ħ framing and the θ₀ match: owner decisions O1–O7 applied (follow-up to #937)

book/README.md

@@ -59,9 +59,9 @@ The full compiled manuscript:
 
 - [`ch05-gauge-structure.md`](ch05-gauge-structure.md) — **The Emergence of Gauge Structure.** The minimal substratum, $d = 3$ from three independent self-consistency filters, the observer-level lattice-wave branch and its staggered factorization, the exact decomposition of the six geometric link directions into $(3,2,1)$, H-link/H-cust as the named bridges from that geometry to a single physical gauge carrier, and the resulting conditional $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ commutant construction, chirality from the trace-out (with explicit evasion of Nielsen-Ninomiya), and the six anomaly conditions forcing $Y_Q = 1/6, Y_u = 2/3, Y_d = -1/3, Y_L = -1/2, Y_e = -1$ with $|q_p| = |q_e|$ as a theorem.
 
-- [`ch06-matter-content.md`](ch06-matter-content.md) — **The Matter Content and Quantitative Predictions.** Three chiral fermion generations from staggered taste structure, one composite Higgs doublet from the singlet taste, gauge couplings with the structural input $1/\alpha_0 = 23.25$ and the no-GUT prediction, and the framework's parameter-free quantitative retrodictions: Cabibbo $\lambda = 1/(\pi\sqrt{2})$ at $0.04\%$, Koide $\theta_0 = 2/9$ at $0.02\%$, six fermion masses to better than $1\%$, all three PMNS angles within $1.1\sigma$, joint $y_t(M_\text{Pl}) = 1$ and $\lambda(M_\text{Pl}) = 0$ matching $m_t$ and $m_H$ from $v$ alone.
+- [`ch06-matter-content.md`](ch06-matter-content.md) — **The Matter Content and Quantitative Predictions.** Three chiral fermion generations from staggered taste structure, one composite Higgs doublet from the singlet taste, gauge couplings with the structural input $1/\alpha_0 = 23.25$ and the no-GUT prediction, and the framework's parameter-free quantitative retrodictions: Cabibbo $\lambda = 1/(\pi\sqrt{2})$ at $0.04\%$, Koide $\theta_0 = 2/9$ at $0.001\%$, six fermion masses to better than $1\%$, all three PMNS angles within $1.1\sigma$, joint $y_t(M_\text{Pl}) = 1$ and $\lambda(M_\text{Pl}) = 0$ matching $m_t$ and $m_H$ from $v$ alone.
 
-- [`ch07-gravity.md`](ch07-gravity.md) — **Gravity and Cosmology.** Boundary-mode dispersion, the framework's $\hbar$ derivation from thermal self-consistency at the cosmological horizon, the Bekenstein–Hawking entropy with the $1/4$ factor recovered as a consistency relation of the horizon calibration (GW250114 confirms the classical area theorem), dark-sector phenomenology including $a_0 = cH/6$, the conditional dissolution of the $10^{122}$ cosmological constant discrepancy (descriptive distinction established, gravitational decoupling conditional on the open coupling step), and a cycle-averaged Page curve on the full product space of the black hole and its radiation, conditional on the scrambling condition H-scramble and the decoupling condition H-decouple. Discusses the open empirical tension with DESI DR2's apparent phantom crossing and the framework's prediction $w(z) \approx -1$.
+- [`ch07-gravity.md`](ch07-gravity.md) — **Gravity and Cosmology.** Boundary-mode dispersion, the framework's action-scale relation for $\hbar$ from thermal self-consistency at the cosmological horizon, the Bekenstein–Hawking entropy with the $1/4$ factor recovered as a consistency relation of the horizon calibration (GW250114 confirms the classical area theorem), dark-sector phenomenology including $a_0 = cH/6$, the conditional dissolution of the $10^{122}$ cosmological constant discrepancy (descriptive distinction established, gravitational decoupling conditional on the open coupling step), and a cycle-averaged Page curve on the full product space of the black hole and its radiation, conditional on the scrambling condition H-scramble and the decoupling condition H-decouple. Discusses the open empirical tension with DESI DR2's apparent phantom crossing and the framework's prediction $w(z) \approx -1$.
 
