PUH Theorem 295 (Derivation, Calibration, and Bracket) — The Cascade Endgame: Snap Ignition at the Interior Threshold Φ* = 1/128; the Freeze Exponent as a Volume-Vanishing Exponent Calibrated on Five Instances; the Quaternionic Bracket p ∈ [6, 19] fr

Photonic Universe Hypothesis (PUH) — Derivation, Calibration, and Bracket. THE PRE-DECLARATION. Before the exponent papers were read, the decision rule was logged to disk: reduce the terminal flow to ds/dt = −M|F′| along the collapse ray (licensed by T294's exact rank-1 tension collapse); write M ~ δ^a, |F′| ~ δ^b; pre-declare KC-A — if the Snap sits at the degenerate endpoint AND a+b ≥ 1, Snap-as-event (T288/T287) is inconsistent with the gradient-flow cascade (T246/T247). VERDICT: KC-A NOT TRIGGERED. T247's own correction puts the freeze in the MOBILITY, not the functional — tension is MAXIMAL at the core (9E_P snap cap), so b = 0 — and the Snap is INTERIOR: with λ(Φ) = 9E_P·Φ (minimal model) and T238's λ* = 9E_P/128, the Snap fires at Φ* = 1/128, under one percent folding, mobility essentially full. The fold's anatomy: EXPONENTIAL IGNITION (dΦ/dt ≈ ΓΦ, Γ ∝ R_sub; Snap delay τ* = Γ⁻¹ ln(Φ*/Φ₀), LOGARITHMIC in the photon seed; minimal-model-tagged; dimensional bridge for Γ's absolute scale named as gated) → SNAP EVENT at Φ* → POWER-LAW FREEZE u ~ t^(−1/(p−1)). THE EXPONENT MADE WELL-POSED (Theorem 295.2). The freeze exponent p IS the volume-vanishing exponent of the Casimir sublevel set Vol{|P_d| < ε ∀d} ~ ε^p at the top real nilpotent stratum of the folding real form. T251's filed values are its two easy cases: split (transversal complete intersection, p = 8) and compact (definite ball, p = 124). The quaternionic case is hard for an identifiable reason: E8(−24)'s real cone sits in NON-REGULAR complex orbits where invariant differentials die; the quantity is of real log-canonical-threshold type. THE DATA LOCK (Đoković, Asian J. Math. 5, 561–584 (2001); full text fetched, tables used). E8(−24) has 36 nonzero real nilpotent orbits; the maximal O³⁶ lies in the complex Bala-Carter class E6, dimension 216 (confirming Kostant–Rallis 112−4 = 108 doubled by Sekiguchi; the dim_R = dim_C fact literature-verified via arXiv:1402.6796); centralizer 32; explicit representative tabulated (Table 2 row 36). From the kernel filtration (Table 5 row 36): 32 ad-Jordan blocks [23, 17×7, 15, 11, 9×7, 3, 1×14], all four sanity sums passing (count 32; total 248; Σ(s−1) = 216; largest block 23 → m_max = 22 = weighted-Dynkin cross-check). Process note on the record: a first transcription dropped the filtration's final entry, closed at 247, and was caught by these sanity sums before use. CONSEQUENCES: SIX Casimir differentials survive at the top stratum (degrees 2, 8, 12, 14, 18, 20); TWO die (24 at slice order ≥ 2 — a weight-24 coordinate squared; 30 at order ≥ 3), by Kazhdan-weight arithmetic on the slice weights [24, 18×7, 16, 12, 10×7, 4, 2×14]. THE RULE, CALIBRATED (Theorem 295.3; calibrated, not proven). Exponent = Σ over transverse cutting directions of 1/(vanishing order of the binding invariant). FIVE instances: split-E8 = 8 (filed); compact-E8 = 124 (filed); so(3,2) = 2 predicted, 1.986 measured; so(5) = 5 analytic; so(4,1) = 2.25 = 1 + 1/2 + 1/2 + 1/4 predicted and CONFIRMED BY EXACT QUADRATURE of the measured integer-clean local model (I₂ = √32 q₃ − 2z²; I₄ = 16q₁² + 16q₂² + 32q₃² + 2z⁴) — flat at 2.250 across seven decades. THE PLATEAU EPISODE, FILED AS METHOD: Monte-Carlo probes first read ~1.85–2.0; the discrepancy was HELD until diagnosed — the quartic direction's 1/4 cost switches on only below ε ~ γδ⁴ (near 1e-5 for the probe's δ), so the probes read a 2.0 plateau; the capped model REPRODUCES the plateau, the uncapped model gives the asymptote, and a deep probe's raw count ratios (2.13/2.27/2.30/2.31) agree. Lesson filed beside T294's exact-arithmetic principle: finite-window probes read the plateau; exact quadrature of the measured local model is the decisive instrument. DISCLOSED: the codimension candidate p = 32 that this collaboration itself proposed at the investigation's start (Kostant–Rallis + Sekiguchi + codimension identification) was KILLED by this same calibration — so(4,1) has codimension 4 but exponent 2.25 — our own candidate, excluded by our own test. THE BRACKET (Result 295.4). Applying the calibrated rule to the locked data: 32 slice directions; up to six surviving linear cuts (rank r_alive ≤ 6, slice-independence unverified without structure constants); all remaining directions at orders ≥ 2, cost ≤ 1/2 each. Hence p_quat ∈ [r_alive, 16 + r_alive/2]; at r_alive = 6: p_quat ∈ [6, 19]. EXCLUDES compact 124 AND the codimension 32; the split value 8 survives inside. THE EXACT VALUE IS A NAMED SUCCESSOR: leading-form ranks on the 32-dim slice at Đoković's explicit representative — a structure-constant computation with every input now tabulated. THE THIRD COLUMN (Result 295.5). T252's exact halt law ran on two branches T265 rules out. The realized branch: H ~ t^(−p/(p−1)) with exponent n ∈ (19/18, 6/5] = (1.056, 1.2] — a THAWING BAND, steeper than matter-only (n = 1), far from Λ (n → 0), bracketing the old split 8/7 ≈ 1.143. Freeze law u ~ t^(−1/(p−1)) with exponent between 1/18 and 1/5. Caveats carried from T252: onset scales as u₀^(−(p−1)) in cascade units and the cascade clock is underived, so the present epoch's position relative to onset is not settled here. The DESI-class evolving-w arbiter is REFRAMED onto a bounded band. KILL-CONDITIONS: (i) any further calibration instance disagreeing with exact quadrature falsifies the rule; (ii) if the successor finds r_alive < 6 the bracket shifts per the stated formula, and if it finds p_quat outside [r_alive, 16 + r_alive/2] the rule fails at E8(−24) and the bracket falls; (iii) thawing dark energy established with n outside (1.056, 1.2] — or thawing firmly excluded with the epoch shown post-onset — falsifies the realized-branch identification; (iv) KC-A stays armed retroactively against any future relocation of the Snap to the endpoint. NOT CLAIMED: exact p_quat; a proof of the rule; the dimensional bridge for Γ; Φ* = 1/128 beyond the minimal model (interior CHARACTER robust, the number inherits linearity); r_alive = 6; the epoch's onset position.