A registered result in its source context
Original ledger classification: Exact [E] census/theorem/controls + honest kill ('free => regular' false) (Python-only per GATE.WOLFRAM.02).
Recorded statement: The ST31 degree-8 round + the unified d/4-theorem (one module, 30 checks, ~10 s; discovery probe st31_degree8_probe.py 30/30; exact Fraction/Z[i] arithmetic). ORDER-8 LANDSCAPE [E]: 10560 order-8 elements, 4800 with free census {8:30},…
Statement, assumptions and sources
Paper context: input 5 tfpt 3 e8 audit bootstrap
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E8.ST31D8.01
Original ledger classification: Exact [E] census/theorem/controls + honest kill ('free => regular' false) (Python-only per GATE.WOLFRAM.02).
Original claim text
The ST31 degree-8 round + the unified d/4-theorem (one module, 30 checks, ~10 s; discovery probe st31_degree8_probe.py 30/30; exact Fraction/Z[i] arithmetic). ORDER-8 LANDSCAPE [E]: 10560 order-8 elements, 4800 with free census {8:30}, only 480 with x^2 central; the regular witness has |C| = 192 (Springer), chi_v = (x^2 + i)^2 EXACT (all eigenvalues primitive zeta8, doubled), v^2 a GENERATOR of mu4; HONEST KILL: 'free => regular' is FALSE at d = 8 (4320 free elements with x^2 noncentral; witness chi = Phi_8, |C| = 32) -- and already at order 4 (510 noncentral free 4-elements, none regular). THE d/4-THEOREM [E, central]: for every degree d in {8, 12, 20, 24} the Springer-regular d-clocks satisfy x^{d/4} = +-i Id, a GENERATOR of mu4, never +-1 (one formula: all exponents of G31 are 3 mod 4); regular sets = 2 Galois classes each with sign split J vs J^3; THE CONVERSE is exact at d in {8, 20, 24} (extras 0) with the UNIQUE exception at d = 12: exactly 1280 extra elements = the classes of the COMPILER CLOCK c and its Galois twin -- chi_c = (x - i)^2 (x^2 + ix - 1) is half the regular-12 Springer block plus a doubled i, all eigenvalues satisfy lambda^3 = -i (exactly why c^3 = J^3 is central): the compiler-clock classes are the ONLY non-regular clocks in all of G31 whose d/4-th power generates the center. CONTROLS [E]: full order census (max 24); free orders = the divisors > 1 of the degrees; degrees = free orders divisible by 4 above 4 (honest: NOT the maximal free orders). No marker moves.
Original classification detail
D4-THEOREM: every Springer-regular d-clock has x^{d/4} = +-i Id (generator of mu4; exponents 3 mod 4); converse exact at d in {8, 20, 24}, unique exception at d = 12 = the compiler-clock class (chi_c = (x-i)^2(x^2+ix-1), c^3 = J^3); order-8 landscape complete; 'free => regular' killed; no marker moves.
Dependencies as recorded
E8.ST31.01; E8.ST31DEG.01; E8.ZIQUOTIENT.01; v629; v633; v634; v647
Open the original ledger source, row 657Location recorded in the ledger
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- E8 audit and bootstrap
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- Front matter / supportE8 audit and bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
Navigation through a shared verification script
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- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-10-18: Project separation and retained physics historyResearch changelog; TFPT — Introduction and reading guide
Registered verification modules
- v654_st31_degree8.pyRegistered verification module · core
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E8.ST31D8.01: die ST31-Grad-8-Runde + der vereinheitlichte d/4-SATZ — **JEDE Springer-reguläre d-Uhr von G31 erfüllt x^{d/4} = ±i·Id, einen ERZEUGER von μ₄ (nie ±1; eine Formel aus der Exponenten-Systematik: alle Exponenten ≡ 3 mod 4); DIE UMKEHRUNG IST EXAKT BEI d ∈ {8, 20, 24}, UND DIE EINZIGE AUSNAHME BEI d = 12 IST DIE COMPILER-UHR-KLASSE** (1280 Extra-Elemente = Klasse von c und Galois-Zwilling; χ_c = (x−i)²(x²+ix−1) = die HALBE reguläre 12er-Springer-Box, alle Eigenwerte λ³ = −i — exakt darum ist c³ = J³ zentral: die Compiler-Uhr-Klassen sind die EINZIGEN nicht-regulären Uhren in ganz G31, deren d/4-Potenz das Zentrum erzeugt); **Ordnung-8-Landschaft komplett (10560 Elemente, 480 regulär = 2 Galois-Klassen, χ = (x²+i)² exakt) und der ehrliche KILL: „frei ⇒ regulär“ ist FALSCH** (4320 freie 8er nichtzentral; schon bei Ordnung 4: 510, keins regulär); Grade = freie Ordnungen mit 4 | d, d > 4 (30 checks, ~10 s)
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