 - [`ch08-juno.md`](ch08-juno.md) — **JUNO and the Neutrino Sector.** Focused case study of the framework's parameter-free PMNS prediction $\sin^2\theta_{12} = 1/3 - 1/(4\pi^2) = 0.3080$ and the associated sum rule $2\sin^2\theta_{12} + \sin^2\theta_{23} = 7/6$, made before JUNO's first measurement and confirmed at $0.07\sigma$ against the post-JUNO global fit. Discriminates against the TM1/TM2 column-preservation patterns at sub-0.005 sum-rule precision.
 

book/appendix-a-prediction-status.md

@@ -77,7 +77,7 @@ The reference table combines Standard Model observables, coupling-sector inputs
 | 8 | $m_d/m_s$ | $\lambda^2$ | L | 1 | Structural $\lambda$; GST empirical | Ch 6 §6.5 |
 | 9 | Jarlskog $J$ | $\eta/(12\pi^6)$ | E | 2(b) | Empirical $\eta$ | Ch 6 §6.4 |
 | 10 | Koide $Q$ | $2/3$ | C | 1 | **<0.001%** | Ch 6 §6.5 |
-| 11 | Koide $\theta_0$ | $C_2/d^2 = 2/9$ | C | 1 | Within 1% | Ch 6 §6.5 |
+| 11 | Koide $\theta_0$ | $C_2/d^2 = 2/9$ | C | 1 | 0.001% | Ch 6 §6.5 |
 | 12 | $\sin^2\theta_{12}$ | $1/3 - 1/(4\pi^2)$ | L | 1+2(b)ᶠ | **0.07$\sigma$ JUNO** | Ch 8 |
 | 13 | $\sin^2\theta_{13}$ | $4/(18\pi^2)$ | C | 1 | Within 1% | Ch 8 |
 | 14 | $\sin^2\theta_{23}$ | $1/2 + 1/(2\pi^2)$ | L | 1+2(b)ᶠ | Within 1$\sigma$ | Ch 8 |
@@ -95,7 +95,7 @@ The reference table combines Standard Model observables, coupling-sector inputs
 
 **ᶠ** *Layer 2(b) (sin²θ_12, sin²θ_23, PMNS sum rule).* The Layer 1 component is structurally protected against the $A_1$ perturbation by the cosines-dressing of the TBM sum rule (cubic geometry alone). The remaining Layer 2 component is the $A_2$ Wilson-coefficient ratio $b_{23}/(b_{12}+b_{13}) = 4/3 = 4(d-1)/(2d)$, equivalent to the substrate-level relation $b_{23}^\text{sub} = (4/3)(b_{12}^\text{sub}+b_{13}^\text{sub}) + 1/9$. The ratio's 2(a)-tractable period — the label meaning the residual question was sharply specified and admitted definite outcomes from substrate calculations — has ended: the calculation was performed. It does *not* mean the answer is forced by representation theory alone — the May 6, 2026 audit (§B.2.2) established that the natural Higgs-democratic spurion does not produce this ratio (instead giving $b_{12}:b_{13}:b_{23} \approx 17:1:1$ in the heavy-quark hierarchy, off by factor ~24 from the required 4/3), and the relation requires substrate amplitudes with the non-natural scaling $\delta M_{ij}^\text{sub} \propto m_{\max(i,j)}$ plus a channel-specific enhancement factor $K_{23}/K_{12} \approx 2.8$. The substantive content is bijection-specific — Layer 2(b) — and the sharply specified verification path has since been executed: the §7.3 routing computation excludes the second-order route (the two-hop operator is a pure spin-taste monomial, orthogonal to every mass channel), leaving the recorded $O(\lambda^3)$ construction as the one unpursued analytical route. The empirical match at 0.07$\sigma$ (JUNO) and 0.6$\sigma$ (sum rule) confirms the bijection has the required structure; the $7/6$ sum rule remains the direct experimental test.
 
