E47 Exact-to-Machine Precision — spectral core, mint lock. Mini AI Labs visual system.

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VISUAL SYSTEM · MINI AI LABS

Near-black laboratory. Cyan live · bronze Λ-ghost · mint lock. No type on stills.

Record: CITY-LABS-VISUAL-SYSTEM-20260915.

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Reproducibility audit · 2026-09-14

The canonical Drive folder currently contains the theorem sheet, registry crosswalk, and monograph, but no runnable validator, environment lock, machine-readable result certificate, or hash manifest. The exact E0 identities remain intact. E1 support comes from the separate repaired Proof Forge witness, independently rerun today at 120/120 PASS with rank(P)=47, gap 11663.999999999955, and ||Q||2=186624.00000000017. Witness SHA-256: df6ff23f6dc7ff07cc42f102a9c9411ca350050ec6efe29370dcc13114f62a92. JSON SHA-256: bd95e1de4af544c369797d57f94e2c8958c16d6bd398c86ed39132e5b6439010. The three-document folder is therefore not a self-contained standalone E1 package.

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Canonical synthesis: exact representation theory → spectral kernel → orthogonal projector → positive contraction generator → analytic convergence law → explicit tolerance contract → machine-scale residuals.

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Canonical identity

$$ 125 \to 47 \to P_{47} \to K^2 \to 11664:186624 \to \kappa_+=16 \to \varepsilon_=1/99144 \to \rho_=15/17 $$

Machine floor: εmach = 2⁻⁵² ≈ 2.220446049250313 × 10⁻¹⁶.

Core object

• V = V₂⊗³, dim V = 125.
• C = (J₁ + J₂ + J₃)², spec(C) = {0,2,6,12,20,30,42}.
• K = (C − 6I)(C − 30I).
• E₄₇ = ker K = E₆ ⊕ E₃₀, dim E₄₇ = 47.
• P₄₇ = P₆ + P₃₀.
• Ωc = 47/125 = 0.376.
• λmin⁺(K²) = 11664, λmax(K²) = 186624, κ₊ = 16.
• Γ = I − K²/99144, with complement factor ρ = 15/17.

Precision ladder

exact integer/rational arithmetic → exact representation-theoretic spectrum → exact spectral projector → analytic spectral gap and contraction bounds → 10⁻⁷–10⁻⁸ spectral classification → 10⁻¹⁰ runtime assertions → 10⁻¹² normalized certification contract → 10⁻¹⁴–10⁻¹⁵ observed structural residuals → 2.220446049250313 × 10⁻¹⁶ float64 floor.

Canonical Drive package