarity defect. PMNS $\theta_{23}$ sits at the maximal-mixing octant boundary — $> 4\sigma$ from the upper-octant best-fit solution, $\sim 1$–$2\sigma$ from the lower-octant best-fit, with the global posterior bimodal pending T2K-II / DUNE octant arbitration (NuFit-6.0, Esteban, Gonzalez-Garcia, Maltoni, Martinez-Soler, Pinheiro & Schwetz 2024).

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R.5 Neutrino Sector

The Neutrino sector provides the "Risky Predictions" required to distinguish GCT from standard particle physics.

[!WARNING] Neutrino Mass Tension: The Planck 2018 + DESI BAO 2024 combined constraint yields Σm_ν < 0.072 eV (95% CL), and DESI DR2 tightens the ΛCDM-context bound to Σm_ν < 0.064 eV. These bounds put the GCT prediction of 0.085 eV in active tension, not in a falsified state. The registered lower exclusion boundary is Σm_ν < 0.075 eV at definitive precision (above the normal-hierarchy oscillation floor near 0.059 eV) or Σm_ν > 0.15 eV; the 0.072/0.064 eV ΛCDM bounds are adjudicated through the joint P.4/P.6 GCT-native likelihood gate, whose native $w(z)$ amplitude is expected to return a bound near 0.064 eV rather than rescue the 0.0853 eV prediction. This is an active falsification risk in the GCT framework.

*** CRITICAL NOTE: The 0.0853 eV neutrino floor is in $\Lambda$CDM-context tension with current model-dependent cosmological bounds. Reassessment requires a GCT-native likelihood that jointly fits the registered dark-energy fingerprints; invoking $w<-1$ alone is insufficient to claim compatibility.*

[!IMPORTANT] Neutrino Floor and Dark-Energy Scope

The $\Sigma m_\nu = 0.0853$ eV prediction is in tension with DESI DR2's $\Sigma m_\nu < 0.064$ eV $\Lambda$CDM bound. Any relief requires a GCT-NATIVE CMB+BAO+LSS likelihood that is sign-opposite to the DESI-preferred CPL quadrant (DESI: $w_0>-1$, $w_a<0$; GCT: $w(z)\leq-1$ throughout). Generic dynamical-DE rescue is not available without an explicit GCT-native likelihood; this is registered as Open Problem O.13 closure-path C1. The neutrino bound and the multi-channel EoS are NOT co-immune.

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R.6 Astrophysics & Cosmology

This sector establishes macroscopic falsification targets tracking vacuum geometry properties and biological thresholds.

Operational thresholds for the DESI DR2 sign-opposite tension row. Row P.6 has two separate decision rules. The state-level CPL-fit gate fires only if DESI DR2 + Euclid DR1 (and subsequent registry-approved likelihood updates) establish a $3\sigma$ or stronger $w_0>-1,\ w_a<0$ trajectory with no biogenic-channel fingerprint. The biogenic-channel fingerprint gate is the Roman/Stage-V low-redshift test of the Class-2 envelope $|\Delta w_{\rm bio}(z=0.28)|\in[2,5]\times10^{-5}$ with the asymptote-from-below sign; excluding that envelope at covariance-aware sub-$5\times10^{-5}$ precision falsifies the registered single-channel fingerprint. The current protocol_de_multichannel.py score is a diagnostic registered-menu closure failure, not a covariance-aware likelihood threshold.

[!NOTE] §R.6.1 Illustrative-only callout — bare biogenic-driving overshoot pre-suppression magnitude. The bare biogenic-driving formula $\rho_{DE}^{\rm bare} \sim \lambda_{bio} \cdot \dot{\mathcal{I}} / I_0$ evaluated before holographic-displacement suppression produces a value approximately $10^{220}$ times the observed $\rho_\Lambda = 5.4 \times 10^{-10}$ J/m³. This figure is Tier 4 illustrative only — its purpose is to make the size of the holographic-displacement task explicit, not to predict a state-level number. The absolute magnitude of $\rho_\Lambda$ is a state-level quantity per §R.0 and is NOT a derivation output of the framework. The callout format prevents table-extraction misreading; the load-bearing mechanism claim is the holographic-displacement reduction (Ch12 §12.1), not the pre-suppression overshoot magnitude. See §R.7 row 3 for the mechanism-vs-magnitude scope split. | $m_{phason}$ | $\rho_\Lambda = \tfrac{1}{2} m_{phason}^2 \langle\Phi_\perp\rangle^2$ (V2 Ch14 §14.1.3 condensate form; calibrated against $\rho_\Lambda$ for $\langle\Phi_\perp\rangle$ pinned by the App M condensate equation) | $1.71 \times 10^{-5}$ eV | $\rho_\Lambda = 5.4 \times 10^{-10}$ J/m³ | Calibrated; conditional on $\langle\Phi_\perp\rangle$ closure (App H O.1) | Consistency Check | 3 | | $E_{bio\_phason}$ | Phason Zeno activation gap | $\sim 1.6$ eV | Unknown macro coherence threshold | Pending | Prediction | 2 | | Protocol D — Anaesthetic Isotope Shift (P.13; mirrors App V Table V.1 P.13 + engine protocol_isotope_experiment.py) | Spin-Entropy Modulation via bound-water $^{17}$O hyperfine (Ch16 §16.3.2b + App F §F.4); central branch $0$ at $n_{rp}=0$ pending O.21, with conditional O.21/O.33 $f_{bound}$ sensitivity band $[1.0, 2.0] \times 10^{-3}$ at $n_{rp}=1$ | Central branch predicts no LORR shift. Conditional sensitivity branch predicts $\Delta C_{50}/C_{50} \in [-0.20\%,-0.10\%]$ between $^{17}$O-substituted and $^{16}$O-control bound-water populations (lower anesthetic threshold under $^{17}$O substitution; engine convention (C50_H2_17O - C50_H2_16O)/C50_H2_16O = -f_bound); the disfavored $n_{rp}=2$ branch reaches signed $[-0.40\%,-0.20\%]$ and requires geometry re-evaluation | Drosophila $^{17}$O LORR assay window after systematics-gate redesign; no current power claim is registered because the §16.3.4 systematic budget rejects the operative gate | No operative falsification gate under the §16.3.4 systematic budget. $\sigma_{\rm syst,total}\approx0.0577$ (57.65× the 0.10% naive gate), and halving S2/S4 reduces $\sigma_{\rm syst}$ by only 1.20×; until a systematics-validated gate is specified, the Tier 2 framework-level mechanism is preserved and the quantitative gate is deferred. | Prediction | Tier 2 mechanism (Spin-Entropy Modulation via $^{17}$O hyperfine) + Tier 4 quantitative-gate operationally unfalsifiable under registered systematic budget |

* The PTA anisotropy peak deviation $\Delta\Gamma_{\max} = \phi^{-18}\cdot\max_\theta |P_6(\cos\theta)| = \phi^{-18}\cdot 1 \approx 1.73 \times 10^{-4}$ (Ch21 §21.1.2) is derived from the rigorous linear phason-graviton coupling formula. The square-root (phonon) scaling does not apply to the phason-graviton vertex. Related published quantities are the linear Legendre projection $\hat C_6 = (4\pi/13)\phi^{-18} \approx 1.67 \times 10^{-4}$ and the squared angular-power-ratio $C_6/C_0 = (4\pi/13)\phi^{-36} \approx 2.90 \times 10^{-8}$ — distinct from the peak deviation. Published comparisons must specify which normalization.

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R.7 Epistemic Summary

• Sector Closure Status: The bare gauge and lepton-mass sectors are determined by the 5-postulate-plus-1-dimensional-anchor subsystem enumerated in Parameter Ledger §0.1; the current native-RGE endpoint audit adds A2, $\alpha_2^{-1}(M_{\rm GUT})\approx24$-$25$, as a Tier 3 calibrated gauge-flow boundary, and corrected lepton/Higgs/QED comparison rows add A3, measured low-energy $\alpha(0)$. The present gauge + lepton + native-RGE + precision-comparison accounting is 5 postulates + 3 anchors. The hadronic, biophysical, and cosmological sectors carry additional calibrated parameters per Parameter Ledger §0.2–0.4. See Parameter Ledger §0.5 for the aggregate accounting and §0.6 for the terminal-target ambition.
• Systemic Reliability: No aggregate precision score is claimed across the mixed Tier 2/Tier 3/Tier 4 rows. The scorecard is row-local: precision statements include or exclude calibrated handles, active tensions, and external anchors only where that row explicitly says so.
• Iron Rule Note: The Electron Mass Planck-link row ($m_e$) carries a Tier 2 framework + Tier 3 integer anchor + Tier 4 physical-link conjecture disposition (see §R.1 row 1 + README + Ch07 §7.2.2). The $\phi^{-107}$ exponent is a canonical exact arithmetic identity of $H_3$ ($|n| = 107 = 1^2 + 5^2 + 9^2 = \sum m_i^2(H_3)$, unique to $H_3$ among finite irreducible Coxeter groups) identified with the electron-mass scaling under the Tier 4 K-theoretic gap-label-action physical-link conjecture from the $V$-side Coxeter invariant to the 6D trace-image projection $\tau_*(K_0(\mathcal{A}))$ pending Open Problem O.14; exact arithmetic on a Tier 4 chain does not promote the chain. The Standard Planck Mass convention ($M_P$) is a Tier 3 phenomenological choice forced by empirical closure — not uniquely derived from the 6D geometry (which prefers $\bar{M}_P$). The ~1006 ppm precision (engine output against CODATA 2022) is conditional on this Tier 3 anchor; an alternative figure of 753 ppm sometimes cited informally corresponds to a non-CODATA $M_P/\alpha$ calibration and is not the canonical engine output.
• Anomalies Under Disposition Per Row:

