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TOMOGRAPHIC BRIDGE CLOSURE · 2026-08-09

The Kartekeya Intertwining Bridge Theorem — canonical monograph

The typed bridge is now exact when a structure-preserving map is supplied. For unitary $U$,

$$ \rho_D=U\rho_EU^\dagger,\qquad M_D=UM_EU^\dagger,\qquad P_D=UP_EU^\dagger. $$

Born probabilities and kernel occupancy are preserved:

$$ \operatorname{Tr}(\rho_DM_D)=\operatorname{Tr}(\rho_EM_E),\qquad \operatorname{Tr}(P_D\rho_D)=\operatorname{Tr}(P_E\rho_E). $$

For group-theoretic tomography, the complete measurement frame, including multiplicity labels for repeated irreps, must be transported or independently declared. For independently constructed graph or reconstruction operators, the gate remains the measured intertwining residual $\delta_{\mathrm{int}}=\|G_DU-UK^2\|_F/\|K^2\|_F$.

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Tomography boundary update · KT-E47-2026-07-20

The E47 projector is certified, but the visual sinograms, radial profiles, isosurfaces, and tomographic reconstructions in Kartekeya plates are not machine data unless generated from an explicit forward operator, measurement model, reconstruction algorithm, and dataset. They remain design visualizations or H-class models pending those artifacts.

Certified algebraic core

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Related visual audit

The cyclic-root, persistence-classifier, and graph-Laplacian plates are reviewed in Computational Visual Validation Atlas — Plate Audit II. Their finite computations are preserved, while any identification with canonical $E_{47}$ remains open.

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Paired historical branch: Babylonian Mathematics — Sexagesimal Computation, Square Roots, and the Newton–Heron Lineage records the separate scalar root-finding lineage and explains why it is adjacent to, but not identical with, spectral projection.

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🌀 Canonical research boundary

Spectral tomography is treated here as an inverse-problem framework: observations are mapped through declared forward operators into a reconstructed latent field, then analyzed by graph, spectral, geometric, and projection methods.

The exact $E_{47}$ projector is a finite-dimensional algebraic object. It may be tested as a feature selector, invariant monitor, regularizer, or reduced-order controller. It does not by itself create physical tomographic data, define a semantic graph, or prove that a reconstructed field is quantum-mechanical.

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Executive finding

The accessible Drive corpus contains a substantial mathematical scaffold for a spectral-tomography program:

• exact finite-dimensional spectral decomposition on $V_2^{\otimes3}$;
• a rank-47 orthogonal projector and positive-semidefinite contraction dynamics;
• Hamiltonian, Lagrangian, variational, geometric, and Bohmian restrictions to the protected sector;
• graph-Laplacian and curvature language;
• NumPy, SciPy, SymPy, and QuTiP validation suites;
• visualization plates proposing sinograms, multi-domain projections, graph fields, isosurfaces, spectral gaps, and reconstruction metrics.

The most defensible compression is

$$ \boxed{ \text{observations} \rightarrow \text{forward operators} \rightarrow \text{latent reconstruction} \rightarrow \text{graph discretization} \rightarrow \text{spectral filtering} \rightarrow \text{validation} } $$

The exact algebra and projector dynamics are already reproducible. The displayed tomographic benchmarks remain a validation program until their raw data, graph construction, reconstruction code, manifests, and archived outputs are attached.

1. Exact algebraic core

Let $V_2$ denote the five-dimensional spin-2 irreducible representation of $SU(2)$ and define