The Gaussian Code Bridge: E₈ over ℤ[i], the Extended Hamming Code, and a Four-Bit Information Layer

Hamann, Stefan; Rizzo, Alessandro

Note N1Working note

The Gaussian Code Bridge: E₈ over ℤ[i], the Extended Hamming Code, and a Four-Bit Information Layer

Construction A, the μ₄ complex structure, and the code that builds the lattice returning as its message space — exact algebra, machine-certified

Build the E₈ lattice by Construction A over the extended Hamming code [8,4,4], placed equivariantly with respect to four fixed coordinate pairs, and let J be the complex structure that rotates each pair (J² = −1, μ₄ = ⟨J⟩). Then L becomes a unimodular Hermitian ℤ[i]-lattice of rank 4, and reduction modulo the ramified Gaussian prime 1+i produces a canonical four-bit quotient L/(1+i)L ≅ F₂⁴. The note proves: the 240 roots avoid the zero class — by the two-line norm argument |(1+J)x|² = 2|x|² against the doubly-even minimum 4 — and distribute exactly 15×16 over the fifteen nonzero classes, each class a union of exactly 4 of the 60 Gaussian root lines. On this quotient the pair 3-cycle σ acts as a family 3-cycle with a fixed anchor bit — the information-bit action of the Reed–Muller code RM(1,3) — with residual identification gauge of order exactly 18, and the sixteen coordinate roots ±2eᵢ form precisely one class, the σ-fixed label F₁+F₂+F₃. The classes cut across the Construction-A codeword fibers: the code that builds the lattice returns, after Gaussian reduction, as its message space. A quartic companion theorem types the invariant-theoretic shadow: the restrictions of the Weyl-invariant root power sums F₈, F₁₂, F₂₀, F₂₄ to ker(J−i) are a system of basic invariants of the complex reflection group G₃₁ (Chevalley), while the vanishing of the complementary degrees {2,14,18,30} is proved but honestly typed as the trivial μ₄-orbit factor, with no G₃₁ content. Every statement is certified by two exact-arithmetic probes (26/26 and 22/22 checks, no floats) and 65 kernel-checked Lean 4 theorems, with must-fail controls. No claim beyond the stated algebra is made.

Inputs

›The extended Hamming code [8,4,4], the four μ₄ coordinate pairs, and the Construction-A lattice L = A(C*) ⊂ ℤ⁸ — pure exact algebra, no physical anchor.

Contribution

›The equivariant placement census: of the 30 placements of the [8,4,4] code in F₂⁸, exactly two are invariant under both the in-pair swap π_J and the pair 3-cycle π_σ (machine census, probe check I0.1).
›The four-bit quotient: L/(1+i)L ≅ F₂⁴ with the 240 roots avoiding the zero class and distributing exactly 15×16 over the fifteen nonzero classes (each class = 4 of the 60 Gaussian root lines).
›The information layer: σ acts on the quotient as the information-bit action of RM(1,3) — a family 3-cycle with a fixed anchor bit, residual identification gauge of order exactly 18; the sixteen coordinate roots form the one σ-fixed class F₁+F₂+F₃.
›The quartic companion: F₈, F₁₂, F₂₀, F₂₄ restricted to ker(J−i) are basic invariants of G₃₁; the vanishing of degrees {2,14,18,30} is the trivial μ₄-orbit factor 1+(−i)^d+(−1)^d+i^d = 0 — proved AND honestly typed as carrying no G₃₁ content.
›Machine certification: gaussian_code_bridge_probe.py (26/26) and quartic_half_probe.py (22/22, exact arithmetic, no floats), promoted as v689/v690, plus TfptCarrier/GaussianCodeBridge.lean and TfptCarrier/QuarticHalf.lean (44+21 = 65 kernel-checked theorems, no sorry, no native_decide).

Not claimed here

›No claim beyond the stated algebra is made: this is an exact lattice/coding-theory statement about E₈ over ℤ[i], not a new physical readout, and it moves no status marker.

Falsification surface

›Must-fail controls are part of the result: non-equivariant placements, non-integral complex structures, and ℤ[i]⁴ each kill or trivialize the structure exactly as demanded; any of the 65 Lean theorems failing to kernel-check falsifies the note.

