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Thermodynamics In The Quantum World Under A Signed Ledger: A −+− / +−+ Contract With Zeros-Stay-Zeros

William Richardson

Published Jan 18, 2026• 0 citations
Publisher

Abstract

This manuscript reframes a coherence-based “work vs heat” split for coherently driven quantum systems (cavity/laser/atom setting) as a signed-ledger contract that is replayable and falsifiable: each trial is written into {-1,0,+1} with explicit alternation backbones (-+- and +-+) enforced by Strong-to-Weak, missingness handled by STLF triad filtering under a strict zeros-stay-zeros rule, and saturation controlled by SoZ to prevent late-time null tails from being misread as reversibility or “pure work.” The note adds a portability layer via odd-null and sign-flip pairing, supplies a 4S3 leave-one-out control overlay to prevent threshold travel and hidden knob chasing, and specifies minimal token fields (contract_id, corridor_hash, window_hash, seed_hash) so independent reruns can reproduce the same acceptance decisions and falsifiers (flip-only, disable-L, withhold-node, no-test-band).

DOI
10.5281/zenodo.18293905

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This manuscript reframes a coherence-based “work vs heat” split for coherently driven quantum systems (cavity/laser/atom setting) as a signed-ledger contract that is replayable and falsifiable: each trial is written into {-1,0,+1} with explicit alternation backbones (-+- and +-+) enforced by Strong-to-Weak, missingness handled by STLF triad filtering under a strict zeros-stay-zeros rule, and saturation controlled by SoZ to prevent late-time null tails from being misread as reversibility or “pure work.” The note adds a portability layer via odd-null and sign-flip pairing, supplies a 4S3 leave-one-out control overlay to prevent threshold travel and hidden knob chasing, and specifies minimal token fields (contract_id, corridor_hash, window_hash, seed_hash) so independent reruns can reproduce the same acceptance decisions and falsifiers (flip-only, disable-L, withhold-node, no-test-band).
A theoretical and computational framework separating accepted protocol support, record-quality diagnostics, replicated entropy moments and implementation-level bookkeeping. A bounded positive effect induces a completed quotient Hilbert space; integer density-matrix moments are calculated through permutations on identical product copies; signed entropy increments telescope along a fixed reduction. Version 3.2 clarifies support versus distinguishability, finite-dimensional assumptions, higher-order observables, finite-copy uncertainty, numerical rank, logarithm units, additive ledgers and the limits of return diagnostics. A failed chosen inverse does not alone establish irreversible dynamics, and zero entropy residual does not establish state or apparatus return. The companion provides new reproducible central-spin and 500-state SWAP checks, an explicitly defined dephasing control, the recovered historical ledger script and its rerun, editable equations and manuscript sources. Missing historical central-spin and entropy-fit generators are disclosed. The associated Finite Record Protocol Calculator v0.3.0 supplies instrument/record diagnostics, Python and JavaScript parity checks, strict validation and a reduced-state entropy-path interface. This is the first deposit of this manuscript series, superseding working drafts v1.0 through v3.1. All examples are analytic or simulated; no laboratory measurements, empirical anchor, independent Born-rule derivation, emergent geometry or thermodynamic cost inferred from the accepted effect alone are claimed. The manuscript uses CC BY 4.0. Executable software and source code in the companion retain separate rights; see LICENSE_SCOPE.txt and software_dependency/RIGHTS.txt.
QFE-1 is a falsifiable quantum-thermodynamics research proposal for testing whether an explicit quantum battery can gain useful non-equilibrium free energy from a single equilibrium thermal reservoir while every declared auxiliary resource, including control, memory, coherence, timing and correlations — is returned to its reference condition and externally supplied work is independently metered. Version 2.0 adds a formal weak-coupling free-energy bound, a quantitative superconducting-circuit reference regime, a thermal master-equation baseline, an analytical null calculation showing energy gain without work gain, explicit false-positive controls, strong-coupling safeguards, a staged experimental programme, and pre-registered falsification criteria. No experimental violation of thermodynamics or discovery of perpetual motion is claimed.