1. Hierarchy Problem: Dispositioned via holographic screening ($10^{-40}$ force gap).
2. Hadron Mass Gap: Dispositioned via the Baryonic Triad eigenvalue ($15 + \phi^{-1}$), with row-specific hadronic residuals and open QLQCD closure paths retained above.
3. Vacuum Energy: Tier 2 mechanism (Cosmology of Zero + Biogenic Driving with holographic-displacement suppression); the absolute magnitude of $\rho_\Lambda$ is a state-level quantity under §R.0 and is NOT a derivation output of the framework — it is a measurement input. The §R.6 $\rho_{DE}$ row reports the bare biogenic-driving overshoot before holographic suppression (an illustrative Tier 4 pre-suppression number, $10^{220}$ vs the observed $\rho_\Lambda = 5.4 \times 10^{-10}$ J/m³) to make the size of the holographic-displacement task explicit; it is not framed as a state-level prediction. The mechanism explains why $\rho_\Lambda$ is small and dynamic (Biogenic Driving + holographic restriction; Ch12 §12.1); the absolute value is a state-level measurement input. The multi-channel shape-proxy partition closure path to the DES-Dovekie recalibrated DESI CPL fit has been evaluated explicitly in GCT_Physics_Engine/src/protocol_de_multichannel.py and CLOSURE-FAILS for the registered five-channel menu across the swept cosmological range — the shape menu is a convex combination of channels all carrying $w_c \leq -0.9$, not a physically evolved multi-fluid background model, so $w_{\rm total}$ cannot reach the DES-Dovekie $w_0 = -0.803$ (Ch14 §14.6.3, App H O.13 amplitude gap; C1/C2/C3 remain open); the state-level DESI CPL-fit reconciliation is therefore a live external-data tension awaiting DESI + Euclid arbitration rather than a closed disposition.
4. Dark Matter: Dispositioned via the 3.55 keV Lattice Fracture signature and Topological Glass, subject to the XRISM and linewidth gates listed in App V.

Summary: Selected gauge/lepton quantities reach sub-percent residuals under the disclosed Tier 2 mechanisms plus Tier 3 specific anchors of Parameter Ledger §0.1, with the electron exponent additionally inheriting the O.14 Tier 4 physical-link conjecture. The broad internal look-elsewhere statistical disposition of the combined $\phi$-eigenvalue convergence is given in §R.9 and the Prediction/Postdiction Firewall; the tier labels and open-problem conditionalities that qualify each row are enumerated in §R.0–§R.2 and App H.

---

R.8 Falsification Criteria

GCT would be definitively falsified if ANY of the following are observed:

1. Lepton Hierarchy Violation:

• Pole-mass consistency gate: the displayed corrected muon formula $m_\mu = m_e \phi^{11}(1+5\alpha+\phi^8\alpha^2)$ remains within its registered tolerance under the §R.2.1 loop-order discipline.
• Falsification: If precision measurements, under the same pole-mass and radiative-correction convention, drive the corrected-formula residual outside the registered 3σ gate. The gate is not a free prediction of $\phi^{11}\pm0.5\%$.
• Current Status: Muon 21 ppm residual (within tolerance); tau 50.85 ppm TENSION at the 50 ppm registry gate pending O.5/O.26b closure. See §R.2.1 Loop-Order Discipline caveat: the bare Tier 2 geometric precision is ~0.25%; the ppm precision is obtained against PDG pole-mass targets that already include SM 2-loop electroweak + 3-loop QED corrections (~10-40 ppm), so the residual implicitly assumes GCT higher-loop corrections agree with SM higher-loop structure to ppm precision.

2. Dark Matter Line Absence:

• Prediction: Emission line at 3.50-3.60 keV from galaxy clusters
• Falsification: Protocol C terminal no-line non-detection operates on the Bulbul-level empirical-derived floor $\Gamma \lesssim 2.0\times10^{-28}\ \mathrm{s}^{-1}$ in a $\geq94$ Ms equivalent stacked exposure, obtained by scaling the XRISM 3.75 Ms reference stack limit $\Gamma\lesssim1.0\times10^{-27}\ \mathrm{s}^{-1}$ by $\sqrt{3.75/94}$. The 26 Ms / $\Gamma\lesssim3.8\times10^{-28}\ \mathrm{s}^{-1}$ stack remains the morphology-linewidth milestone, not the terminal no-line falsifier. The background-limited $F_{3\sigma}\approx1.1\times10^{-8}\ \mathrm{ph/cm^2/s}$ calculation at Resolve $A_{\rm eff}=200\ \mathrm{cm^2}$ and residual background $6.6\times10^{-4}\ \mathrm{cnt/s/keV}$ corresponds to $\Gamma\lesssim1.2\times10^{-30}\ \mathrm{s}^{-1}$, but that is a theoretical ceiling, not the operative gate, unless aperture overlap, continuum subtraction, K-XVIII line-model degeneracy, and stack-efficiency losses are controlled. The remaining branches are $W_{\rm int}>20$ eV FWHM after instrument/kinematic deconvolution and morphology $\Delta\chi^2\geq9$ vs smooth-$\rho$ templates in mass-weighted stress-gated above-$\sigma_{\rm crit}$ cluster regions. See Ch15 §15.1.4 + App V P.3 + falsifiability_registry P.3 + App FM P.3 for the full preregistration parity.
• Current Status: SENSITIVITY-LIMITED. The current 3.75 Ms stack is $\sim5\times$ too shallow relative to the registered sensitivity floor; the $W_{\rm int}$ test is not yet decisive. Confirmation regime is $W_{\rm int} \leq 5$ eV (within XRISM Resolve $\sim5$ eV FWHM instrument floor); $W_{\rm int}<1$ eV is a narrow-line result consistent with sterile-neutrino kinetics before cluster-virial, instrumental, and stacking convolution, while the Milky-Way halo blank-sky null (Dessert, Rodd & Safdi 2020, Science 367:1465) imposes an independent flux/background-modeling constraint. Linewidth alone is not decisive; morphology, flux floor, background model, and K-XVIII subtraction remain load-bearing.

3. Neutrino Mass Floor Violation:

• Prediction: $m_1 = m_e \phi^{-36} \approx 0.0153$ eV (absolute floor)
• Falsification: $\Sigma m_\nu < 0.075$ eV at definitive precision (below the prediction's exclusion boundary while staying above the normal-hierarchy oscillation floor) OR $\Sigma m_\nu > 0.15$ eV (above the inverted/high-mass branch)
• Current Status: DESI 2024 $\Sigma m_\nu < 0.072$ eV and DESI DR2 $\Sigma m_\nu < 0.064$ eV place the 0.0853 eV floor in TENSION, not compatibility and not falsification; the GCT-native $w(z)$ likelihood path is expected to remain near $\Sigma m_\nu \approx 0.064$ eV because the registered dark-energy amplitude is too small to reproduce the generic CPL relaxation.

4. Proton Mass Formula Breakdown:

• Prediction: Current operative formula $m_p = m_e \cdot \phi^{15+\phi^{-1}}$ gives a residual of $\sim0.015\%$ against PDG ($m_p = 938.272...$ MeV). This is a Tier 2 mechanism (icosahedral cage structure) + Tier 3 sheet-exponent handle, with closure pending AKN-action (O.32) and QCD-scale radiative corrections. The exponent additive $+\phi^{-1}$ is a candidate Berry-phase identification (R.9.1), not a closed derivation; ppm-level display is not used as evidential precision for this offset-fitted handle.
• Falsification: If future precision finds deviation $>100$ ppm that survives QLQCD-1L closure
• Current Status: Open precision closure (~0.015% match attributed to uncalculated QCD-scale radiative corrections per §R.9.1; falsification gate fires on a deviation surviving QLQCD-1L closure)

5. Weinberg Angle at Unification:

• Prediction: Bare $\sin^2\theta_W = \phi^{-3} \approx 0.236$ at crystallization scale
• Falsification: If two-loop RGE with threshold matching yields $>5\%$ discrepancy
• Current Status: Bare angle yields 2.1% error relative to Z-pole → consistency check, not strong support (RGE-scale running and convention choices remain load-bearing)

6. 3.55 keV Photon from Atomic Transitions:

• Prediction: Protocol C detects anomalous 3.55 keV photon from isotope decay
• Falsification: Deconvolved intrinsic linewidth $W_{\rm int} > 20$ eV after instrument, kinematic, K-XVIII, and stacking systematics are inside the frozen budget; confirmation is $W_{\rm int} \leq 5$ eV, with $W_{\rm int} < 1$ eV as the strong narrow-line discriminator.
• Current Status: Awaiting decisive intrinsic-linewidth measurement

