Artian's Entropy and Second-Law Reference Theorem: Completed-Record Persistence, Record-State Recurrence Separation, Typed Capacity Geometry, Anchored Modular Charge, Clausius Production, Surface Entropy, and Page-Capacity Consequences

Abstract

Two entropy coordinates, one finite Second-Law cone This release separates the cumulative record written by completed source events from information lost through an observer's access map. The combined law is \[ \Sigma_{\rm QTT}[T_1,T_2] =k_B\big[N_{\rm rec}(T_2)-N_{\rm rec}(T_1)\big] +k_B\sum_e\left[ D(\rho_e\Vert\omega_e) -D(\Phi_e\rho_e\Vert\Phi_e\omega_e) \right]\ge0. \] The source coefficient is fixed before entropy is evaluated. The orientation-preserving signed-permutation orbit has \[ |\mathcal O_3^+|=3!\,2^2=24, \] and its typed capacity sum is \[ 24\frac{\pi}{6}\ell_A^3 =\ell_A^3\int_{S^2}d\Omega =4\pi\ell_A^3. \] One A2 spend and 24 A3 response fronts therefore give the exact net incidence coefficient \(24-1=23\). Version 3.0 adds a record-state recurrence theorem. If the projected microstate returns but a legal record has been written, \[ x(T_2)=x(T_1), \qquad N_{\rm rec}(T_2)>N_{\rm rec}(T_1), \] then the full record-bearing states are unequal. The result separates projected recurrence from record erasure without altering the standard Poincare theorem on a fixed finite-measure full state space. The companion multiplicity bound is also made explicit: \[ 0\le\ln W_n\le\sum_e\ln d_e \le N_{\rm rec}\ln d_{\max}. \] Completed history, compatible multiplicity, and accessible information are therefore related but not interchangeable counts. Reader doorway: The Equation on the Tombstone Main Book: 10.5281/zenodo.17527179 Website: quantumtraction.org