Artian's Casimir Boundary-Access Framework: Gauge-Invariant Plate Projectors, the Standard Coefficient, Three-2pi Non-Equivalence, and a Conditional Finite-Address Test

Abstract

How two conducting plates reshape the photon spectrum without retuning the photon The ideal parallel-plate Casimir coefficient is familiar. The sharper question is what mathematical operation a conducting boundary performs on the physical photon space. This paper constructs that operation after Gauss reduction. If \(\Gamma_B\) is the perfect-conductor trace, the admissible one-photon space and its plate operator are \[ \mathcal H_B=\ker\Gamma_B\cap\mathcal H_\gamma^{\rm phys},\qquad P_B^2=P_B=P_B^\dagger,\qquad M_B=\Gamma_s(P_B). \] The plate therefore changes the accessible photon domain while the source Maxwell law remains fixed. Explicit TE/TM counting gives two physical polarizations; the single TM \(n=0\) term is independent of plate separation after matched subtraction. The Abel--Plana cut then yields \[ \boxed{ \frac{E_{\rm Cas}}{A} =-\frac{\hbar c}{L^3} \left(\frac{\pi^2}{6}\right) \left[\int_0^\infty\frac{t^3\,\mathrm dt}{e^{2\pi t}-1}\right] (2_{\rm pol}) =-\frac{\pi^2\hbar c}{720L^3} } \] and \[ \boxed{\frac{F_{\rm Cas}}{A}=-\frac{\pi^2\hbar c}{240L^4}}. \] The repeated number \(2\pi\) is then placed under an operator-level audit. Abel--Plana summation, Bisognano--Wichmann modular flow, and A7 completed-address capacity use the same scalar normalization in different typed objects: \[ 2\pi_{\rm AP}=2\pi_{\rm BW}=2\pi_{\rm A7} \quad ot\Rightarrow\quad \mathcal O_{\rm AP}\cong\mathcal O_{\rm BW}\cong\mathcal O_{\rm A7}. \] A QTT bridge therefore requires an explicit reduced intertwiner: a quotient algebra, a unital star-homomorphism, state preservation, and modular-flow compatibility. Scalar coincidence alone is not that proof. The paper also derives a finite-address branch from the exact source stencil. For plates aligned with one native source rail, and only after declaring photon exposure and excluding an independent order-\(\ell_A^2\) boundary-local coefficient, the branch gives \[ \boxed{ \frac{E_{\rm Cas}^{\rm cond}}{A} =-\frac{\pi^2\hbar c}{720L^3} \left[1+\frac{\pi^2}{14}\left(\frac{\ell_A}{L}\right)^2 +O\!\left(\frac{\ell_A^4}{L^4}\right)\right] } \] with the corresponding force law \[ \boxed{ \frac{F_{\rm Cas}^{\rm cond}}{A} =-\frac{\pi^2\hbar c}{240L^4} \left[1+\frac{5\pi^2}{42}\left(\frac{\ell_A}{L}\right)^2 +O\!\left(\frac{\ell_A^4}{L^4}\right)\right]. } \] The standard coefficient is recovered without a fitted parameter. The finite-address coefficients are closed within the printed constructor class; their laboratory use remains conditional on a photon-exposure and orientation map. Real-material tests must additionally pass through Lifshitz response, temperature, geometry, patch-potential, calibration, and covariance rows before a QTT verdict is allowed. Scientific status: CLOSED: SIGMA-GAUGE-REDUCED-PEC-PLATE-PROJECTOR RECOVERED: SIGMA-CASIMIR-IDEAL-COEFFICIENT CLOSED: SIGMA-THREE-2PI-NON-EQUIVALENCE PENDING: SIGMA-CASIMIR-BW-A7-REDUCED-INTERTWINER CONDITIONAL THEOREM: SIGMA-FINITE-ADDRESS-CASIMIR-COEFFICIENT PENDING: SIGMA-CASIMIR-PHOTON-EXPOSURE-ORIENTATION-MAP Version: 2.0 Concept DOI: 10.5281/zenodo.20099862 Author: Ali Attar Website: quantumtraction.org Main book: Quantum Traction Theory: Main Book v10.01 Related public anchors: Maxwell source theorem Finite-address wave theorem A6 necessity and independence theorem A7U framework Casimir lexicon entry Access derivation map The reconstruction ZIP contains the LaTeX source, local style, independent arithmetic verifier, generated certificate, three-round adversarial review, visual audit, and SHA-256 manifest.