Quantum Work Network

A Rust reference implementation for numerically computing how much of the energy driven through a small quantum network arrives at a receiver (the load) as extractable work (ergotropy), and how that changes under phase noise. The project is developed as a single, incremental research and implementation history from Milestone 1 through Milestone 15A, via finite-chain (N=3..7) event-structure diagnostics, a descriptive decomposition of the load's coherent ergotropy (M12), a short-horizon counterfactual branch experiment (M13), coupling-control switching-work accounting (M14), and a single end-point ideal local extraction (M15A). After M15A, a supplementary comparison extended the C2 protocol to t=10 and compared it against a long-window C0 baseline on the common saved grid; it is not assigned a new Milestone number.

Project overview

The physical system is a chain of two-level sites (qubits) with one 3-level load attached at one end. One end site is driven by an external pulse to inject energy; the propagation of that energy along the chain and its accumulation in the load are tracked as the time evolution of a density matrix under a Lindblad master equation. The recurring question across all milestones is narrow and concrete: for the same situation, how does the locally extractable work at the load change when phase noise (dephasing) is present — measured directly, within a finite-size, finite-time scope.

Each milestone builds on the previous one without modifying earlier code, CSVs, or reports. Later stages generalize the chain length (N=3 -> 5 -> 7), separate two noise conventions (fixed-per-site vs fixed-total), and finally compare N=3 and N=7 at equal drive input energy and analyze the shape of the resulting work curves.

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Scope and limitations

What this repository provides are "values directly observed for this physical model, these initial conditions, this finite time, and this Rust implementation." In particular, this project does not:

• prove any quantum advantage or that quantum beats a classical method;
• establish universal scaling laws, exponential/power-law decay, thermodynamic limits, or N->infinity behavior;
• claim a general advantage of longer chains, or universal laws of noise position or protection strength;
• treat curve-transform fits or decay fits as proofs of a physical mechanism — these are descriptive shape diagnostics only;
• treat W_time_area as cumulative extracted work, or usable_fraction as an independent performance gain, or XGamma as lost work / dissipated energy / efficiency / damage;
• assert causal explanations not written in the reports.

Throughout, observed results (what happened) and candidate interpretations (why it might happen) are kept distinct, and the fixed-total-dephasing comparison (Milestone 10) is not conflated with the equal-input comparison (Milestone 11).

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Physical model and numerical conventions

Frozen in Milestone 1 and maintained thereafter.

Model. N two-level sites (|0> empty, |1> excited) plus one 3-level load. Defaults (hbar=1): inter-site coupling J=1, site-load coupling g=0.25, all angular frequencies = 1. Chain onsite Hamiltonian omega * sum_i |1><1|_i. Drive site = 0 (one end), load coupling site = N-1 (other end).

Coherent drive. H_drive(t) = Omega f(t){ exp(-i omega t) sigma_1^+ + exp(+i omega t) sigma_1^- }, envelope f(t)=sin^2(pi t / tau) for 0<=t<=tau (else 0), defaults tau=3.2, Omega=0.2.

Phase noise. L_phi,j = sqrt(gamma_phi/2) sigma_z,j on each chain site; no direct noise on the load. Time evolution: d rho/dt = -i[H(t), rho] + sum_k D[L_k] rho.

Basis and vectorization. Tensor order |q1, q2, ..., qN, load> (rightmost load index varies fastest; for N=3, |q1, q2, q3, load>). Column-major vectorization vec(rho) = [rho(0,0), rho(1,0), ..., rho(0,1), ...]^T, so vec(A rho B) = (B^T kron A) vec(rho). For N=3 the 24 x 24 density matrix maps to a length-576 vector and a 576 x 576 Liouvillian.

Liouvillian.

L = -i (I kron H - H^T kron I)
    + sum_k [ L_k* kron L_k
            - 1/2 I kron (L_k^dagger L_k)
            - 1/2 (L_k^dagger L_k)^T kron I ]

Collapse operators are passed with coefficients already included (e.g. sqrt(gamma_phi/2) sigma_z).

Local ergotropy (extractable work).

W(rho_L) = Tr(rho_L H_L) - min_U Tr(U rho_L U^dagger H_L)

computed from the load's reduced state.

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Milestone roadmap

Numerical details for every stage live in the individual MILESTONE_*.md reports and CSVs; this section gives only the arc and the representative outcomes.