7. Lorentz Violation Upper Limits:

• Prediction: Discrete lattice effects appear at $E_{QG} \sim E_{Planck}$
• Falsification: If LIV tests constrain $E_{QG} > 10^3 E_{Planck}$, discrete structure falsified
• Current Status: Current limits $E_{QG} > 10 E_{Planck}$ → COMPATIBLE

8. Phantom Dark Energy Crossing:

• Prediction: $w(z)$ remains below $-1$ throughout the modeled low-redshift branch; $z\approx0.28$ is the V7' sensitivity marker, not a transition redshift.
• Falsification: DESI DR2 + Euclid DR1 producing a 3σ+ CPL detection of $w_0 > -1$ AND $w_a < 0$ (quintessence-today + more-phantom-into-past trajectory, opposite to GCT's phantom-asymptote-from-below), with no biogenic-channel fingerprint, falsifies the registered five-channel state-level menu. It does not by itself prove a no-go theorem against every possible GCT dark-energy channel; that stronger statement remains O.13.
• Current Status: EXTERNAL-DATA TENSION (single-channel external-data tension; multi-channel shape-proxy partition CLOSURE-FAILS for the registered five-channel menu). (Epistemic note: DES-Dovekie recalibrated DESI DR2 + CMB (arXiv:2511.07517) reports $(w_0, w_a) = (-0.803 \pm 0.054,\ -0.72 \pm 0.21)$, sign-opposite to GCT's phantom prediction, with joint dynamical-DE significance 3.2σ. The tension is calibration-robust: the official DESI DR2 release reports the CPL departure from $\Lambda$CDM at $3.1\sigma$ for BAO+CMB and across $2.8$–$4.2\sigma$ when supernova samples are added, with every calibration in the same sign-opposite $w_0>-1,\ w_a<0$ quadrant; the $3.2\sigma$ recalibrated figure is the registered central anchor, not the upper edge of the adverse evidence. The GCT single-channel biogenic-DE engine output is $(w_0, w_a) \approx (-1.0038, +0.0136)$ — sign-opposite on both CPL parameters. The multi-channel shape-proxy partition (Copeland-Sami-Tsujikawa 2006, Ch14 §14.6.3) implemented in GCT_Physics_Engine/src/protocol_de_multichannel.py evaluates the closure path explicitly: (a) the 4032-point sensitivity sweep over the five parameters $(\Omega_{\Lambda,0}, f_{\rm ext}, f_{\rm chiral}, f_{\rm pert}, z_{\rm chiral})$ — with $\Omega_{\Lambda,0}$ swept across $\{0.50, \ldots, 0.75\}$ — returns a best-fit diagonal diagnostic score $\approx 4.861$ against the DES-Dovekie target, flat to within $0.0446$ diagnostic-score units across the entire $\Omega_{\Lambda,0}$ grid; this score uses diagonal DESI errors only and is diagnostic only, not a full likelihood distance, physical multi-fluid background model, or closure threshold. (b) Analytically, the convex-combination ceiling closes the remaining range including $\Omega_{\Lambda,0}\to0$ because every channel in the registered five-channel menu carries $w_c(z) \leq -0.9$, so $w_{\rm total}(z) \leq -0.9$ for every redshift and channel-fraction choice — the DES-Dovekie central value $w_0 = -0.803$ lies above this ceiling and is unreachable. The convex-combination ceiling is partly tautological: every channel in the registered decomposition was selected with $w_c \leq -0.9$ by construction (the perturbation channel uses the Tier-3 calibrated $w_{\rm pert} = -0.9$), so the closure failure reflects the current channel menu rather than an unconditional convex-combination theorem at the framework level. Constructing a thawing-quintessence channel with $w_c > -0.803$ — App H O.13 closure path C1, with C2/C3 as alternate channel-menu revisions — would open the path; every dark-energy channel the framework currently supplies is phantom-directed. The law-level Tier 2 biogenic-DE mechanism (sign-determined $w_{\rm bio}(z) \leq -1$ from a single channel) is preserved as a mechanism-level claim, but the state-level reconciliation with the DES-Dovekie CPL fit via the registered five-channel shape menu is not available.)

Summary Table:

Note on conditional rows. P.20 down-quark is POSTDICTION-CONSISTENT conditional on O.5: the primary $\phi^{\phi}$ output sits inside the registered shell-resonance gate, while rigorous convergence proof and algebraic-field identification remain open. P.36 is a separate consistency-check row inside its 200,000 ppm cross-route convention band. See App FM for the full per-prediction status, row codes, and tolerance derivations.

---

R.9 The Look-Elsewhere Effect and Statistical Significance

[!IMPORTANT] Canonical Statistical Significance Reference

Observable	Raw σ	Trial Factor	Quoted σ	Method	Status
Lepton exponent pair (N=11, N=17)	~2.6σ	full $\mathbb{Z}[\phi]$ multi-base sweep (Lucas-vs-non-Lucas exponent assignments)	~2.6σ	Broad internal look-elsewhere LEA	Canonical headline (broad internal look-elsewhere) — matches the §R.9.2 footnote calculation
Lepton exponent pair (N=11, N=17) — Conservative-SM-texture-window conditional auxiliary	~2.6σ	~25,000× (Conservative full-SM-texture window $N \in [4, 50]$)	~3.1σ	Window-conditional LEA	Conditional auxiliary on the SM-texture-window prior; not a free headline
Lepton exponent pair (N=11, N=17)	~2.6σ	~153× ($N \in [8, 25]$ depinning-threshold window)	4.4σ	Single-base LEA, conditioned on framework prior	Conditional auxiliary on the depinning-threshold prior (Ch08 §8.1.1)
Proton mass (15+φ⁻¹)	~4σ	~100×	~2.7σ	Single-base LEA	Auxiliary per-target

The Cross-Base Monte Carlo joint test (Muon+Tau+Proton+α) is outside the canonical Master Statistical Table scope by design: the bare-control test is auxiliary and does not produce a canonical sigma headline. App R §R.9.4 treats the cross-base MC test-construction-conditional disposition as a qualitative structural-compatibility check, not as a canonical sigma figure for the Master Statistical Table.

The canonical cited figure is ~2.6σ (broad internal look-elsewhere, lepton exponent pair under the full $\mathbb{Z}[\phi]$ multi-base sweep — see §R.9.2 footnote). The 3.1σ figure is conditional on the Conservative SM-texture window prior $N \in [4, 50]$; the 4.4σ figure is conditional on the framework's depinning-threshold prior $N \in [8, 25]$ — the $N=8$ lower bound is itself a structural postulate per Ch08 §8.1.1, and citing 4.4σ as a free headline without naming the conditioning prior is significance laundering under the Firewall governance discipline. The Cross-Base Monte Carlo joint test is a test-construction-conditional structural-compatibility check (conditional on the four-observable choice, four tolerance bands 0.1%/0.1%/0.5%/0.3%, and irrational-base sampling distribution [1.05, 4.0]) and is documented separately at §R.9.4; it is never cited as a free-standing headline significance. The Abstract uses $\sim 2.6\sigma$ as the canonical headline; all conditional figures are cited only with their conditioning priors named at the point of citation.

Critics may argue that the mass formulas (e.g., $m_\mu \sim \phi^{11}$) are numerological coincidences found by searching a large parameter space. However, the GCT search space is rigorously constrained by the Symmetry Octaves of the icosahedron ($N=6, 12, 18$).

• Constraint: Resonance is only permitted at $N \times 6 \pm 1$ (the "near-octave" stability points).
• Result: The Muon ($N=11 = 2\times6-1$) and Tau ($N=17 = 3\times6-1$) fall exactly on these predicted nodes.
• Significance: Conditional on the $6N\pm1$ symmetry-octave cadence prior, the probability of two fundamental masses falling randomly onto this specific geometric cadence is calculated as $P < 10^{-4}$; the broad internal look-elsewhere lepton-pair headline remains $\sim 2.6\sigma$.

R.9.1 The Proton Mass Formula

The formula $m_p = m_e \cdot \phi^{15 + \phi^{-1}}$ yields a match to observation with relative error of ~0.015%.

Search Space Analysis: Consider the space of simple formulas of the form:

• $m_p = m_e \cdot \phi^N$ (pure power)
• $m_p = m_e \cdot \phi^{N + \phi^{-1}}$ (power + correction)
• $m_p = m_e \cdot \phi^{N} \cdot (1 + \alpha \phi^{-k})$ (multiplicative correction)

For integer $N \in [10, 20]$ and considering that:

1. The base exponent 15 is topologically motivated (3 quarks × 5-fold symmetry)
2. The correction term $+\phi^{-1}$ yields sub-100-ppm precision

Probability Estimate:

• Probability of random integer $N$ yielding $<1\%$ precision: $\sim 1/100$
• Probability of additive correction improving to $<0.02\%$: $\sim 1/100$
• Combined probability: $\sim 1/10^4$

Statistical Significance: The match is significant at approximately the ~2.7σ level (LEA-corrected; raw ~4σ) for a randomly chosen formula. (Note: the lepton-pair canonical headline is ~2.6σ broad internal look-elsewhere over the full $\mathbb{Z}[\phi]$ multi-base sweep; the 3.1σ figure is conditional on the Conservative full-SM-texture window choice $N \in [4, 50]$ and is reported as a conditional auxiliary — see §R.9.2 and the Prediction/Postdiction Firewall Master Statistical Table).

