Artian's Entropy and Second-Law Reference Theorem: Completed-Event Ledger Growth, Anchored Modular Charge, Clausius Production, Surface Entropy, and Page-Capacity Consequences

Abstract

Where irreversibility enters a finite quantum record The local and global statements usually grouped under "the Second Law" are different mathematical objects. This paper keeps them separate and then shows how they combine. At one address, the anchored modular charge is \[ Q_w=2\pi D(\rho_w\Vert\omega_w), \] and every declared CPTP access map produces the nonnegative loss \[ \Sigma_{{\rm acc},w} = k_B\left[ D(\rho_w\Vert\omega_w) -D(\Phi_w\rho_w\Vert\Phi_w\omega_w) \right] \ge 0. \] The source ledger supplies a second theorem. One funded A2 endurance spend is followed by the A3 packet of twenty-four repair fronts. The signed incidence of every closed event is therefore \[ \boxed{ \Delta N_{\rm act} =(-1+24)n_{12}+R_{12} =23n_{12}+R_{12} \ge0 }. \] Completed source records and access-relative modular loss occupy different coordinates of one positive-cone ledger. Both are normalized by the same A7 completed bundle and valued by the same additive entropy unit. The resulting master law is \[ \Delta\mathbf S_{\rm QTT} = \begin{pmatrix} k_B\Delta N_{\rm rec}\\ \Sigma_{\rm acc} \end{pmatrix} \in\mathbb R_{\ge0}^{2}, \] and, in the constructor class with no additional relative weight, \[ \boxed{ \Sigma_{\rm QTT}[T_1,T_2] = k_B\Delta N_{\rm rec} +k_B\sum_e\left[ D(\rho_e\Vert\omega_e) -D(\Phi_e\rho_e\Vert\Phi_e\omega_e) \right] \ge0 }. \] The same finite address budget fixes the horizon surface coefficient without using Hawking radiation as a constructor: \[ Q_{\Sigma}=8\pi\ell_A^2, \qquad Q_{\rm bundle}=2\pi, \qquad \boxed{ S_H=k_B\eta_H\frac{A}{4\ell_A^2} \le k_B\frac{A}{4\ell_A^2} }. \] For an independently declared unitary transfer, the paper prints the required fine-grained capacity bridge and obtains the Page-capacity envelope \[ \boxed{ S_{\rm out}^{\rm fine}(T) \le \min\{S_{\rm transferred}(T),S_{\rm remaining}(T)\} }. \] Without such a transfer channel, no evaporation trajectory is assumed. The paper also gives a visibility-based decoherence map, a finite local modular bound, the Clausius and fluctuation-theorem reductions with their premises, and no-retune transport and blackbody consistency rows. Version: 2.0 Concept DOI: 10.5281/zenodo.20045306 Author: Ali Attar Website: quantumtraction.org Main book: Quantum Traction Theory: Main Book v10.01 Scientific status: GREEN: SIGMA-A2-A3-FINITE-INCIDENCE-CLOSED GREEN: SIGMA-A7-COMPLETED-RECORD-PERSISTENCE-CLOSED GREEN: SIGMA-LOCAL-MODULAR-PRODUCTION-CLOSED GREEN: SIGMA-QTT-TWO-TERM-SECOND-LAW-CLOSED GREEN: SIGMA-HORIZON-SURFACE-COEFFICIENT-CLOSED CONDITIONAL: SIGMA-PAGE-CAPACITY-ENVELOPE STANDARD RECOVERY: THERMAL-TRANSPORT AND BLACKBODY ROWS Related public anchors: Artian Boltzmann weighting Information-loss and horizon source theorem Einstein-Hilbert coefficient theorem Derivation Atlas QTT Lexicon