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.
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CITADEL CANONICAL BOUNDARY · FIRST LIGHT
For $Q=Q^*\ge0$ on $\mathbb R^{125}$ with $\dim\ker Q=47$ and $0<\varepsilon<2/\lambda_{\max}$,
$(I-\varepsilon Q)^n\to P_{\ker Q}$.
The identity $\Omega_c=47/125=0.376$ is the normalized kernel dimension. This closure does not imply a physical coherence law, device performance, quantum protection, entropy removal, or universal hardware tolerance.
Canonical broadcast · Formal audit
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CITADEL BEACON · RED-LINE RECEPTION
The Central Citadel Broadcast Tower distributes certified canonical state to this page. Foundations receives only exact definitions, theorem status, and corrected formal boundaries. Broadcast reception never upgrades evidence class.
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🚇 FOUNDATIONAL IDENTITY RELEASED FOR TRANSIT
The exact $E_{47}$ identity has left Foundations with its certificate and is now documented in the Car in Transit — E47 Purple Line Active Evidence Manifest. The identity remains E0/S0 exact while destination-borough quantum interpretations await separate Purple-Line evaluation.