Critical Caveat on Radiative Corrections: While the base $\phi^{15}$ is topologically justified by the 15 gluonic bonds in the baryonic triad, the additive correction term $+\phi^{-1}$ is a candidate Berry-phase identification rather than a closed derivation. The remaining $\sim0.015\%$ error represents uncalculated QCD-scale radiative corrections.

R.9.2 The Lepton Mass Hierarchy

§LEA Lepton Exponent Anchor (LEA) — The corrected lepton-mass postdiction is built from the geometric base $m_\ell = m_e \cdot \phi^{N_\ell}$ plus tier-disclosed radiative corrections: muon $(1+5\alpha)$ + $\phi^8\alpha^2$ (Tier 2 phason-channel multiplicity + Tier 3 $+5\alpha$ pole-mass normalization and loop discipline, see App R lepton row), tau $(1-3.6\alpha)$ with tau O.26b pending. The bare pure-$\phi^N$ residuals are $-3.75\%$ (muon) / $+2.69\%$ (tau); the corrected residuals are $\sim21$ ppm (muon) / $\sim51$ ppm (tau against PDG 2024). The $\sim2.6\sigma$ joint significance is defined for the corrected-formula postdiction only; under the strict Firewall the corrected lepton pair is a Tier 2 mechanism + Tier 3 calibrated-radiative-correction postdiction, not a closed exponent prediction from the current postulate set.

The uncorrected per-pair probability is $p_{single} \approx 4 \times 10^{-8}$ (raw per-pair Z-score $\sim 5.4\sigma$ with no multiple-comparison correction). Tolerance-band scope on $p_{single}$: the $4 \times 10^{-8}$ figure is computed against the canonical headline tolerance band that the muon residual sits at $\sim 21$ ppm and the tau residual at $\sim 51$ ppm — i.e., the implicit band per-pair is $\sim 10^{-4}$ on the relative residual. The single-trial probability scales approximately linearly with band width: a band 10× wider yields $\sim 4 \times 10^{-7}$ ($\sim 4.9\sigma$ raw); a band 10× tighter yields $\sim 4 \times 10^{-9}$ ($\sim 5.9\sigma$ raw). The raw $5.4\sigma$ figure is therefore an auxiliary tied to the canonical-residual tolerance band; the canonical $\sim 2.6\sigma$ headline applies the full $\mathbb{Z}[\phi]$ multi-base sweep correction independent of the tolerance band choice and is the operative broad internal look-elsewhere significance (under the Firewall §3 Master Statistical Table and the §R.9 canonical headline disposition). The $\sim 3.1\sigma$ figure is a conditional auxiliary that applies the Conservative SM-texture trial factor over the window $N \in [4, 50]$ across ~25,000 trial pairs, yielding $p_{global} \approx 10^{-3}$ — this is conditional on the SM-texture window-prior choice and is not the canonical headline; citing $\sim 3.1\sigma$ as headline without naming the Conservative-window prior is significance laundering under the Firewall governance discipline. Restricting further to the framework's depinning-threshold prior $N \in [8, 25]$ (153 pairs) gives the conditional auxiliary: $p_{global}^{cond} = 153 \times p_{single} \approx 6 \times 10^{-6} \implies \text{4.4σ (LEA-corrected, conditional on $N \in [8, 25]$)}$

Statistical Significance: Broad internal look-elsewhere canonical headline: ≈2.6σ (full $\mathbb{Z}[\phi]$ multi-base sweep across Lucas-vs-non-Lucas exponent assignments, under the §R.9 canonical-headline disposition and the Firewall §3 Master Statistical Table). The ~3.1σ figure is the Conservative-SM-texture-window-conditional auxiliary ($N \in [4, 50]$, ~25,000 trial pairs), and the ~4.4σ figure is the depinning-threshold-window-conditional auxiliary ($N \in [8, 25]$, 153 pairs); both are reported in §8.5 with their conditioning priors named at the point of citation under the Firewall governance discipline. The $6N \pm 1$ symmetry-octave selection rule (V3 Ch08 §8.3.1), given the muon's $N = 11$, structurally selects $N = 17$ as the next octave $11 + 6 = 17$ — this is the operative structural restriction. The Galois Rationality Constraint (Postulate G, App E §E.5) restricts mass exponents to $\mathbb{Z}[\phi]$ generally but does not reduce to Lucas numbers specifically (17 is not in the Lucas sequence $\{2, 1, 3, 4, 7, 11, 18, 29, \ldots\}$); figures derived from a "Lucas-only" restriction are not supported by the framework's own structural postulates. Justification: The specific indices (11, 17) are not arbitrary:

• $N=11$: Corresponds to the second symmetry octave ($2 \times 6 - 1$)
• $N=17$: Corresponds to the third symmetry octave ($3 \times 6 - 1$) This $6N \pm 1$ pattern is a definitive signature of icosahedral resonance.

R.9.3 The Fine-Structure Constant

The formula $\alpha^{-1} = 360 \times \phi^{-2}$ yields $137.5077$, with residual error $3442$ ppm.

Parameters:

• $360$: 600-cell antipodal edge count ($E = 720/2 = 360$) [Tier 2 topological count; its role as the exact fine-structure multiplier remains layered with the Tier 3 O.19/O.5 magnitude closure]
• $\phi^{-2}$: Golden angle mismatch
• $N=144$: 12th Fibonacci number (dodecahedral saturation)
• Status: Tree-level baseline; Phason Anti-Screening (0.34%) is the 1-loop target.

Compatibility diagnostic. For $N \in [100, 200]$, achieving $<50$ ppm precision gives the rough scan fraction $P \approx 1/2000$ under that discrete interval. This is a conditional diagnostic over a chosen integer handle range, not a headline significance for the fine-structure row, because the interval, 600-cell multiplier, bilayer factor, and residual target are framework-conditioned handles rather than a prior-free experiment.

QLQCD 1-Loop Pathway (41.6 ppm): The 41.6 ppm residual is the theoretical target for the QLQCD 1-loop phason vacuum polarization diagram (Appendix Z.5). Because phasons are bosonic, the loop generates geometric anti-screening. While the topological sign (+) and boundary form are rigorously established (Tier 2), the exact numerical extraction of the icosahedral winding number ($C_{ico} \approx 1.12$) from the continuous aperiodic hull remains an open challenge. The 3442 ppm and 41.6 ppm quantities function as residual targets, not calibration inputs; the back-solved $C_{ico}$ extraction is diagnostic until O.19 closes. This 41.6 ppm gap is formally classified as a Tier 3 open computational item (QLQCD-1L), awaiting exascale non-perturbative lattice integration.

R.9.4 Cross-Base Joint Compatibility Analysis [Tier 3 post-hoc consistency check, NOT a Bayesian prior or sequential pre-registration]

Scope statement. This subsection reports a fixed-sampling compatibility diagnostic for the already-identified $\phi$ base. It is not a Bayesian prior, not a sequentially pre-registered likelihood-ratio test, and not an independent headline significance.

The standard LEA analysis (§U.5, §8.5, and the Firewall §3 Master Statistical Table) applies a trial-factor penalty within the φ-family and yields the canonical broad internal look-elsewhere headline of ≈2.6σ for the lepton exponent pair, with the higher figures (3.1σ Conservative-SM-texture-conditional, 4.4σ depinning-threshold-conditional, 5.4σ single-trial) reported only as conditioning-prior-named auxiliaries.

However, the deeper question — why φ at all? — requires a different test. protocol_global_significance.py addresses this directly with a Cross-Base Monte Carlo over N_samples = 100,000 random irrational bases uniformly drawn from [1.05, 4.0], testing each base for simultaneous agreement with all four GCT targets (Muon, Tau, Proton, α) within their respective tolerances (0.1%, 0.1%, 0.5%, 0.3%).

Result: The number of non-φ bases simultaneously satisfying all four bare-control targets is $n_{hit} = 0$ (zero out of 100,000 sampled). The null probability is bounded above by $P_{null} \le 0.5/100,000 = 5 \times 10^{-6}$ under this fixed diagnostic construction, but the scorecard does not convert this post-selected sparse-control result into a sigma-equivalent headline. The same sparse bare-control test does not make φ itself pass all four targets: bare φ misses the muon and tau tolerance bands. The manuscript-cited corrected φ formulae do pass the four physical comparison bands, while no sampled non-φ bare-control base does.

Cross-base disposition (test-construction-conditional Monte Carlo): no sampled non-φ bare-control base passes all four bands; corrected φ passes the physical comparison bands; bare φ does not. This is an auxiliary structural-compatibility check, not a unique-base theorem and not a headline significance.

