The Folding Real Form Is Indefinite (T264): The Cascade Requires a Non-Definite Killing Form, Excluding the Compact E8(-248) and Its Frozen t^(-1/123) Branch, with the Residual Split-versus-Quaternionic Question Named but Not Resolved
Year: 2026
Type: preprint
Abstract: Photonic Universe Hypothesis (PUH) — T264 — STRUCTURAL NARROWING NOTE. T251 reduced the late-time freeze to a binary fixed by the metric signature of the folding configuration space: SPLIT-type (indefinite Killing form) gives p=8, u(t)~t^(-1/7); COMPACT-type (definite) gives p=124, u(t)~t^(-1/123) (essentially frozen). T251 left the choice open as the sharpened equation-of-state question. This note removes part of that residual on PHYSICAL, not equation-of-state, grounds. THE SIGNATURE IS PHYSICAL: in PUH the Killing form is the metric on the space of condensate configurations, so it sets the kinetic-energy sign structure of the folding modes. A DEFINITE (compact) form makes every one of the 248 directions same-sign and oscillatory — bounded, with NO unbounded descent direction. An INDEFINITE form has opposite-sign directions — unbounded descent directions. THE CASCADE NEEDS DESCENT: the PUH cascade is a directed, irreversible, downhill flow from unfolded Phase I to folded Phase III (T206/T239/T247). It REQUIRES unbounded descent directions, which a definite form lacks. EXCLUSION: the compact real form E8(-248) is therefore excluded as the folding form, removing T251's p=124, t^(-1/123) essentially-frozen branch — on physical grounds, independent of the equation of state. The framework's freeze is one of the LIVE (non-frozen) alternatives. THE RESIDUAL (named, not resolved): split E8(8) vs quaternionic E8(-24), both indefinite. Two opposed arguments: - Cascade geometry favours SPLIT: T251's clean p=8 used a real saturation cone of full dimension dim-rank=240 — the defining signature of the split form E8(8) (128 noncompact directions). The quaternionic form (112 noncompact) has a smaller cone => a third exponent, neither 8 nor 124. - Weak-sector chirality favours QUATERNIONIC: T212 derives parity violation from L-R asymmetry (SU(2)_L unbroken, SU(2)_R broken at rebound scale, chirality from the rebound). A parity-asymmetric vacuum may prefer a chiral real form; the quaternionic E8(-24) (maximal compact E7xSU(2)) carries such structure where the maximally-symmetric split form does not. Neither is decisive: the cascade argument assumes the folding cone IS the full real nilpotent cone (inherited T175/T238); the chirality argument is heuristic. CLARIFICATION (prevents an error): T212's E7xSU(2) is the COMPLEX maximal subalgebra giving the form-agnostic branching 248=(133,1)+(1,3)+(56,2) — NOT the maximal compact subgroup of the quaternionic form. Same name, different structure. So T212's gauge decomposition does NOT by itself select the quaternionic form; the chirality argument is separate and weaker. Contribution: excludes the compact/frozen branch on physical grounds (a real narrowing, no equation of state needed); names the split-vs-quaternionic residual one step narrower than T251. Built with full corpus searchable.
Author: Martell, Brian
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166048327
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