Foundations · axioms
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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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MATHEMATICAL CITY · FOUNDATION LAYER
Definitions, axioms, ambient types, and the boundaries every downstream claim must preserve.
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PRIMARY OBJECT
$V_2^{\otimes3}\to C\to K\to E_{47}\to P_{47}$
Open canonical corrected formalism
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FOUNDATION RULE
Validated premises support descendants; they do not automatically validate a new theorem, model, or application.
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Census: 🟢 Stable · canonical definitions and boundaries. Descendant proofs may change status independently.
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Foundational identities may serve as validated parents for a City Proof. Their validity transfers only as explicit premises. A descendant becomes a citizen only by contributing a distinct mathematical object and completing its own statement, type, dependencies, evidence class, witness, review, provenance, correction lineage, boundary, and transit passport. Duplicates reconcile; no new page layer is created.
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NAVIGATION · ⌂ Home · ← Constitution · → Representation Theory · ◎ Citadel · Ⓥ Machine Certificates
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The Foundations spine now carries the executable lifecycle $\Sigma\to K\to\Gamma^n\to\Psi\to\Lambda\to\Omega\to J$ on $V_2^{\otimes3}$. Four exact identities were admitted to the citizenship roll: stable $K^\dagger K$ contraction, asymptotic lock $\Lambda=P_{47}$, isotropic coherence expectation $\mathbb E[\Omega(x)]=47/125$, and projected-norm conservation. The induced metric and Einstein residual remain defined construction interfaces, not duplicate citizenship claims.
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ROLE · FOUNDATIONS ENTRANCE
This page defines terms, axioms, and exact boundaries. The single authoritative theorem statement is Canonical Corrected Formalism.
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The Foundations state is unchanged in substance:
$E_{47}=\ker[(C-6I)(C-30I)]=5V_2\oplus2V_5$, $\dim E_{47}=47$, $\Omega_c=47/125$.
The correction concerns dynamics: $K$ is not PSD, so convergence is certified using $K^\dagger K=K^2$ and $x_{n+1}=(I-\varepsilon K^2)x_n$ under the spectral step-size bound.