Milestones 1-3: foundations

Static modules first (operators, partial trace, ergotropy) with conventions frozen (M1); then the Liouvillian and an accuracy-first dense matrix-exponential propagator (M2, 26 regular tests); an opt-in 576 x 576 integration smoke test confirming the full 24-dimensional model composes (M2.1); state diagnostics and signed-power accounting (M3); and an explicit fix for signed-power integration that splits sign-reversal intervals at the linearly interpolated zero crossing (M3.1, e.g. [(0,1),(1,-1)] -> energy_net=0, energy_in=0.25, energy_out=0.25). The DenseExponentialPropagator is a correctness baseline; efficiency is deferred.

Milestones 4-6: comparison and extraction

M4 — single-shot transport from an initial coherent state: across all 16 conditions matched to the same time and same load energy, noise-free A > phase-noisy B in ergotropy (matched ratio 1.229-49.318); differences are coherence-derived; zero physical-check failures. M5a implements and verifies the time-dependent RK4 propagator (step-halving convergence 7.80e-7 -> 4.99e-8 -> 3.19e-9, ratio ~15.6; 47 passed / 0 failed / 1 ignored). M5b confirms a finite pulse from vacuum generates nonzero load ergotropy (A 5.5424e-2, B 3.0302e-3; sanity check).

M5c (central comparison). With A/B load energy at t=10 matched to relative difference 4.001e-5 (Omega_B=0.431953125), ergotropy A 5.2798e-2 / B 8.2846e-3, A/B = 6.373; all 10 success conditions PASS. Only the final load energy is matched — Omega and total input energy are not — so this is neither an equal-input-cost comparison nor a noise-only causal comparison.

M6a is an implementation cross-check (not a new physical discovery): reconstruct the 5c final state and recover gross extracted work = load ergotropy via the ideal local unitary (18/18 checks pass; post-extraction ergotropy zero; gross-work A-B ratio 6.373).

Milestone 7: noise location and protection (fixed N=3)

M7a places noise on a single site: at t=10 the minimum W is at the entrance (site1) and the minimum usable fraction at the exit (site3); middle noise costs less than either end. M7b (no new time evolution) describes when the noise-vs-noise-free difference appears persistently (E and W begin at a diagnostic-level t=2.25; usable fraction is threshold-dependent). M7c ideally removes noise from selected sites; both-ends protection (leaving only middle noise) gives the largest recovery in W, usable fraction, and E, with protection non-additivity classified as positive_nonadditivity. M7d sweeps end-site gamma 0.5->0 (center fixed): recovery is discretely monotonic non-decreasing, maximum sensitivity interval 0.05->0.00. All of 7c/7d use ideal noise removal — not real devices, cost, or imperfect protection; curvatures are not called critical exponents or phase transitions.

Milestones 8-9: chain-length extension and validation

M8a changes only chain length N=3 -> N=5 (regression within 2e-9); W_max ratio N5/N3 free 0.3965, noisy 0.3620. M8b probes N=7 dense feasibility and finds it infeasible with the current dense method (~23.7 h to t=10). M8c replaces only the sigma_z dephasing term with an exact component-wise DiagonalDephasingKernel (not an approximation; 140 PASS / 0 FAIL), giving ~29.96x speedup and updating N=7 t=10 to feasible_candidate.

M9a runs N=7 noise-free to t=10 (W_max 2.2436e-2 at t=7.71; peak_resolved). M9b runs N=7 all-site noisy (gamma=0.5 per site, fixed-per-site, so total noise grows with N: N3=1.5, N5=2.5, N7=3.5): the noise-free "N7 W_max > N5 W_max" feature does not survive here (N7/N5=0.369).

M9c (fixed-total-noise) fixes the summed site gammas at TOTAL_GAMMA=1.5 to partially disentangle "increasing N" from "increasing total noise." Its history, in order:

1. An initial diagnostic found an eigenvalue-solver failure: one minimum-eigenvalue point at t=0.02 became NaN, failing positivity and finite_values, so MILESTONE_9C_REPORT.md recorded numerical_issue_stop as an interim status.
2. An exact diagnostic (MILESTONE_9C_DIAGNOSTIC.md) showed rho itself finite at t=0.02 (trace ~1, Hermiticity error 0, reproducible); the NaN came from non-finite solver output recorded uniformly by the CSV formatter. Independent Complex Schur gave all-finite eigenvalues.
3. The robust re-verification (MILESTONE_9C_VALIDATION.md) uses Hermitianized SymmetricEigen as primary and falls back to Hermitianized Complex Schur only at the two failing times (t=0.01, t=0.02); the fallback is in the diagnostic layer only and does not change the time evolution.
4. Positivity was determined at all 1001/1001 times (primary 999 success / 2 failure, fallback 2/2, solver_failure 0; worst selected minimum eigenvalue -5.278e-18).
5. The comparison against the existing 9c trajectory had maximum difference 0 (1001 times, tolerance 1e-12).
6. Final state completed_comparison_with_fallback_diagnostic.
7. The N=3, N=5, N=7 comparison under fixed total phase noise may therefore be adopted as a formal result.