-**Distinctive empirical content.** Three predictions stand out as the framework's strongest empirical content. (i) JUNO $\sin^2\theta_{12} = 1/3 - 1/(4\pi^2) = 0.3080$ at **0.07$\sigma$** against the post-JUNO global fit — a parameter-free retrodiction; its exact, freedom-free form makes JUNO's design-lifetime precision ($\pm 0.0014$) a pre-registered forward test. (ii) the conditional Milky Way crossover, whose radius depends on the baryon budget. (iii) Koide $Q = 2/3$ to better than **0.001%** — the sharpest framework prediction across all 23, matching observation at four-significant-figure precision.
+**Distinctive empirical content.** Three predictions stand out as the framework's strongest empirical content. (i) JUNO $\sin^2\theta_{12} = 1/3 - 1/(4\pi^2) = 0.3080$ at **0.07$\sigma$** against the post-JUNO global fit — a parameter-free retrodiction; its exact, freedom-free form makes JUNO's design-lifetime precision ($\pm 0.0014$) a pre-registered forward test. (ii) the conditional Milky Way crossover, whose radius depends on the baryon budget. (iii) Koide $Q = 2/3$ to better than **0.001%** — the sharpest framework prediction across all 23 in relative precision, matching observation at four-significant-figure precision.
 
 ### A.4.2 Gravitational-sector observables
 
@@ -212,7 +212,7 @@ The framework's cumulative empirical record can be summarized as follows.
 
 **Most distinctive content.** Two predictions distinguish the framework from standard accounts most sharply: the substratum-emergent operator distinction (with empirical signatures in B1, B2, B13, B14) and the partially-quantum regime $\mathcal{F}(r) = 2r - r^2$ (with empirical exposure in F1). Neither has analog in standard quantum mechanics or effective field theory.
 
-**Sharpest single match.** Koide $Q = 2/3$ to better than 0.001% (entry 10) — the framework's sharpest individual empirical match.
+**Sharpest single match.** Koide $Q = 2/3$ to better than 0.001% (entry 10) — the framework's sharpest individual empirical match in relative precision.
 
 **Precision and empirical standing.** The comparisons have different evidential roles: Koide and Cabibbo supply numerical retrodictions, the fitted coupling entries are retrodictions by construction, $A\cdot B$ is an inconclusive finite-size extrapolation, and the neutrino mass sum is constrained by upper bounds. The quoted precision belongs to each comparison separately.
 

book/appendix-b-derivations.md

@@ -13,7 +13,7 @@ The appendix's content is organized by mathematical structure rather than by cha
 
 *B.3 The trace-out as a Jordan-Chevalley projection.* The framework's content on the Standard Model gauge structure relies on a structural property of the trace-out operation: it extracts the semisimple part of the evolution matrix's Jordan-Chevalley decomposition and discards the nilpotent monodromy. This appendix gives the full algebraic derivation: period formula, Jordan-Chevalley decomposition, $q$-independence of the Weil-Deligne conductor, additive decomposition over gauge irreps, and the precise sense in which the trace-out is a projection.
 
-*B.4 The gap equation for $\hbar$.* The framework derives Planck's constant $\hbar$ from a thermal self-consistency argument between the classical horizon temperature (substratum-level) and the KMS temperature (emergent QFT-level). This appendix gives the full four-step derivation: spatial locality and boundary dependence, deep-sector gauge equivalence, dimensional determination, and thermal self-consistency producing $\hbar = c^3 \epsilon^2 / (4G\sigma)$.
+*B.4 The gap equation for $\hbar$.* The framework relates Planck's constant $\hbar$ to the discreteness scale through a thermal self-consistency argument between the classical horizon temperature (substratum-level) and the KMS temperature (emergent QFT-level). This appendix gives the full four-step derivation: spatial locality and boundary dependence, deep-sector gauge equivalence, dimensional determination, and thermal self-consistency producing $\hbar = c^3 \epsilon^2 / (4G\sigma)$.
 
 *B.5 Anomaly cancellation and unique hypercharges.* The framework's content on the Standard Model matter content derives the unique hypercharge assignment from the six anomaly conditions of $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$. This appendix gives the explicit calculation: the six anomaly conditions, the algebraic constraint structure, and the unique solution producing $(Y_Q, Y_u, Y_d, Y_L, Y_e) = (1/6, 2/3, -1/3, -1/2, -1)$ with no free parameters.
 