[!IMPORTANT] Scope discipline on the cross-base figure. The $P_{null}$ bound is conditional on the §R.9.4 test construction — specifically on the choice of four target observables (Muon, Tau, Proton, $\alpha$), the four tolerance bands (0.1% / 0.1% / 0.5% / 0.3%), the sparse bare-control formula family, and the irrational-base sampling distribution $[1.05, 4.0]$. None of these choices is pre-registered as an independent constraint, and each was made after observing the φ-family formulas. A test that draws a different observable set, different tolerance ladder, different formula family, or different sampling distribution can produce a different significance from the same null hypothesis; the bound is therefore a structural compatibility diagnostic of the four-observable corrected-φ prediction set against a fixed bare-control sampling scheme, not a broad global significance and not a sigma-equivalent scorecard result. It is reported alongside the canonical $\sim 2.6\sigma$ broad internal look-elsewhere lepton-pair headline (§R.9.2) as an auxiliary consistency check, NOT as a headline significance. Pre-registration of the observable set + tolerance ladder + formula family + sampling distribution, or an exhaustive §R.9.5 base/formula sweep, is the closure path that would convert this from an auxiliary to a broader global figure; that closure is registered as a forward-looking item under §R.9.5 below.

No sampled non-φ base satisfies the four bare-control constraints within the §R.9.4 tolerance bands, but the protocol does not prove φ unique: bare φ misses the lepton bands, the corrected φ formulae carry the physical comparison, and exhaustive formula/base closure remains pending §R.9.5 closure. The ~2.6σ broad internal look-elsewhere lepton-pair canonical figure (with 3.1σ Conservative-SM-texture-conditional and 4.4σ depinning-threshold-conditional as conditional auxiliaries) remains the operative significance headline.

Cross-reference: protocol_global_significance.py, output protocol_global_significance_results.json::cross_base_monte_carlo.

Remaining Epistemic Debt: We acknowledge that certain elements (e.g. higher-order radiative background models) are phenomenological fits pending microscopic derivation. The geometric skeleton for the lepton drag coefficients, electron exponent, proton Berry phase, Higgs VEV Absolute Pipeline mechanism, Strange Quark $12\alpha$ drag, and CKM $\phi^{-1}$ exponent carries Tier 2 mechanism status where stated, while the Jackiw-Rebbi $2\pi\phi^{-1}$ CP Phase is Tier 3 in the PMNS-sector magnitude row pending the holonomy-to-observed-PMNS map. Specific coefficients and ppm/percent residuals remain Tier 3 or Open-Problem-conditional per the row-level dispositions above (O.5 / O.14 / O.26b in particular; O.20 is closed at the 1440 enumeration level).

R.10 Registry Disposition Mirror

This block is the Appendix R source-of-truth mirror for narrative disposition text. Each line is emitted from GCT_Physics_Engine/falsifiability_registry.json::entries[*].disposition_text so Appendix R and Appendix FM preserve identical registry dispositions.

