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Original ledger classification: Exact identity/lattice-grade compression [Identity/Lattice]: all four rungs re-certified from scratch in exact arithmetic; unifies note_e8_gaussian_code/v689 (census, norm doubling), v803 (non-split jet), v783/v798/v819 (metaplectic zeta8, Clifford coset, RM-CSS phase bit); no marker move (Python-only per GATE.WOLFRAM.02)

Recorded statement: GAUSSIAN.RAMLADDER.01: the (1+i) ramification ladder -- the four documented roles of the ramified Gaussian prime pi2 = 1+i are ONE machine-checked object with four rungs, each re-certified from scratch, ONE module from one probe (33/33…

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Paper context: input 5 tfpt 3 e8 audit bootstrap

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GAUSSIAN.RAMLADDER.01

Original ledger classification: Exact identity/lattice-grade compression [Identity/Lattice]: all four rungs re-certified from scratch in exact arithmetic; unifies note_e8_gaussian_code/v689 (census, norm doubling), v803 (non-split jet), v783/v798/v819 (metaplectic zeta8, Clifford coset, RM-CSS phase bit); no marker move (Python-only per GATE.WOLFRAM.02)

Original claim text

GAUSSIAN.RAMLADDER.01: the (1+i) ramification ladder -- the four documented roles of the ramified Gaussian prime pi2 = 1+i are ONE machine-checked object with four rungs, each re-certified from scratch, ONE module from one probe (33/33 checks, verdict RAMIFICATION-LADDER-EXACT; discovery probe gaussian_ramification_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM, ~5 s). RUNG 1, NORM DOUBLING: N(pi2) = 2 and (pi2^2) = (2); on the Gaussian E8 lattice (the unique mu4/sigma-equivariant Construction-A placement, 2 of 30, rebuilt) (1+J)^T (1+J) = 2 I_8 EXACTLY; min L = 4 by exhaustive census, the zero class of L/(1+i)L carries NO root (0/240 directly), 240 = 15 x 16 over the nonzero classes (each a union of 4 of the 60 Gaussian lines), elementary divisors (1,1,1,1,2,2,2,2), index 16 = N(pi2)^4. RUNG 2, THE FOUR-BIT ADDRESS: L/(1+i)L ~= F2^4 with the deck J acting TRIVIALLY on all 240 roots. RUNG 3, THE NON-SPLIT JET: Z[i]/(2) ~= F2[eps]/(eps^2) (ring iso UNIQUE among all 24 bijections); W = L/2L FREE of rank 4 over the dual numbers (256/256 combinations distinct); the self-extension 0 -> V -> W -> V -> 0 exact with deck = JET UNIT 1 + eps, and of ALL 65536 linear sections EXACTLY 0 are deck-equivariant -- the no-canonical-origin obstruction IS non-splitness (v803); plus-type form census 136/120, Arf 0, the 120 root pairs ARE the q = 1 shell (240 = 15 x 8 x 2). RUNG 4, THE METAPLECTIC LIFT: zeta8 = (1+i)/sqrt2; (1+J)^2 = 2J; (S H)^3 = zeta8 I per binary factor and (T3 K3)^3 = i I = THE DECK (v798); exact Z[zeta8] BFS census: |C2/mu8| = 11520 = 16 x 720 with ZERO Galois-mixed classes, |C2| = 92160 = 2 x 46080, C2 = K u zeta8 K (K = U G31 U^-1, v783 chart theorem cited [E]); witness parities chi(H(x)I) = 1 (the RM-CSS zeta8 phase bit, v819 S4 -> v798 [E]), chi(H(x)H) = chi(CZ) = 0. Controls fire (split ring F2 x F2 admits 0 ring isos -- 2 must ramify; split self-extension admits 65536/65536 sections; scrambled nilpotent breaks ker = im). A compression: the four corpus roles (note_e8_gaussian_code Thm 1/v689, v689, v803, v783/v798/v819) are ONE ladder -- the same ramified prime read at (pi2), (pi2^2) and pi2/sqrt2. Exact integer/Fraction/sympy; no floats, no RNG. Python-only per GATE.WOLFRAM.02.

Original classification detail

RAMIFICATION-LADDER-EXACT: the four documented roles of pi2 = 1+i -- norm doubling / empty zero class, the 4-bit address, the non-split jet with deck = 1 + eps, the metaplectic zeta8 with the Clifford index-2 coset and the RM-CSS phase bit -- are ONE ladder (33/33 exact); defect-free controls (split ring, split extension, scrambled nilpotent) all fire

Dependencies as recorded

v689; v783; v798; v803; v819

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Registered verification modules

  • v833_gaussian_ramification_ladder.pyRegistered verification module · core
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    GAUSSIAN.RAMLADDER.01: die (1+i)-Verzweigungsleiter — die vier dokumentierten Rollen des verzweigten Gaußschen Primelements π₂ = 1+i sind EIN maschinell geprüftes Objekt mit vier Sprossen, jede von Grund auf re-zertifiziert — **RAMIFICATION-LADDER-EXACT (33/33): SPROSSE 1 (Norm-Verdopplung): N(π₂) = 2, (π₂²) = (2); auf dem Gaußschen E8-Gitter (1+J)ᵀ(1+J) = 2I₈ EXAKT; min L = 4 per Zensus, die Nullklasse von L/(1+i)L trägt KEINE Wurzel (0/240), 240 = 15 × 16, Elementarteiler (1,1,1,1,2,2,2,2), Index 16 = N(π₂)⁴; SPROSSE 2: L/(1+i)L ≅ F₂⁴ mit dem Deck J TRIVIAL wirkend — der 4-Bit-Adressraum; SPROSSE 3 (der nicht-spaltende Jet): Z[i]/(2) ≅ F₂[ε]/(ε²) (Ring-Iso EINDEUTIG unter allen 24 Bijektionen), W = L/2L frei vom Rang 4 über den dualen Zahlen, Deck = JET-EINHEIT 1 + ε, und von ALLEN 65536 linearen Schnitten sind EXAKT 0 deck-äquivariant — die Kein-kanonischer-Ursprung-Obstruktion IST Nicht-Spaltung (v803); Plus-Typ-Form 136/120, die 120 Wurzelpaare SIND die q = 1-Schale; SPROSSE 4 (der metaplektische Lift): ζ₈ = (1+i)/√2, (1+J)² = 2J, (SH)³ = ζ₈I, (T₃K₃)³ = iI = DAS DECK (v798); exakter Z[ζ₈]-BFS-Zensus: |C₂/μ₈| = 11520 = 16 × 720, NULL Galois-gemischte Klassen, C₂ = K ∪ ζ₈K (v783 [E]); Zeugen-Paritäten χ(H⊗I) = 1 (das RM-CSS-ζ₈-Phasenbit), χ(H⊗H) = χ(CZ) = 0. Kontrollen feuern (der SPALT-Ring F₂×F₂: 0 Isos; die spaltende Selbsterweiterung: 65536/65536 Schnitte). Exakt Integer/sympy; keine Floats, kein RNG.**

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