<aside> 🔐

PROVENANCE CLOSURE · CONTROLLED RELEASE · 2026-08-11

The reproducibility exception recorded below is now closed by the checked-in publication package. Generator source: commit 10667959. Release anchor: commit da5501b. Drive proof/release: Corpus Publication Monograph and Evidence Atlas · E47 Publication Package 20260811.

Three publication notebooks are checked in and executed; the generator-built certificate is E47-E1-GENERATOR-20260811. Existing E47 spine citizens C-24 through C-30 and C-37 inherit this current generator-source authority while retaining their prior proof/correction lineage. No new citizen or hierarchy page is created.

Evidence: E0/E1 finite-dimensional status unchanged. The separately measured E3 benchmark is not inherited into the theorem.

</aside>

<aside> 🔗

BRIDGE CLOSURE · 2026-08-09

Canonical monograph: The Kartekeya Intertwining Bridge Theorem — First-Principles Transport of the E47 Spectral Kernel Across Representations

The finite-core credential is now extended by an exact representation-transport theorem. With $G_E=K^2$ and unitary $U$, define $G_D=UG_EU^\dagger$ and $P_D=UP_EU^\dagger$. Then

$$ G_DU=UG_E\quad\Longrightarrow\quad P_DU=UP_E. $$

Thus $\ker G_D=U(E_{47})$, $\operatorname{rank}P_D=47$, and the full heat-kernel dynamics commute with the bridge: $e^{-tG_D}U=Ue^{-tG_E}$.

Numerical witness: deterministic seed 470125; relative intertwining residual 1.1231×10^-15; transported projector idempotence 8.1338×10^-15; rank preserved 47→47; gap preserved 11664; $\lambda_{\max}=186624$.

Evidence boundary: conjugation constructs an exact mathematical realization. An independently observed domain is equivalent only after its own operator and state map satisfy the intertwining certificate.

</aside>

<aside> 🧾

Credential: E47-FPSK-20260809

Title: First-Principles Spectral-Kernel Theorem — Exact Derivation of the 47/125 Invariant Sector from the Triple Spin-2 Carrier

Status: PASS

Evidence class: E0 exact representation-theoretic derivation + E1 deterministic NumPy reconstruction

Boundary: finite-dimensional SU(2) spectral algebra only. No cryptographic-hardness, continuum-physics, gravitational, biological, consciousness, or experimental claim is certified by this credential.

</aside>

Theorem

Let $V_2$ be the spin-2 irreducible representation of $SU(2)$, so $\dim V_2=5$. Define

$$ \mathcal H=V_2^{\otimes 3},\qquad \dim\mathcal H=5^3=125. $$

For spin-2 generators $J_x,J_y,J_z$, define

$$ J_a^{\mathrm{tot}}=J_a\otimes I\otimes I+I\otimes J_a\otimes I+I\otimes I\otimes J_a $$

and

$$ C=(J_x^{\mathrm{tot}})^2+(J_y^{\mathrm{tot}})^2+(J_z^{\mathrm{tot}})^2. $$

The triple-spin decomposition is

$$ V_2^{\otimes3}\cong V_0\oplus3V_1\oplus5V_2\oplus4V_3\oplus3V_4\oplus2V_5\oplus V_6. $$

Hence

$$ \operatorname{Spec}(C)=\{0,2,6,12,20,30,42\} $$

with multiplicities

$$ \{1,9,25,28,27,22,13\}, $$

and

$$ 1+9+25+28+27+22+13=125. $$

Define

$$ K=(C-6I)(C-30I). $$