PUH Theorem 296 (Exact Computation) — The Quaternionic Freeze Exponent p = 63/4 from a Certified E8 Structure-Constant Build: Kazhdan Weights of the Slodowy Slice, a Compact-G2 Reductive Centralizer, and T252's Realized Column H ~ t^ (−63/59)

Photonic Universe Hypothesis (PUH) — Exact Computation. THE ITEM. T265 stated it verbatim: "The exact quaternionic exponent is not computed here. It requires the dimension of the real nilpotent cone of E8 (−24). " T295 identified the machinery — the freeze exponent of T247's mobility law IS the volume-vanishing exponent of the Casimir sublevel set at the top real nilpotent stratum, of which T251's split p = 8 and compact p = 124 are the two easy cases — calibrated a working rule on five instances, bracketed the answer at p ∈ 6, 19, and named the successor computation. This paper performs it. THE BUILD AND ITS GATE. A complex E8 Chevalley basis was constructed from scratch (Bourbaki simple-root coordinates, Frenkel–Kac cocycle; 240 roots, 120 positive). Validation exact: Jacobi identity with ZERO failures on 600 random basis triples; Killing normalisation tr (ad²) = 120 exact. Đoković's representative of the maximal real orbit was then built and — as a PRE-DECLARED, BLOCKING gate logged before any derivation — required to reproduce his published kernel filtration. It reproduces ALL SIXTY-NINE graded numbers: total dim ker (ad E) ʲ and both graded columns, for every j from 1 to 23, closing at 248. Structure constants and root map are certified against an independent published source. THE STRUCTURE. The six roots of the representative form an E6 BASE (Cartan matrix computed; determinant 3), so E is the principal nilpotent of a regular E6 subalgebra — confirming Đoković's type label by computation. The sl2-triple was built with H the E6 co-Weyl vector 2ρ∨ = (16, 22, 30, 42, 30, 16), H = (12, 14, 20, 28, 22, 16, 6, −8) in coroot coordinates, and verified by exact integer equalities. A lemma falls out: ε (α, −α) = −1 for EVERY E8 root, from (α, α) = 2. THEOREM 296. 1. The Kazhdan weights of the 32-dimensional Slodowy slice, computed as the adH spectrum on ker (ad F), are 24, 18×7, 16, 12, 10×7, 4, 2×14 — exactly the multiset T295 derived from Đoković's block sizes, now re-derived from our own certified basis by a different route. THEOREM 296. 2. Along a weight-w coordinate, Id contributes order 2d/w when w | 2d; the vanishing order is the minimum over the eight degrees. Verified by truncated-polynomial traces: the combinatorial minimum is ATTAINED in every class — seven for seven, no leading coefficient vanishing. w=24 linear by I₁2; w=18 order 2 by I₁8; w=16 linear by I₈; w=12 order 2 by I₁2; w=10 order 4 by I₂0; w=4 linear by I₂; w=2 order 2 by I₂. COROLLARY (double-derivation): the order-1 classes are exactly w = 24, 16, 4 — THREE linear cuts — independently reproducing the rank rₐlive = 3 measured by exact elimination over two primes (with the dead differentials d = 24, 30 confirmed identically zero and all alive rows confirmed to vanish on the orbit tangent). Weight combinatorics and structure-constant elimination are unrelated computations returning the same number. THEOREM 296. 3 (the definiteness closure). The rule's cost 1/m per direction presumes the binding form is DEFINITE — a real-form property T295 could not settle. It collapses to one question. The m = 0 part of ker (ad F) is the reductive centralizer: verified dim 14, Lie rank 2, hence type G2 (matching the Bala–Carter datum for the E6 orbit in E8). The slice decomposes under it as 1 + 7 + 1 + 1 + 7 + 1 + 14, and both the 7 and the adjoint carry a UNIQUE invariant quadratic form — so definiteness everywhere reduces to: compact G2 or split G2 (2)? The triple is verified normal (E, F wholly in p; H wholly in k; k of dim 136, p of dim 112), and the measured grading of the adjoint block is 14 in k, 0 in p. Since ZK of a normal triple complexifies the maximal compact of the real reductive centralizer, the maximal compact is all of G2: the centralizer is COMPACT G2, not split G2 (2) (which would have shown 6 and 8). Every invariant form on the slice is therefore definite and the calibration's convention is SATISFIED, NOT ASSUMED. Two literature numbers are reproduced free from our own structure constants: z (E) splits 28 in k and 4 in p, giving KC-orbit dimension 136 − 28 = 108 (the Kostant–Rallis count, 112 minus real rank 4), and complex orbit dimension 248 − 32 = 216 (Đoković Table 1). The slice grading splits 28/4 likewise: all structure (both sevens, the adjoint) in k, all four trivial reps in p. RESULT 296. 4. p = 1 + 7 (1/2) + 1 + 1/2 + 7 (1/4) + 1 + 14 (1/2) = 63/4 = 15. 75 — inside T295's published bracket 6, 19, and inside the revised range from T295's own stated shift formula once rₐlive was measured at 3 rather than assumed at 6. The pre-registration held at every grade. RESULT 296. 5 — three quantities stop being bands. Freeze law u ~ t^ (−4/59). T252's REALIZED COLUMN, missing since T265 ruled out both integrated branches, is filled: H ~ t^ (−63/59), exponent 1. 0678 — steeper than matter-only (1), far from Λ (→0), just below the old split 8/7, and INSIDE T295's PRE-DECLARED thawing band (1. 056, 1. 2] near its lower edge; the band was declared before the number existed. KL exponent via T263's bridge: θ = 1 − 1/ (2p) = 61/63 ≈ 0. 9683. KILL-CONDITIONS: (i) the rule stays falsifiable by any calibration instance disagreeing with exact quadrature of its local model — 63/4 falls with the rule; (ii) any independent recomputation disagreeing with the 69-number gate invalidates the build and everything downstream; (iii) if the Kostant–Sekiguchi input is misapplied and the centralizer is split G2 (2), the order-2 blocks lose definiteness and the exponent falls to roughly 7. 25 — a large, checkable difference, which is why the grading measurement is exhibited; (iv) thawing dark energy established away from exponent 1. 068, or thawing excluded with the epoch shown post-onset, falsifies the realized-branch identification. NOT CLAIMED: a proof of the rule (calibrated on six instances now, not derived; the quantity is real-log-canonical-threshold type, resolution-grade in general) ; independence from the literature (the ZK/maximal-compact statement is the single link not computed here) ; a derived cascade clock (still open, so the epoch's position relative to onset remains unsettled) ; the dimensional bridge for the ignition rate (T295's open item, untouched) ; or observational confirmation — 63/4 is a prediction and Result 296. 5 states the test.