<!-- APP_R_REGISTRY_DISPOSITIONS_START -->

• P.1 / muon_mass: POSTDICTION-CONSISTENT alignment check. Tier 2 harmonic-ladder mechanism + Tier 3 integer/residual: the ~21 ppm residual is conditional on SM-equivalent radiative-correction discipline, and bare Tier 2 geometric precision is the uncorrected harmonic-ladder scale. This settled PDG/CODATA row is a postdictive alignment audit, not prospective detection evidence.
• P.1b / tau_mass: Tier 2 mechanism + Tier 2 integer-pair (D=18, N=5) + Tier 3 combination rule c = -D/N pending O.26b. Engine residual is about 50.9 ppm against the 50 ppm scorecard target, so the row emits AT-GATE TENSION pending O.5/O.26b/QLQCD-1L. The registered tri-state gate is 0-50 ppm PASS, >50-55 ppm AT-GATE TENSION, and >=55 ppm hard FAIL under the current convention.
• P.1c / electron_mass: Tier 2 mechanism + Tier 3 dimensional/integer anchor + Tier 4 K-theoretic physical-link conjecture pending O.14. The phi^{-107} arithmetic identity is Tier 1 as an H_3 Coxeter fact, but the electron-mass derivation chain inherits the O.14 physical-link conjecture. The Dixmier-trace type-check protocol is expected to report type_check_pass=true, dixmier_trace_pass=false, status=open_problem_O.14 until the Connes-Moscovici residue computation closes.
• P.2 / proton_mass: Tier 2 mechanism + Tier 3 sheet/exponent handle pending AKN-action closure. Lattice QCD definitively confirms m_p value outside 500 ppm of m_e phi^(15+phi^-1). The ~0.015% residual is the current tree-level mechanism precision, not hadron-sector closure from the current postulate set; ppm display is suppressed for this offset-fitted handle.
• P.3 / dm_line_width_morphology: P.3 Protocol C XRISM 3.55 keV conditional preregistration design. Operative forward observables are Gamma/exposure sensitivity, W_int, and stress-aperture morphology; W_int is the deconvolved intrinsic line-width residual after instrument and registered kinematic/turbulence deconvolution, and centroid agreement is not scored here. GCT is falsified at decisive sensitivity if any registered branch holds: (a) no line at terminal Bulbul-level empirical-derived Gamma <= 2.0e-28 s^-1 in >=94 Ms equivalent exposure, (b) W_int > 20 eV FWHM, or (c) morphology fails stress-gated rho_above_sigma_crit^2 by Delta chi^2 >= 9 against both smooth rho and smooth rho^2 templates inside the numerically frozen sigma_crit = 0.53 keV cm^-3 aperture. The 26 Ms / Gamma <= 3.8e-28 s^-1 stack remains a morphology-linewidth milestone. Frozen target cells are Bullet bow-shock/Mach shell, Coma merger or radio-relic shells, A2142 cold-front/merger shell, A3667/A2256/A2319 shock or merger shells, plus only off-core Perseus/A2029/A1795 annuli whose pre-line pressure/shear proxy passes the same cut; relaxed cool cores are sub-threshold calibrators unless their pre-line maps exceed the cut. The ideal background-limited calculation gives F_3sigma ~= 1.1e-8 ph/cm^2/s and Gamma <= 1.2e-30 s^-1, but this is non-operative unless a real stack demonstrates those losses are controlled. The preregistration must include a per-cell ARF/RMF/background/effective-mass exposure table; nominal equivalent exposure alone is not decisive. Until that table is frozen, the terminal no-line branch is preregistered but not executable as a completed decisive falsifier; the package supports sensitivity-limited constraints plus morphology/linewidth milestones. Preregistered budget: S1 RMF/ARF residuals < 2%; S2 K-XVIII subtraction < 1e-5 s^-1; S3 gain drift < 0.3 eV; S4 split-stack scatter < 0.5 eV; S5 random-effects width < 1 eV; S6 cross-exposure calibration < 2%. Low-shear dSph branch remains as registered in falsification_second_branch.
• P.3-centroid-context / dm_line_3p55keV: Centroid agreement at 3.5486 keV is a Tier 3 postdiction-context row. It may be reported only as context for a line that has already passed the blinded W_int and morphology gates; it cannot by itself confirm or falsify P.3.
• P.4 / neutrino_sum: Joint with P.6: the 0.0853 eV floor remains in TENSION with DESI 2024 LambdaCDM Sigma m_nu < 0.072 eV and DESI DR2 LambdaCDM Sigma m_nu < 0.064 eV. The registered exclusion band is Sigma m_nu < 0.075 eV or Sigma m_nu > 0.15 eV at definitive precision; Sigma m_nu > 0.10 eV at 3sigma over Planck+DESI+CMB lensing keeps the biogenic-DE survival branch active, while definitive values in [0.05,0.10) are handled by App NS. A GCT-native CMB+BAO+LSS likelihood with the GCT-specific w(z) is required for quantitative assessment and is expected to return about 0.064 eV rather than 0.0853 eV; generic w0wa relaxation cannot be imported because the GCT biogenic prior is sign-opposite to the DESI-preferred dynamical-DE quadrant.
• P.4b / neutrino_m1_floor: m_1 below normal-hierarchy minimum massless floor (0.0589 eV implied) inconsistent with GCT phi^(-36) suppression
• P.5b / g2_electron: Electron g-2 is a kinematic-threshold null at this loop order: the Tier 2 mechanism says a below-threshold lepton does not activate the cage-depinning vertex, while the numerical threshold E_res = m_e·φ^11 inherits the Tier 3 N=11 exponent. Falsified if any future measurement establishes |Δa_e^anom| > 10⁻¹² at >3σ ROBUSTLY across α-input choices (Cs-α2 + Rb-α2 + matter-wave gradiometry agreement). Current state: Cs-α gives 0.96σ agreement; Rb-α gives 3.94σ tension; status PENDING-α-RESOLUTION until α-input ambiguity resolves.
• P.5 / g2_muon: Tier 3 revival path requires a reopened >3sigma experiment-theory gap near the GCT correction scale under R-ratio/lattice arbitration. A consolidated high-HVP lattice-dominant WP2025-class baseline collapses the anomaly and is adverse evidence, not a survival path. Current WP2025 baseline registers as adverse evidence because Delta a_mu^GCT ~= 2.51e-9 overshoots the consolidated gap by ~3.4sigma.
• P.5c / no_verifier: Long-horizon theoretical extrapolation / consistency check: GCT computes Delta_a_tau ~ 2.50655e-9 from the same 5-channel A_5 averaging applied to the above-threshold tau using alpha_CODATA (the tau at 1776.93 MeV sits 17.5x above the N=11 activation threshold). Current PDG bound |Delta_a_tau| < 1e-1 is ~8 OOM weaker than the value; FCC-ee tau-g-2 program target sensitivity ~1e-6 is ~3 OOM short. Operationally UNFALSIFIABLE at near-term and FCC-ee-target sensitivities, so this is not an operative falsifiable prediction until a plausible discriminator exists. Closure of the form-factor-universality sub-problem of O.26 would tighten the extrapolation; the channel-count universality across leptons is the load-bearing Tier 3 assumption.
• P.6 / no_verifier: ACTIVE TENSION (DES-Dovekie recalibrated DESI registered-menu CLOSURE-FAILS); Euclid/Roman reserved for robustness and fingerprint arbitration. Single-channel biogenic-DE engine output (w_0, w_a) ~ (-1.0038, +0.0136) is sign-opposite to the DES-Dovekie recalibrated DESI DR2 + CMB target (arXiv:2511.07517): (w_0, w_a) = (-0.803 +/- 0.054, -0.72 +/- 0.21), with joint dynamical-DE significance 3.2sigma. CLOSURE-FAILS evidence: registered 5-channel menu (protocol_de_multichannel.py) gives w_0 in [-1.009, -0.991] across scan band (near-phantom quadrant); the DES-Dovekie central value w_0 = -0.803 +/- 0.054 remains in the quintessence-today quadrant and above the registered channel ceiling. Multi-channel shape-proxy partition (Copeland-Sami-Tsujikawa 2006; Ch14 §14.6.3) evaluated via 4032-point sweep over (Omega_Lambda,0, f_ext, f_chiral, f_pert, z_chiral) in protocol_de_multichannel.py returns best-fit diagonal diagnostic score ~4.861 against the DES-Dovekie target, flat to within 0.0446 diagnostic-score units across the swept Omega_Lambda,0 range; this is diagnostic only and not a covariance-aware likelihood distance, physical multi-fluid background model, or closure threshold. The closure failure applies to the registered five-channel shape-proxy menu, not to a physical likelihood-weighted multi-fluid prediction; w_total(z) is a convex combination of channels all carrying w_c <= -0.9, so it cannot reach the DES-Dovekie quintessence-today region for any choice of channel fractions, including any Omega_Lambda,0. A physical prediction requires continuity-evolved rho_c(z) and the DESI covariance treatment. The state-level CPL-fit gate fires if DESI/Euclid/Roman produce a 3sigma+ CPL detection of w_0 > -1 AND w_a < 0 (quintessence-today + more-phantom-into-past trajectory, opposite to GCT's phantom-asymptote-from-below) with no biogenic-channel fingerprint; this falsifies the registered five-channel state-level shape-proxy menu, not a no-go theorem over all possible GCT dark-energy channels. The law-level w(z) <= -1 phantom-direction claim from a single channel remains independently testable via Roman Space Telescope joint-bin analysis across z in [0.05, 0.5].
• P.6b / no_verifier: Tier 2 frozen-phason-vacuum mechanism + Tier 3 operative biogenic-DE quartic mass scale m_phason_operative = 1.7e-5 eV conditional on O.1 closure. The Weinberg candidate M_P phi^-147 ≈ 2.2 meV is separate O.37 reconcile-or-discard material; hbar H0 is the Hubble energy scale, not a direct phason mass constraint.
• P.7 / no_verifier: |C_hat_6| < 1e-4 at 3sigma in NANOGrav 15yr + PPTA DR3 + EPTA DR2 combined matched-filter analysis (this falsification clause is the linear-Legendre form; see P.7d for the dedicated row, P.7c for peak deviation, P.7e for squared form)
• P.8 / no_verifier: F1 = four-conjunct joint null. Protocol A-Prime F1 is falsified if and only if all four conjuncts hold after frozen S0-S7 acceptance checks pass: (a) NO peak is detected in the Branch A 112 +/- 10 MHz window, (b) NO peak is detected in the Branch B 42 +/- 10 MHz window across the full required scan band 32-200 MHz, with <=5 kHz spacing and the required scan statistic M = 201,600, alpha_local ~= 1.34e-8, N_T2,scan ~= 5.1e3 shots per frequency/bin cell, and ~=1.03e9 shots across the full six-cell scan, (c) NO anti-Zeno sign at the 150 MHz control, and (d) NO chirality contrast against achiral control. Frozen NV ensemble: N_caps=50 caps x N_NV=20 implanted NV centers per cap = 1000 NV-center observations; shot noise scales as 1/sqrt(N_caps*N_NV), and the 20 NVs per cap are pooled for depth-jitter mitigation. Branch A is the primary GCT 100-MHz prediction; Branch B is the geometric sibling for O.12-negative outcomes (eta_Zeno suppression of the primary line shifts the peak to the 42 MHz sub-harmonic). A single-branch null does not falsify. Measured per-cap P_CISS and post-transfer CISS retention are required before unblinding. Peak-amplitude magnitude is conditional on App H O.23 + O.24 closures and is NOT a separate falsifying threshold.
