<aside> 🧪

CANONICAL CONSOLIDATED INTAKE · 2026-08-19

Drive authority: Revised E47 Multi-Branch Formalism — Consolidated Python / NumPy / SciPy / SymPy Validation.

</aside>

Spectral → spatial → computational chain

$$ V_2^{\otimes3}\to K^2\to E_{47}\to D=L_{\mathrm{FCC}}\otimes I+I\otimes K^2\to\ker D. $$

The combined carrier has dimension $125\times125=15625$ and

$$ \dim\ker D=47. $$

The FCC spectral gap is

$$ \lambda_2(L_{\mathrm{FCC}})=\frac{15-3\sqrt5}{2}\approx4.14589803375, $$

so spatial homogenization controls the asymptotic rate.

14-qubit embedding correction

$$ 15625\hookrightarrow2^{14}=16384, $$

with 759 padded modes. The earlier $t=0.01$ high-fidelity claim is superseded: the seeded conditional kernel fidelity is about $0.009388$. The same filter reaches $1-F<10^{-14}$ at approximately $t=4.151$ and machine-level kernel fidelity by $t=5$.

Repaired Hodge bridge

The original triangle used a missing edge and gave $\|\partial_1\partial_2\|_2=\sqrt2$. With the repaired edge set,

$$ \partial_1\partial_2=0,\qquad \beta_1=1, $$

and $T_\epsilon=P_{47}W_\epsilon\Psi_H$ transports the harmonic mode into $E_{47}$ at machine precision.

Persistence control

The 8-agent ring has $\beta_1=1$, $\Delta t_{\mathrm{safe}}=0.175$, $v^*=0.07$, and $\dim E_{\mathrm{swarm}}=9$. The current geometry does not activate the collision-safety rows, so that part remains unexercised rather than failed.

Correction ledger