The Thermodynamic Cost of Knowing

The Thermodynamic Cost of Knowing: Observation as Irreversible Payment This preprint is a formal technical note establishing a rigorous, machine-verified bridge between Quantum Information Theory (Englert-Greenberger duality) and Classical Thermodynamics (Landauer's Principle). Abstract The UMST/DUMSTO Thermodynamic Gate has been extended to enforce physical quantum boundary conditions. Every mathematical proposition presented in this note is mechanically verified in Lean 4 with Mathlib. Across 53 core modules (58 .lean files), the underlying formalization contains 515 theorems, 33 lemmas, 6 explicit axioms, and 0 sorry gaps. We formally prove that every fraction of a bit of which-path information causes a proportional destruction of interference, and this destruction carries an exact Landauer thermodynamic cost. The Core Result: The Principle of Maximal Information Collapse The central theorem of this framework demonstrates that information extraction cannot be decoupled from thermodynamic dissipation. Specifically, the acquisition of which-path information strictly binds the residual coherence capacity of the quantum state, forcing a minimum Landauer erasure cost when the environment is reset. When an observer extracts which-path information from a quantum system, the residual coherence capacity is strictly bounded by: Residual Coherence = 1 − [ MI_extracted / (k_B T ln 2) ] ∈ [0, 1] This formally verifies that observation is an irreversible thermodynamic payment, anchoring the abstract axioms of quantum measurement (collapse) to the physical constraints of heat generation. Formally Proved Englert Complementarity: V² + I² ≤ 1 Which-Path Collapse: V → 0 after projection (Lüders channel) Cost-Coherence Identity: Q = k_B T ln(2) · (1 − Residual Capacity) Landauer Bound: Erasure cost matches thermodynamic heat dissipation Quantum Mutual Information: I(A:B) = S(A) + S(B) − S(AB) ≥ 0 General Residual Coherence: measurement-update coherence bound for arbitrary bases Erasure Channel: complete depolarization entropy bound Verification Stack & Source Availability While this document details the theoretical and mathematical framework, the complete proof source code is publicly available via the associated tytolabs GitHub Repository. Formal Proof: Lean 4 + Mathlib (53 modules, 515 theorems, 33 lemmas, 6 explicit axioms, 0 sorry). Property Testing: Haskell/QuickCheck (14 physical admissibility properties). Numerical Simulation: Python (stdlib + QuTiP) — 87 unit tests, 4 simulations. Cross-Language Specs: Coq/Rocq (9 modules, 0 Admitted) and Agda (11 modules) parity translations.