• P.8b / no_verifier: Direct detection: absence of any coherent 360-400 GHz line at >3sigma from active neural tissue under MEA-correlated gamma-burst conditions after cryogenic TES + high-Q cavity upgrade; persistence of the same line in anesthetized control at unchanged amplitude; or line centroid outside 380 +/- 20 GHz at >3sigma. Current-sensitivity direct detection is not an operative gate because the cavity-integrated signal remains several orders of magnitude below the thermal floor after N^2 enhancement and 1 ms Dicke integration (absolute margin set by the radiance-to-power etendue conversion, bracketed multimode-to-mode-matched, routed to Open Problem O.10). Operative modulation-differential gate: blinded frozen-pipeline absence of anaesthetic-cycle-correlated 360-400 GHz power at >3sigma across N_shots >= 1e5 anaesthetic cycles.
• P.8c / no_verifier: Protected-subspace LGI branch fails if the blinded primary endpoint K_3(nu_c) <= 1 at 3sigma after Holm correction across the registered branch window (112 +/- 10 MHz if O.12 positive; 42 +/- 10 MHz if O.12 negative), the 150 MHz control, and deuterated-Trp contrasts. Registered prediction: Tier 2 qualitative sign pattern K_3(nu_c) > 1, K_3(150 MHz) <= 1, and deuterated-Trp suppression of the nu_c excess; the branch value and nominal effect size K_3 = 1 + phi^-9/pi = 1.0042 (Delta K_3 ~0.0042) are Tier 3 pending O.12/O.23. Sample size target n >= 1.7e7 valid triplets per condition across at least 12 independently prepared samples; family-wise alpha = 0.01; exploratory frequency scans cannot replace the registered branch/150 MHz contrasts.
• P.9 / no_verifier: Roman-successor mission (or DESI Y10+ extended spatial-mapping program) finds no dipole amplitude in the registered [0.6, 1.5]e-4 delta_w/w band aligned within +/-30 degrees of Shapley (l ~ 311 deg, b ~ 30 deg) at > 3sigma. Predicted amplitude tied to V7' Class-2 envelope cosmic-mean |Delta w(z=0.28)| in [2, 5] x 10^-5 (Ch14 §14.6.3) times realistic biogenic-overdensity factor delta_bio in [1, 3] at Shapley scale (Local Sheet to Laniakea); the predicted amplitude is below DESI Y3 spatial-mapping sensitivity and functions as a Roman-successor amplitude target.
• P.10 / no_verifier: Alias row only. See P.8 for the authoritative Protocol A-Prime thresholds, scan statistic, sample-size budget, Branch A/B joint-null, S0-S7 acceptance, 150 MHz anti-Zeno control, chirality-contrast requirement, measured per-cap P_CISS acceptance floors, post-transfer CISS retention, and 50 caps x 20 NV ensemble. Readiness and verifier/package-freeze status are identical to P.8. A single-branch null remains sensitivity information, not framework falsification.
• P.11 / no_verifier: Operational preregistration: TB+EB cross-correlation estimator; beta_birefringence > 0.1 deg at 3sigma; gray zone 0.05-0.1 deg sensitivity-limited; Planck PR4 70% galactic mask plus foreground residual model; Bonferroni correction over 6 azimuthal bins; instrument polarization-angle uncertainty <= 0.05 deg; OSF/Zenodo freeze before unblinding.
• P.12 / no_verifier: Operational preregistration: rotation-measure dispersion vs frequency estimator; 6-fold azimuthal correlation amplitude > 0.05 rad^2; gray zone 0.025-0.05 rad^2 sensitivity-limited; CHIME/FRB 1000-burst sample plus SKA-Mid 1-year sample; Galactic Faraday rotation residual <= 5 rad/m^2; OSF/Zenodo freeze before unblinding.
• P.32 / strong_cp_theta_bar: GCT predicts theta-bar = exactly 0 from icosahedral chirality-preserving cycle closure (all 5/3/2-fold I_h rotations close at multiples of 2pi exactly under cut-and-project preservation). Current bound |d_n| < 1.8e-26 e·cm → |theta-bar| < 1e-10. Positive detection of |theta-bar| > 1e-10 at >3sigma in next-gen nEDM (SNS, n2EDM) would falsify the mechanism. Theorem-grade promotion requires QLQCD-1L closure (non-perturbative proof of anomaly-free topological quantisation).
• P.12b / ckm_jarlskog: No current Tier 2 GCT prediction for the CKM Jarlskog invariant J. The candidate J_GCT proportional to phi^-n awaits first-principles closure (Open Problem O.7, bundled with QLQCD-1L research debt O.5). The delta_CP = 360 deg * phi^-1 phase ansatz is independent of J and stands as Tier 3 in the PMNS-sector magnitude row pending the holonomy-to-observed-PMNS map and data-tension resolution.
• P.13 / protocol_d_mc_systematics: No operative falsification gate under the §16.3.4 systematic budget. Central branch n_rp=0 gives f_bound=0 and no predicted LORR signal pending O.21. Sensitivity branch n_rp=1 gives f_bound=[1.0,2.0]e-3 and signed endpoint shift [-0.20%,-0.10%]. The systematic-only SD is 0.0577, 57.65x the 0.10% naive gate; halving S2/S4 reduces sigma_syst by 1.20x and does not recover power. Until a systematics-validated gate is specified, the Tier 2 framework-level mechanism is preserved and the quantitative gate is deferred.
• P.13c / verify_p13c_nmr_polarity: Executable preregistration verifier bound to P.13c. Primary statistic: 99% bootstrap CI (10^4 resamples) of Delta T_2 = T_2(17O) - T_2(16O). Decision rule (sign-aware): (a) the 99% bootstrap CI excludes zero AND (b) abs(Delta T_2) > 3*sigma_measurement AND (c) sign(Delta T_2) = +1, meaning T_2 in 17O is longer than in 16O. GCT CONFIRMED on (a)+(b)+(c) jointly only if the pilot variance gate satisfies the returned max_total_per_preparation_sigma_fraction_for_80pct_power. GCT FALSIFIED if (a)+(b) hold but (c) fails; statistically significant negative Delta T_2 falsifies the sign-positive prediction. A sign-positive but sub-5% significant effect is reported as positive-sensitivity / effect-size revision, not as confirmation or failure. CI overlap is inconclusive. Operational preregistration: n=24 independent biological preparations per isotope condition, three repeated NMR acquisitions per preparation, two-sided alpha=0.01, beta=0.20, power target for a 5% active-state polarity reversal, pilot variance acceptance below the verifier-returned sigma ceiling, dominant 17O quadrupolar-relaxation systematic bounded to sigma<0.2 ppm under the frozen relaxation model, S1-S8 systematics required, preregistration package/code/synthetic-data budget frozen before unblinding, and first decisive run targeted for 12-18 months after protocol freeze and isotope-logistics validation. Current status remains OPEN_CONDITIONAL until blinded lab data land; the verifier executes the frozen n=24/99%-CI/0.2-ppm/systematic/alpha-beta power design.
• P.13b / alpha_inverse: alpha^-1 bare outside 5000 ppm of 360 phi^-2 (3442 ppm residual is the QLQCD-1L phason anti-screening debt — tightening below 3442 ppm requires QLQCD-1L closure, not a falsification opportunity)
• P.14 / weinberg_bare: sin^2 theta_W bare outside phi^-3 by more than 5% — would falsify either the Gram length ratio (T1.b) or the cut-and-project dimension (T1.a). M_Z endpoint remains Tier 3 import / A2-anchored RGE flow, not falsification target for the bare value.
• P.15 / weinberg_gut: Machine-precision Gram boundary scalar rho_G differs from phi^-2, or canonical normalized Cartan share rho_G/(1+rho_G) differs from 0.27639320225. The physical bare Weinberg angle remains the separate volume-coupling value sin^2 theta_W = phi^-3.
• P.16 / newton_g: Tier 2 thermodynamic mechanism + Tier 4 Planck-link conjecture (inherits O.14) + Tier 3 dimensional anchor; 2274 ppm postdiction. Falsification gate: G outside 5000 ppm of the m_e-anchored Jacobson thermodynamic chain.
• P.17b / higgs_mass_bare: m_H^bare = v · sqrt(2 lambda_bare) = v/2 from lambda = 1/8 (Tier 2 geometric packing identity). Bare prediction 123.11 GeV vs PDG 125.10 GeV: +1.6% gap (1.99 GeV; ~16000 ppm). PASS if bare-to-physical gap stays within 5% (consistent with expected SM 1-loop radiative correction sign and rough magnitude). Falsified if observed m_H drops below the bare prediction at >3sigma, or if the full GCT 1-loop computation (TP-E) yields a correction with the wrong sign or magnitude to close the gap. Tier 2 promotion of the physical-mass prediction requires TP-E delivering a derived correction matching PDG within ~100 ppm.
• P.17 / higgs_vev: Higgs VEV v = m_mu * 1440 * phi is a Tier 2 mechanism with A3 measured-alpha inheritance and a Tier 3 calibrated 1440-handle residual. The ~181 ppm row is inherited from the muon-mass closure times the chosen 1440*phi factorization, not an independent geometric closure of v; O.20 factorization and O.5/O.14/O.15/O.19 remain upstream.
• P.18 / cabibbo_angle: Cabibbo has a Tier 2 bare prediction sin(theta_C)^bare = phi^{-3}; the corrected s_12 = phi^{-3}(1-5 alpha) value is Tier 2 bare + Tier 3 lepton-to-hadron transfer pending O.5 and is a live 3.6 sigma CKM tension under the registered PDG comparison.
• P.19 / up_quark: m_u outside PDG 2024 m_u = 2.16 +/- 0.07 MeV at mu=2 GeV for the registered m_e * geometric exponent; comparison inherits a Tier 3 QCD scheme/running bridge until QLQCD matching closes
• P.20 / down_quark: POSTDICTION-CONSISTENT conditional: the primary protocol reports $m_d = m_u \phi^{\phi} = 4.716$ MeV, +0.33% against the PDG central value and inside the 110,000 ppm shell-resonance band. Tier 2 FK-determinant mechanism + Tier 3 $\phi^{\phi}$ infinite-volume-limit identification conditional on O.5; closed-cage deep-tail mean $\det_{FK}/\phi^{\phi}=0.9976$ (sample std $0.0253$), with sequence-mean signed error vs PDG $+0.09\%$.
• P.21 / strange_quark: The strange-quark area-law scaling is retained as the mechanism, but the specific 12α drag coefficient is a Tier 3 state-level handle per the Ledger; falsification remains m_s outside the PDG +/- 5sigma gate for this registered expression.
• P.22 / charm_quark: m_c outside +/- 3% of m_u phi^(13+phi^-3) heuristic (Tier 3 promotion path via QLQCD-1L K-theoretic gap label)
• P.23 / bottom_quark: Tier 2 5/4 coefficient + Tier 3 m_c input → Tier 3 absolute prediction (1.15% as state-level residual conditional on O.5); falsification remains m_b outside 4130-4230 MeV at 5sigma.
• P.24 / top_quark: Top-quark absolute mass is Tier 3 as a state-level residual conditional on O.5/QCD pole-mass closure; falsification remains m_t outside the PDG +/- 5sigma gate for the geometric prediction.