The final source of truth for 9c is MILESTONE_9C_VALIDATION.md. The numerical_issue_stop in MILESTONE_9C_REPORT.md is interim; MILESTONE_9C_DIAGNOSTIC.md is the interim diagnostic record.

Milestone 10: fixed-total-dephasing comparison and XGamma

Milestone 10 organizes the fixed-total comparison the later work builds on, distinguishing two noise conventions: fixed-per-site (per-site rate constant, summed total grows with N) and fixed-total (summed rate held at TOTAL_GAMMA, per-site rate TOTAL_GAMMA/N). For N=5, 7 these are not the same total-noise condition.

• M10a compares existing results (no new time evolution), separates the two conventions, and leaves missing fixed-total N=3/N=5 entries as not_available rather than guessing (completed_with_explicit_missing_values).
• M10b newly computes TOTAL_GAMMA=3.0 for N=3, 5, 7 and introduces XGamma (completed_fixed_total_gamma_3_comparison).
• M10c recomputes TOTAL_GAMMA=1.5 under the same XGamma diagnostics; its N=7 trajectory matches the 9c source of truth to maximum difference 0 across 7 quantities x 1001 times. This 10c N=3 series is the reference reused as the N=3 side of the Milestone 11 comparison (completed_with_fallback_diagnostic).
• M10 Final reads only the 10a/10b/10c artifacts. Under both TOTAL_GAMMA = 1.5 and 3.0, the same finite-condition rankings held:

Metric	Ranking (both TOTAL_GAMMA = 1.5 and 3.0)
W_max	N=7 > N=5 > N=3
W(t=10)	N=7 > N=5 > N=3
usable_fraction	N=7 > N=5 > N=3
W_time_area	N=3 > N=5 > N=7
ergotropy arrival	N=3 fastest, N=5 middle, N=7 slowest
XGamma	N=7 > N=5 > N=3

The reading is deliberately not "longer chains are better": under fixed total dephasing the metric rankings differ (longer chains have higher peak/final ergotropy and usable fraction; shorter chains have larger W time-area and earlier arrival). This is a finite-condition descriptive result, not a general law. W_time_area/E_time_area are time-integrals of state quantities, not cumulative work or input; the two TOTAL_GAMMA points are not used to infer any functional form.

XGamma is a dephasing-kernel-weighted coherence-exposure diagnostic, x_gamma(t) = sum_ab Gamma[a,b] |rho[a,b](t)|^2, time-integrated — not lost work, dissipated energy, dephasing power, heat, entropy production, efficiency, or damage.

Why Milestone 11. In Milestone 10 the total dephasing was matched but the integrated drive input energy of N=3 vs N=7 was not, so ranking differences alone cannot isolate the effect of chain length. Milestone 11 fixes that. Note the Milestone 11 matched N=7 trajectory (Omega ~ 0.18748) is a different trajectory from the Milestone 10c N=7 run (Omega = 0.2); the two must not be conflated.

Milestone 11: equal-input N=3 vs N=7 analysis

Equal-input matching (11c-11f). The N=7 drive strength Omega is chosen so its integrated drive input energy equals the N=3 reference (target_E_drive_in = 5.9618618770136536e-2), by an auditable, one-trajectory-at-a-time procedure: a weak-drive initial guess (11c, no new evolution), an upper probe (11d, lower_probe_required), a lower probe confirming a bracket [0.18700000, 0.18770762191709489] (11e, local_input_bracket_confirmed), and evaluation of the interpolated point (11f). At Omega = 0.18748395731510084 the measured input matches the target to relative mismatch 2.8157011099880636e-6, with passing numerical-quality checks (matched_input_found_with_fallback_diagnostic). This matches drive input energy only; chain length, dimension, bond count, and geometry still differ, and no unique root, global uniqueness, monotonicity, or dt convergence of the matching is claimed.

