PUH Theorem 353 (The Step Rule Decides the Cascade) — Greedy Descent from E7 Ends on SU(2)³ with No SU(3) Anywhere, While Minimal Descent Passes Through SO(10) and Ends on Colour; SU(3)×SU(2) Is a Legitimate Centralizer with Exactly the Five U(1)s Required

Abstract

Photonic Universe Hypothesis (PUH) — Computation. THE SET-UP. T175 minimises an energy on a constraint surface built from the eight Casimir invariants; T238 fixes the couplings as ζ_k = S·d_k/I_k^max. Evaluating g(μ) = Σ_k ζ_k I_k(μ), with I_k(μ) = Σ_α (α,μ)^{d_k} over the 240 roots and degrees {2,8,12,14,18,20,24,30}, makes the cascade computable: at each stage the surviving subalgebra is the CENTRALIZER of the accumulated Cartan directions, and the sequence of centralizers is the cascade. TWO THINGS THAT MIGHT HAVE BEEN FREE ARE FIXED BY FILED RESULTS: T265 establishes the folding real form as quaternionic E8(−24), and T304 establishes that the gravitational sl(2,ℝ) within it is UNIQUE UP TO CONJUGACY, the 112 noncompact root directions forming a single Weyl orbit. NEITHER IS CHOSEN HERE. And a third apparent freedom is not one: the equivalent minima at each stage are Weyl-conjugate and every Casimir is Weyl-invariant by construction — verified, I₂ = 120.000000 and I₃₀ = 2.14748376×10⁹ at four conjugate configurations, IDENTICAL TO EVERY DIGIT — so no selector among them is required. RESULT 353.1 (the step rule decides everything). Two step rules applied to the same E7 give chains sharing only their first step. GREEDY (removes most roots): 126 E7 → 60 SO(12) → 26 SU(2)×SO(8) → 24 SO(8) → 6 SU(2)³ → 4 → 2. MINIMAL (removes fewest): 126 E7 → 60 SO(12) → 40 SO(10) → 24 SO(8) → 12 SU(4) → 6 SU(3). THE GREEDY CHAIN CONTAINS NO SU(3) AT ANY STAGE — its endpoint is three disconnected SU(2) factors, confirmed by CONNECTIVITY and not inferred from the root count, since six roots is SU(2)³ when it splits [2,2,2] and SU(3) when it is one connected component. THE MINIMAL CHAIN PASSES THROUGH SO(10), the canonical grand-unified group whose 16-spinor is exactly one generation with a right-handed neutrino, and terminates on SU(3), which is colour — and T257 places colour at precisely this depth, E6 → SO(10) → SU(5) → SU(3)_colour. The divergence occurs at the second step: greedy removes 66 roots to reach SO(12), minimal removes 54. BOTH ARE LEGITIMATE SUBALGEBRAS OF E7; THE STEEPEST STEP LANDS IN THE WRONG FAMILY. RESULT 353.2 (the target is a legitimate centralizer). A scan of 30,000 directions finds 118 whose centralizer is SU(3)×SU(2) (components [2,6]) and 110 whose centralizer is SU(2)⁴ (components [2,2,2,2]). BOTH HAVE EIGHT ROOTS; ONLY CONNECTIVITY DISTINGUISHES THEM, and neither is rare. AND THE ABELIAN COUNT IS NOT IMPOSED: a centralizer of a Cartan direction has rank 8, so a semisimple part of rank 3 leaves FIVE abelian factors — one hypercharge and four more, which is the number spacetime requires. Nothing in the computation was arranged to produce it. So the Standard Model's gauge algebra is available to this construction, and the greedy cascade's failure to reach it is A PROPERTY OF THE PATH, NOT OF THE ALGEBRA. RESULT 353.3 (two criteria select the same branch). THE ENERGETIC CRITERION, REPORTED NOT RELIED UPON: the maximum of g is 0.105946 on SU(3)×SU(2) against 0.105119 on SU(2)⁴ — a margin of eight parts in a thousand, too small to carry the argument alone. THE STRUCTURAL CRITERION, WHICH IS NOT NARROW: T222 derives three generations from a Z₃ structure forced by the McKay correspondence once E6 is selected. SU(3) has centre Z₃; SU(2)⁴ has centre Z₂⁴ and contains NO Z₃ WHATEVER. So the generation count is available on one branch only, and A UNIVERSE THAT CASCADED TO SU(2)⁴ WOULD HAVE NO THREEFOLD FAMILY STRUCTURE AT ALL. The second criterion is independent of this computation entirely — T222 derives the generation count without reference to any cascade, and is unsuperseded. WHAT IS NOT SETTLED. THE FRAMEWORK DOES NOT SPECIFY THE STEP RULE, AND THAT IS THE WHOLE OF THE OPEN WORK. There is a physical argument for the minimal step — a cascade of successive INSTABILITIES, each relieving the minimum tension necessary rather than the maximum available, which is the opposite of steepest descent — but T175's functional contains no step size and nothing in the archive supplies one. And the minimal-descent computation samples random integer directions and takes the one removing fewest roots: THAT SHOWS WHAT IS REACHABLE UNDER A DIFFERENT RULE, NOT WHAT THE FRAMEWORK REQUIRES. So the honest statement is conditional: the cascade reaches the canonical grand-unified chain IF AND ONLY IF its steps are minimal, and whether they are is undetermined. KILL-CONDITIONS: (i) IF THE PHYSICAL STEP RULE IS GREEDY RATHER THAN MINIMAL, the cascade ends on SU(2)³, the framework has no colour and no generations, and this line fails entirely; (ii) if the centralizer construction is not the right notion of "surviving subalgebra" — if the breaking is not by orthogonality conditions on Cartan directions — the whole computation addresses the wrong object; (iii) if the 0.8 percent energetic margin reverses under a more careful treatment, only the Z₃ criterion remains, and that selects an algebra without showing the cascade reaches it; (iv) if T222's Z₃ is not the centre of the colour SU(3) but a different threefold structure, Result 353.3's structural criterion does not apply — AND T234 RECORDS THAT THE ARCHIVE'S THREE Z₃ STRUCTURES ARE, AS PRESENTLY DERIVED, DISTINCT. NOT CLAIMED: that the cascade reaches the Standard Model, since that holds only under an unproven step rule; that the minimal rule is correct; that the sampling is exhaustive, since 30,000 directions is a sample and not a classification; that the real form or gravitational embedding were derived here, since both are taken from T265 and T304; or that the intermediate algebras have been identified beyond their root systems and connectivity.