@@ -170,7 +170,7 @@ extracting the semisimple part, encoding it via magnitudes rather than orders, a
 
 *Main-text reference: Chapter 7 §7.3.*
 
-The framework derives Planck's constant $\hbar$ as the unique self-consistent value matching the classical horizon temperature to the KMS temperature of the emergent quantum field theory. The derivation has four steps; each step is presented with its full content.
+The framework derives the relation $\hbar = c^3\epsilon^2/(4G\sigma)$ between Planck's constant and the discreteness scale by matching the classical horizon temperature to the KMS temperature of the emergent quantum field theory, conditional on H-slope with the horizon and frame conditions of Chapter 7 §7.3; with $\epsilon$ not independently known, it fixes only $\epsilon/\sqrt{\sigma} = 2\,l_p$ (Chapter 7 §7.4), not the value of $\hbar$. The derivation has four steps; each step is presented with its full content.
 
 **Step 1: Spatial locality.** The classical Hamiltonian of the substratum is spatially local. The interaction decomposes as
 $$H_{\text{tot}} = H_V + H_B + H_D + H_{VB} + H_{BD}$$
@@ -208,7 +208,7 @@ containing no $\hbar$. This is computable entirely from classical-horizon thermo
 
 On the exact de Sitter calibration, the emergent QFT of the framework's Part I content lives on a classical background that has a bifurcate Killing horizon with surface gravity $\kappa$ (the bifurcate structure presupposes emergent local boost invariance at the horizon, a dependency carried in [GR §8.5]). Regularity of the Wick-rotated metric at the horizon requires Euclidean period $\beta = 2\pi c / \kappa$; any QFT on this background — including a lattice-regularized one — is periodic in imaginary time with that period only under the H-Hawking conditions, giving a KMS state at temperature
 $$T_Q = \frac{\hbar \kappa}{2\pi c k_B}$$
-with $\hbar$ unknown. This is a theorem *within* the derived QFT, not an external import.
+with $\hbar$ unknown. The periodicity holds under the H-Hawking conditions and the KMS identification requires the emergent state to belong to that class (H-spectrum), so the matching below is a conditional infrared calibration, not a theorem within the derived QFT ([GR §3.2]).
 
 The two temperatures are computed independently — $T_{\text{cl}}$ from the classical substratum alone (no QM), $T_Q$ from the emergent QFT alone (no classical substratum details) — but they refer to the *same physical degrees of freedom*: the boundary modes $V \times B$ across which $H_{\text{int}}$ couples the visible and hidden sectors. Step 1's boundary-only dependence shows that the visible transition data from which the emergent QFT is reconstructed are fixed (at leading order in $\tau_S/\tau_B$) by these same $V \times B$ dynamics. Temperature is a state property; two complete descriptions of the same modes cannot assign different temperatures without logical contradiction. The matching $T_{\text{cl}} = T_Q$ is therefore not an additional assumption but a consequence of the boundary modes being the same physical objects in both descriptions, with corrections at $\mathcal{O}(\tau_S/\tau_B) \sim 10^{-32}$:
 $$\frac{c^2 \epsilon^2 \kappa}{8\pi G \sigma} = \frac{\hbar \kappa}{2\pi c}$$
@@ -468,7 +468,7 @@ The biconditional is the framework's main characterization result. $\square$
 
 - B.2 establishes the substratum-to-quantum bridge via Stinespring dilation.
 - B.3 establishes the algebraic structure of the trace-out as Jordan-Chevalley projection, organizing the framework's content on gauge structure.
-- B.4 establishes the gap equation determining $\hbar$ from horizon thermodynamics, producing the framework's content on gravitational physics.
+- B.4 establishes the gap equation relating $\hbar$ to the discreteness scale through horizon thermodynamics, producing the framework's content on gravitational physics.
 - B.5 establishes the unique anomaly-free hypercharge assignment, fixing the framework's content on Standard Model matter.
 - B.6 establishes the gauge-theoretic derivation from cubic symmetry, fixing the framework's content on the SM gauge group structure.
 - B.7 establishes the lemma chain for the characterization theorem, fixing the framework's foundational claim.