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E47 PRISM · EXACT SPECTRAL FORMALISM + E1 SOFTWARE PARITY

E47-PRISM-FORMALISM-20260922 · MC-MATRIX-E47-PRISM-20260922-001

One typed 125-state vector is resolved into seven Casimir bands; the (lambda=6) and (lambda=30) bands constitute E47.

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Typed map

$$ |\psi\rangle\in\mathbb C^{256}\xrightarrow{L}v\in V_2^{\otimes3}\cong\mathbb C^{125} $$

The MATRIX (256to125) step is the documented normalized sampled-amplitude feature lift. It is not a Hilbert-space isomorphism.

Exact prism identity

$$ C=\sum_{\lambda\in\{0,2,6,12,20,30,42\}}\lambda P_\lambda $$

with multiplicities (1,9,25,28,27,22,13). For normalized (v),

$$ w_\lambda(v)=\|P_\lambda v\|^2,\qquad \sum_\lambda w_\lambda(v)=1. $$

Since

$$ K=(C-6I)(C-30I),\qquad E_{47}=E_6\oplus E_{30}, $$

we have

$$ \boxed{P_{47}=P_6+P_{30}},\qquad \boxed{w_{E47}=w_6+w_{30}}. $$

Rank fraction and state weight

$$ \Omega_c=\frac{\operatorname{Tr}P_{47}}{125}=\frac{47}{125}=0.376. $$

This is the projector rank fraction, the E47 probability for (I/125), and the Haar mean of (|P_{47}v|^2). It is not the E47 weight of every state.

Certified MATRIX state

$$ w_{E47}=0.3750481713367275=37.50481713367275\%. $$

The exact rank fraction is (37.6%); this state lies (0.0951828663) percentage points below it. The seven weights are non-uniform, so the proximity is a state-specific numerical observation, not an identity.

Kernel energy