• P.25 / ckm_s12: CKM s_12 / Cabibbo has a Tier 2 bare face-tunneling prediction and a Tier 3 lepton-to-hadron transfer correction through (1-5 alpha) pending O.5. The corrected value is not an unconditional Tier 2 prediction.
• P.26 / ckm_s23: CKM s_23 outside Tier 3 ansatz envelope; closure path via TP-F (O.5)
• P.27 / ckm_s13: CKM s_13 outside Tier 3 ansatz envelope; closure path via TP-F (O.5)
• P.28 / pmns_theta12: PMNS theta_12 is a Tier 3 axis-sliding ansatz in the joint PMNS sector; the row cannot be promoted independently while theta_23 remains in octant-dependent tension.
• P.29 / pmns_theta13: PMNS theta_13 is a Tier 3 axis-sliding ansatz in the joint PMNS sector; the row cannot be promoted independently while theta_23 remains in octant-dependent tension.
• P.30 / pmns_theta23: PMNS theta_23 is Tier 3 tension: the bare 45 deg prediction sits at the octant boundary, in >4 sigma tension with the upper-octant best fit and closer to the lower-octant branch; DUNE/Hyper-K octant arbitration is required before promotion.
• P.31 / pmns_delta_cp: delta_CP outside the 360 deg phi^-1 parametric phase by > 5sigma in DUNE / Hyper-K would falsify the registered phase mechanism; numerical PMNS-sector magnitude and downstream eEDM disposition remain Tier 3 pending O.7 and the PMNS tension resolution.
• P.33 / cage_node_count_N144: N=144 cage count contradicted by direct structural enumeration of the icosahedral cut-and-project polyhedron (no possible falsification within the math; exact integer identity F_12 = 144 = 12^2, unique Fibonacci-square in cage-relevant range)
• P.34 / phason_stiffness_ratio: K_perp/K_para measurement deviates from phi^-D with D=18 specifically (other small integers D in [12, 24] also pass the postulate but would falsify the specific D=18 exponent); i-AlPdMn phason elastic measurements place D between 14 and 20 (100-1000x gap to fully resolve)
• P.35 / verify_independent/verify_hypercharges.py::hypercharges_and_anomaly_cancellation: Y_R != 2Q for right-handed singlets (would falsify the Gell-Mann-Nishijima identity itself); exact algebraic identity given SM charges and I_3=0 for singlets, Tier 2 attaches to the E_parallel/E_perp split premise
• P.16b / polaron_healing_length: xi = hbar c / (alpha^2 m_e c^2) = a_0/alpha is the Tier 1 Bohr-Compton scale; microtubule lumen identification 2*xi vs lumen ID is the Tier 3 biological-scale match and is falsified if cryo-EM measurement falls outside 13-17 nm at 5sigma
• P.36 / fbio_jacobian: Tier 2 wavefunction-overlap mechanism. Current engine residual is about 107000 ppm inside the 200000 ppm cross-route convention band; the row is a consistency check pending a single declared Bohr-Compton/xi convention rather than a hard biological-overlap falsifier.
• P.37 / f_bound_conservative_geometry: Operative central branch is f_bound=0 under n_rp=0 pending O.21 closure. Sensitivity branch is [1.0,2.0]e-3 under n_rp=1 conditional on O.21. The n_rp=2 branch [2.0,4.0]e-3 is disfavored but registered; the structural stress-test upper edge 6.1e-3 (n_rp=2, n_contact=12) is the geometric maximum and not used downstream.
• P.38 / nu_zeno_van_Hove: Bare-vs-renormalized frequency channel: the engine PASS is protected by construction against the O.12/O.23 branch logic, but the manuscript requires O.23 protected-subspace renormalization before the 30 MHz bare estimate and the 112 MHz observed branch can be treated as the same physical frequency channel. Protocol A-Prime still tests the branch through the registered T_2 feature, with the 5-200 MHz tubulin-EPR Delta_ST/h band disclosed as the S0 systematic floor. This row is subordinate to the authoritative P.8 four-conjunct F1 gate; it is not a separate framework-level single-branch falsifier.
• P.39 / chi_holographic_susceptibility: chi = pi (c/H_0 ell_P)^2 deviates from 2.27e122 by > 5% once H_0 is fixed via CODATA 2022 (67.4 km/s/Mpc)
• P.40 / w_z_phantom_crossing_z028: Stage-V / Roman Year-10 sub-5e-5 w(z) program finds no phantom-directed V7' Class-2 envelope at z in [0.05, 0.5]; OR pipeline-derived CPL triple disagrees with refit using a derived (not ansatz) P_evolve by > factor 2 in z_cross. DESI DR2 CPL central fit is sign-opposite to the GCT single-channel biogenic curve and is treated as O.13 closure path C1, not as direct support.
• P.43 / imp01_biogenic_DE_pipeline: Pipeline-derived (z_cross, w_0, w_a) inconsistent (cpl(a_cross) != -1 within 1e-3); OR sensitivity-sweep over physically-motivated P_evolve ranges produces z_cross outside [0.3, 0.5] at default
• P.43b / de_multichannel_partition: Registered-menu no-go verdict: CLOSURE-FAILS for the five-channel shape-proxy menu (closure paths C1/C2/C3 of App H O.13 open). The 4032-point sensitivity sweep over (Omega_Lambda,0, f_ext, f_chiral, f_pert, z_chiral) returns best-fit diagonal diagnostic score ~4.861 against the DES-Dovekie recalibrated DESI DR2 + CMB CPL target (arXiv:2511.07517), flat to within 0.0446 score units across the swept Omega_Lambda,0 range; DESI covariance is not included, so this is diagnostic only and not a likelihood distance, physical multi-fluid background model, or closure threshold. DES-Dovekie reports (w_0, w_a) = (-0.803 +/- 0.054, -0.72 +/- 0.21) at joint significance 3.2sigma. Sweep envelope is pinned at w_0 in [-1.009, -0.991], w_a in [-0.012, +0.045]. Analytically, the convex-combination ceiling closes the remaining range including Omega_Lambda,0 -> 0 for this menu: every channel in the registered decomposition carries w_c <= -0.9, so w_total <= -0.9 for any choice of channel fractions. Closure requires a GCT-derivable dark-energy channel with quintessence-today behavior over z in [0, 1.5]; every dark-energy channel the framework currently supplies is phantom-directed.
• P.44 / anti_zeno_upper_crossover: Bare-vs-renormalized frequency channel: the engine PASS is protected by construction because the bare Misra-Sudarshan crossover is about 9 GHz while the registered 112 MHz / 42 MHz windows require O.23 protected-subspace lowering of the effective Hamiltonian variance. This row is therefore open-conditional on protected-subspace renormalization, not direct bare-frequency agreement.
• P.1d / alpha_inverse_bilayer_corrected: alpha^-1 != 137.030 within 100 ppm (would falsify the bilayer correction); current 41.6 ppm residual is the QLQCD-1L research debt (App Z.5-Z.6 open program)
• P.45 / yield_strain_epsilon_y: epsilon_y != phi^-5 within 1% (would falsify the AKN tiling-flip derivation)
• P.46 / ch20_liv_delay_and_rm_coefficient: GRB Delta_t > 1e-15 s at E~100 GeV would falsify the quadratic-only LIV claim — but current Fermi/LAT precision is ~1e-1 s, so this gate is currently UNFALSIFIABLE on operational grounds (parallel to HFGW scope-restriction in App FM §FM.X). The genuine Tier 2 LIV claim is the quadratic-only form (linear LIV forbidden by I_h center-inversion); empirical detection requires next-generation gamma-ray timing with ~10^14× current sensitivity. The RM angular-power Tier 3 sub-claim: SKA-era FRB catalog finds no l=6 RM power at phi^-18 amplitude in a > 5000-FRB catalog at > 3 sigma.
• P.47 / polaron_unity_trefoil: Trefoil-case Polaron Unity is Tier 3 conditional pending O.18 fixed-slice reduction + finite-quotient meridian trace + represented-progroup direct-product/operator-factor closure. The general-prime extension remains Tier 3 conditional on O.18 Anderson-Putnam-to-knot-complement extension + canonicity of K -> A_K + finite-quotient meridian trace construction, plus O.35 trivial-amenable-radical, O.36 primary representation hypothesis, and O.32 KO-dimension sign verification.
• P.7c / pta_l6_peak_deviation: Peak l=6 angular correlation deviation ΔΓ_max != phi^-18 = 1.73e-4 within 5% in SKA-PTA matched-filter analysis
• P.7d / pta_l6_linear_legendre: Linear Legendre projection coefficient |C_hat_6| != 1.67e-4 within 5% in SKA-PTA matched-filter; or |C_hat_6| < 1e-4 at >3 sigma combined NANOGrav 15yr + PPTA DR3 + EPTA DR2
• P.7e / pta_l6_squared_power_ratio: Squared angular-power-spectrum ratio C_6/C_0 != phi^-36 = 2.90e-8 within factor 2; or SKA-PTA (SKA-era) finds C_6/C_0 < 1e-9 at >3 sigma
• P.7f / no_verifier: kSZ velocity cross-power C_6^kSZ / C_0^kSZ != phi^-18 = 1.73e-4 within factor 2 in CMB-S4 + Simons Observatory matched-filter combination against DESI / Euclid galaxy catalog peculiar-velocity reconstruction
• P.7g / no_verifier: FRB dispersion-measure scalar power on l=6 spherical harmonic in joint CHIME/FRB + SKA-era catalog ( > 5000 events) fails to detect anisotropy at amplitude consistent with phi^-18 at >3 sigma
• C.IIT / no_verifier: Tier 2 mechanism + Tier 3 sub-graph values: max_k Phi(k <= 5) = 1.124900 > 0 on canonical induced sub-graphs of the icosahedron under the noisy majority-vote TPM. These sub-graph values DO NOT extend to a global Phi_max(k=12) > 0 bound under the IIT Exclusion Postulate; the full k=12 numerical computation is the residual of Open Problem O.28. Numerical values are PyPhi-environment-pinned and should be compared only under the registered PyPhi configuration.
• P.41 / alpha_s_inv_bare: Tier 3 calibrated handle: alpha_s^-1(bare) = 10 phi^2 = 26.18 combines a geometric integer and geometric factor, but the strong-sector boundary is pending O.42 / QLQCD-2 derivation. The bare->PDG gap is documented open problem, not current support.
• P.42 / E_perp_E_para_amplitudes: |E_perp|^2 / |E_para|^2 != phi^-2 (would falsify the H_4/H_3 cut-and-project construction itself)
• S.ckm_unitarity_3x3 / verify_independent/verify_mixing_unitarity.py::ckm_unitarity_3x3: Tier 1 algebraic identity; no falsification band applies — the claim is provable from definitions and trivially fails iff the hypercharge assignments / unitarity constraints are modified.
• S.hypercharges_and_anomaly_cancellation / verify_independent/verify_hypercharges.py::hypercharges_and_anomaly_cancellation: Tier 1 algebraic identity; no falsification band applies — the claim is provable from definitions and trivially fails iff the hypercharge assignments / unitarity constraints are modified.
• S.pmns_unitarity_3x3 / verify_independent/verify_mixing_unitarity.py::pmns_unitarity_3x3: Tier 1 algebraic identity; no falsification band applies — the claim is provable from definitions and trivially fails iff the hypercharge assignments / unitarity constraints are modified.

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END OF APPENDIX R