Open quantum systems admit two natural definitions of heat: the standard formulation δQ_std = Tr[H dρ] and the entropy-basis formulation δQ_ent = Σ_k dλ_k⟨λ_k|H|λ_k⟩. Their difference W* = δQ_std − δQ_ent is a coherence work term whose thermodynamic status is the subject of this paper. We prove three results. (I) Theorem C (irreducibility): there is no function f: ℝ → ℝ such that δQ_ent + f(W*) satisfies the Clausius equality T dS_vN for all density matrices, proved via a Jacobian rank argument with det J = 3(γ+γ_+)²(15+16 ln 2)/320 ≠ 0 for the complete L₋+L₊ Lindblad system. W* is therefore an irreducible thermodynamic quantity. (II) Theorem N (secular-free second law): the entropy production σ_new = −Tr[D[ρ](log ρ − log ρ_ss)] ≥ 0 requires only complete positivity, not the secular approximation. A concrete example demonstrates σ_std = −0.037 < 0 while σ_new = +0.083 > 0 in the non-secular regime. (III) Theorem H (characterization): W*(D[ρ]) = 0 for all ρ if and only if the Bloch transformation satisfies A_xx=A_yy=A_zz, all cross-axis entries zero, A_xy=−A_yx, and b=0 — equivalent to T₁=T₂ in NMR/quantum-optics language, strictly stronger than the secular condition (T₂ ≤ 2T₁). The three-way identity σ_new = σ_ent − W*/T + Tr[D[ρ](log ρ_ss + βH)], valid for any β, reveals W*/T = σ_ent − σ_std as an experimentally accessible quantity.
This article introduces Richardson Single-Trajectory Local Filtering (STLF) as a ledger-first upgrade to standard single-copy local filtering (ScLF) for noisy entanglement tests and other platforms where drift, missingness, and low-yield “wins” can silently distort conclusions. The central change is bookkeeping: the experiment is treated as one sequenced trajectory, every attempt is assigned a global index, and each of three partial channels writes a symbol in at that index. Here are signed hits and is a literal no-hit/no-decision outcome that must remain visible; the rule is strict: zeros stay zeros (no smoothing, skipping, or post-hoc repair). The second change is architectural: STLF replaces the usual two-partial approach with a triad of partials so that outliers, drift, and “fake dominance” (direction rises only because yield collapses into zeros) can be detected and charged rather than hidden. Direction and yield are tracked separately per partial and combined only by an explicit, replayable scoring rule, so a channel cannot claim improvement by going silent. The protocol also specifies falsifiable requirements: the ledger must be replayable, knob adjustments must be declared functions of the written ledger, and the third partial must remain an independent witness rather than a deterministic echo of the other two. STLF is framed to map cleanly into corridor-style containment and token workflows, and it is explicitly designed to work alongside related corridor tools (ternary knob certification and signed-cycle/Strong-to-Weak tagging) by enforcing the same “zeros as first-class outcomes” accounting discipline.
A standalone transition from Hilbert Subspace to TWA and the Operational TWA+ is presented as a standards-facing lineage bridge that separates non-operational oracle ideas from an executable corridor contract: (i) mathematical Hilbert supplies full-state geometry and the classic pre-operational gates (“is this entanglement” and “in the admissible region”); (ii) TWA-style truncation makes the oracle callable at scale but introduces a boundary/drift attack surface (“shortcut acceptance”); (iii) device-independent subspace targets (including [[5,1,3]]) motivate acceptance structure but still require an operational decision program; and (iv) an operational corridor layer freezes acceptance lines, enforces sign/null/ledger separation with zeros-stay-zeros and odd/even discipline, and defines tokens-only replay so repeated window splits and leave-one-out audits remain stable; finally, (v) TWA+ is framed as explicit payback that recovers truncation costs without relaxing the contract by enforcing preregistered z-tension headroom (Tz) and forced flip/unflip parity auditing (ZFlip), then tightening the story using later corridor tools (oscillation fingerprints, disable-L stress, Sum-of-Zeros at infinity, Centered-X outcomes, and 4S3 closure) so the method stays falsifier-first, replayable, and usable even when source work does not publish a full usable number set.