An exactly solvable model of the arrow of time.
Left (Paper I) — a single ring holds both chirality senses at once, spied on by its recording witnesses. Right (Paper II) — two rings sharing witnesses: what the shared records write links the two histories.
A three-site chiral ring — a time-reversal-odd pointer doublet — monitored by a
bath of recording qubits. The dynamics is pure dephasing and conserves the ring
chirality, so the whole model reduces to a two-branch (or, for several rings, a
2^M-branch) structure with closed forms for every observable: coherences,
record distinguishability, entropy production, redundancy, concurrence.
Nothing here is a fit or an approximation. Every analytic claim is checked against brute-force state-vector evolution and against an independently recoded verification suite, at machine precision.
This repository accompanies two papers.
-
Paper I — Fixation of a time-reversal-odd observable by decoherence: record distinguishability as the local arrow of time in an exactly solvable model (manuscript under arXiv submission). Single ring. Code:
src/· verification:src/verify_note.py· figures:figures/paper1/· notes:notes/paper1/. -
Paper II — Collective irreversibility and arrow competition in shared quantum environments (in preparation). Multi-ring extension. Code:
src/multi/· verification:src/multi/verify_multi.py· figures:figures/paper2/· notes:notes/paper2/.
Paper II builds on Paper I: src/multi/ imports the Paper-I closed forms and
its unit tests confirm exact reduction to Paper I at M = 1.
Paper I. Under a branch-relative definition of entropy production, the
irreversibility and the record content are the same quantity:
⟨σ⟩(t) = D(t) exactly (accumulated Kullback–Leibler record distinguishability).
Polarization, redundancy and entropy production co-transition on one master
curve; the pooled decoherence-to-polarization timescale ratio is
t_P / t_S = 0.953 ± 0.005, and the data collapse holds at RMS residual
0.005.
Paper II (verdicts, with the number that decides each):
| Conjecture | Verdict | Evidence |
|---|---|---|
| C1 total current objectifies, individuals don't (DFS) | confirmed | P_indiv = 0.50 (chance) vs P_total = 1.0 |
| C2 additivity deficit = a multi-information | reinterpreted | deficit is a theorem: Δ = I(s₁;s₂|ξ) + C_bwd, verified to 1e-15; extensive Δ = 0.688·N (R² = 0.99995); I/Δ → 1/8 proven at leading order |
| C3 entanglement lives off the records | confirmed | concurrence C(t) monotone in relative distinguishability; C ≡ 1 in the decoherence-free subspace |
| C4 detailed fluctuation theorem | confirmed | ln[p(σ)/p(−σ)] slope = 1.00 ± 0.02 |
| C5 arrow frustration under bath sharing | resolved | one arrow reverses at a disorder-stable threshold; at equal effort the F/M prescriptions are symmetric duals (f* ≈ 0.40) |
The Paper-I data collapse survives bath sharing (RMS 0.018) and breaks
only in the decoherence-free subspace (RMS 0.40) — the break is itself the
signature of where records fail to form.
Full verdicts and figures: notes/paper2/results_multi.md.
pip install numpy scipy matplotlib sympy
python src/verify_note.py # Paper I: re-derives every analytic claim (~1 min)
python src/multi/verify_multi.py # Paper II: independent from-scratch suiteBoth print ALL CHECKS: PASS. verify_note.py runs six exact checks
(time-reversal operator, forward/backward record laws, σ = 0 identity,
integral fluctuation theorem, Bayes posterior); verify_multi.py verifies the
deficit decomposition theorem, the 1/8 constant (symbolically, via sympy),
and the sign survey — none of it importing the production code.
Every figure regenerates from a fixed seed. Output lands in
figures/paper1/ and figures/paper2/ automatically.
Paper I (from the repo root):
| Figures | Command |
|---|---|
| Figs. 1–3 (partial-info plots, co-transition, histograms) | python src/run_baseline.py |
| Figs. 4–6 (collision protocol, finite-size scaling, collapse) | python src/run_next.py |
| Figs. 7–9 (variant B, hysteresis, ratchet) | python src/run_variantB.py |
| Error bars / disorder statistics (Fig. 5b, ~20 min) | python src/errorbars.py |
Paper II (from the repo root):
| Figures | Command |
|---|---|
| Scenarios (a)–(f): DFS, deficit, frustration, fluctuation theorem | python src/multi/run_scenarios.py all |
Phase diagram f*(t) (the paper's figure) |
python src/multi/run_phase_diagram.py |
| Exact deficit decomposition | python src/multi/analyze_c2.py |
Deficit scaling, collapse error bars, f* stability |
run_deficit_scaling.py, run_collapse_errors.py, run_fstar_stability.py |
src/ Paper I — model, exact closed forms, trajectory protocols
verify_note.py six machine-precision checks of the analytic claims
src/multi/ Paper II — multi-ring extension
model_multi.py closed forms (imports Paper I; M=1 reduction is a unit test)
brute_multi.py brute-force state-vector validation (N=8)
verify_multi.py independent from-scratch verification suite
figures/
paper1/ paper2/ generated figures, one folder per paper
notes/
paper1/ model spec, backward-process derivation, manuscripts
paper2/ multi-ring definitions + theorem, results & verdicts
Validation gates: exact reduction to Paper I at M = 1; brute-force agreement
at N = 8; exact additivity (Δ = 0) for disjoint baths — all to machine
precision.
@unpublished{lejri_chiralring_paper1,
author = {Lejri, Mostfa},
title = {Fixation of a time-reversal-odd observable by decoherence:
record distinguishability as the local arrow of time in an
exactly solvable model},
note = {Manuscript under arXiv submission},
year = {2026}
}
@unpublished{lejri_chiralring_paper2,
author = {Lejri, Mostfa},
title = {Collective irreversibility and arrow competition in shared
quantum environments},
note = {In preparation},
year = {2026}
}Mostfa Lejri — Independent researcher — ORCID 0000-0002-0283-6069.
Both papers made extensive use of Anthropic's Claude as a research assistant; every analytic claim is checked by an independently recoded verification suite. See the manuscripts' acknowledgments.
Released under the MIT License.