Download Note N1 View PDF

Highlights

Probe checks26/26 + 22/22Exact arithmetic, no floats — promoted as v689/v690

Lean theorems65GaussianCodeBridge.lean + QuarticHalf.lean, kernel-checked, no sorry

Root distribution15 × 16The 240 roots over the fifteen nonzero classes of F₂⁴

Message spaceRM(1,3)The code that builds the lattice returns as its message space

Equivariant placements2 of 30Machine census: exactly two placements invariant under π_J and π_σ

Key formulas

The four-bit quotient
$L/(1+i)L \cong \mathbb{F}_2^4$
Reduction of the Hermitian ℤ[i]-E₈ at the ramified prime 1+i. [E]
Roots avoid zero
$|(1+J)x|^2 = 2|x|^2 \ \Rightarrow\ 240 = 15 \times 16$
Two-line norm argument against the doubly-even minimum 4. [E]
Basic invariants
$\{F_8, F_{12}, F_{20}, F_{24}\} \ \text{basic for } G_{31}$
Chevalley on the holomorphic eigenspace; degrees {2,14,18,30} vanish trivially (μ₄-orbit factor). [E]

The three objects

The code: the extended Hamming code [8,4,4] — the unique doubly-even self-dual binary code of length 8, permutation equivalent to RM(1,3) — admits 8!/1344 = 30 placements in F₂⁸, of which exactly two are invariant under both the in-pair swap π_J = (01)(23)(45)(67) and the pair 3-cycle π_σ. The lattice: L = A(C*) = {x ∈ ℤ⁸ : x mod 2 ∈ C*} with [ℤ⁸:L] = 16 and minimum 4 (doubly even), attained by exactly 240 vectors — E₈. The complex structure: J rotates each pair, J² = −1, making L a unimodular Hermitian ℤ[i]-lattice of rank 4.

L = A(C^*) = \{x \in \mathbb{Z}^8 : x \bmod 2 \in C^*\}, \qquad [\mathbb{Z}^8 : L] = 2^4 = 16

\langle x,x\rangle \equiv \mathrm{wt}(c) \equiv 0 \pmod 4

The four-bit quotient

Reduction modulo the ramified Gaussian prime 1+i gives L/(1+i)L ≅ F₂⁴. The 240 roots avoid the zero class by the two-line norm argument |(1+J)x|² = 2|x|² against the doubly-even minimum 4, and distribute exactly 15×16 over the fifteen nonzero classes — each class a union of exactly 4 of the 60 Gaussian root lines.

L/(1+i)L \cong \mathbb{F}_2^4, \qquad |(1+J)x|^2 = 2|x|^2

240 = 15 \times 16, \qquad 60 = 240/4 \ \text{Gaussian root lines}

The information layer

On the quotient the pair 3-cycle σ (order 3, commuting with J) acts as a family 3-cycle with a fixed anchor bit — the information-bit action of the Reed–Muller code RM(1,3) — with residual identification gauge of order exactly 18. The sixteen coordinate roots ±2eᵢ form precisely one class, the σ-fixed label F₁+F₂+F₃. The classes cut across the Construction-A codeword fibers: the code that builds the lattice returns, after Gaussian reduction, as its message space.

The quartic companion — G₃₁ and the honest typing

The restrictions of the Weyl-invariant root power sums F₈, F₁₂, F₂₀, F₂₄ to the holomorphic eigenspace ker(J−i) are algebraically independent invariants of the complex reflection group G₃₁ and hence, by Chevalley's theorem, a system of basic invariants. The vanishing of the complementary degrees {2,14,18,30} is proved but honestly typed as the trivial μ₄-orbit factor 1+(−i)^d+(−1)^d+i^d = 0 (d ≢ 0 mod 4) — no G₃₁ content.

\{F_8, F_{12}, F_{20}, F_{24}\}\big|_{\ker(J-i)} \ \text{basic invariants of } G_{31}

1 + (-i)^d + (-1)^d + i^d = 0 \quad (d \not\equiv 0 \bmod 4)

Machine verification

Two exact-arithmetic discovery probes (gaussian_code_bridge_probe.py, 26/26 checks; quartic_half_probe.py, 22/22 checks — no floats anywhere) are promoted to the permanent suite as v689/v690, and the Lean 4 modules GaussianCodeBridge.lean and QuarticHalf.lean carry 65 kernel-checked theorems (no sorry, no native_decide). Must-fail controls — non-equivariant placements, non-integral complex structures, ℤ[i]⁴ — kill or trivialize the structure exactly as demanded.

The Gaussian Code Bridge: E₈ over ℤ[i], the Extended Hamming Code, and a Four-Bit Information Layer

Key formulas

The three objects

The four-bit quotient

The information layer

The quartic companion — G₃₁ and the honest typing

Machine verification

Key formulas at a glance

Cite this document