Shape at equal input (11g). Comparing the two matched trajectories (N=3 from the 10c series, N=7 from 11f): N=3 W_max = 3.0302e-3 at t=5.63; N=7 W_max = 3.3853e-3 at t=7.70 (ratio ~ 1.11717). N=7 has the higher, later, narrower peak but a smaller 0->10 W time-area (the W curves cross twice). W_time_area is a state-quantity time-area, not cumulative extracted work; the higher N=7 usable fraction is a descriptive ratio, not an independent performance gain.

Descriptive shape fits (11h, 11j, 11k). These describe how much of the N=7 curve can be described by transforming the N=3 curve; they are not mechanism proofs, and AIC-like/BIC-like are descriptive complexity comparisons only.

• 11h single amplitude/shift/scale transform: normalized RMSE ~ 0.05485 (A=1.02343645, delta=2.49, s=0.91), leaving structured residuals.
• 11j asymmetric rise/fall transform: normalized RMSE ~ 0.03899 (A=1.05674517, delta=2.32, s_rise=0.965, s_fall=0.515); asymmetric_time_scaling_partially_supported.
• 11k post-peak decay: best low-parameter model is a two-stage exponential, normalized RMSE ~ 0.00810 (switch=8.50, lambda_1=0.15, lambda_2=0.20); the remaining 11j residual is concentrated late (~59.6% of its absolute area in t=9->10). Final judgment late_tail_structure_remains; auxiliary (internal to the decay fit) two_stage_tail_decay_supported.

Design and audit (11i). Milestone 11i is a design and audit, not a measurement. It confirmed that up to that point only aggregate load quantities were stored (no site-resolved populations/currents, full density matrices, mutual information, negativity, or mode occupation), and specified a minimal follow-up: re-run the single matched N=7 trajectory once, saving site-resolved diagnostics (completed_design_with_targeted_recomputation_required). 11i itself introduced no new time evolution. The minimal site-resolved diagnostic it specified was subsequently executed in Milestone 11M (below); 11i remains the design, 11M is its single-trajectory execution. The candidate mechanisms themselves (group velocity, boundary reflection, mode beating, entanglement/correlation fronts) remain undetermined — 11M records time structure descriptively and does not establish any of them.

dt-halving convergence of the matched N=7 condition (11L). For the equal-input matched N=7 condition (Omega = 0.18748395731510084, fixed, not re-matched), the internal time step was halved from dt=0.0025 to dt=0.00125 (8000 RK4 steps, t=0..10, 1001 saved points; one new full trajectory). The 11f formal coarse condition was not recomputed and its SHA-256 matched before and after. Two separate verdicts are reported and kept distinct:

• Matching preservation: PASS. With Omega held fixed, fine E_drive_in = 5.9618450901723317e-2, relative mismatch 2.8157045010698592e-6; fine-vs-coarse input change -2.0217e-13 (relative 3.3911e-12). No re-matching was performed and no new matching root is claimed.
• Physical convergence: PASS. Scalar metrics 23/23 PASS (e.g. W_max coarse/fine 3.3852501213e-3/3.3852501213e-3, W(t=10) 2.3264679067e-3 both, XGamma 5.3283047839e-2 vs 5.3283047838e-2); same-time trajectory quantities 10/10 PASS; W-peak shape PASS (peak time 7.70, half-max width 3.85, both grids). Numerical quality: finite PASS, max trace error 3.775e-15, Hermiticity error 0, solver_failure = 0, with one robust-positivity fallback point (primary 1000 success / 1 failure, fallback 1/1).

Final judgment matched_condition_dt_halving_passed_with_fallback_diagnostic. This is convergence of this one matched N=7 trajectory under a single dt-halving; it does not claim convergence at arbitrary dt, for other matched conditions, for a fine-dt-specific matching root, or for any other N/Omega/noise.

Site-resolved transport and load-local change (11M). Executing the 11i design on a single new trajectory (N=7, TOTAL_GAMMA=1.5, Omega = 0.18748395731510084, dt=0.0025, t=0..10 — no fine-dt re-run, since 11L already confirmed dt convergence), 11M saves per-site populations n_j = <sigma_j^+ sigma_j^->, nearest-neighbor bond currents (I_{j->j+1} = -2 J Im(z_j), positive = drive->load), a site7->load energy current I_7_to_load = <i[H_interaction, H_load]>, and the load's E / W / passive energy, all on the same 1001 times. Continuity and unit checks PASS (max continuity residual 1.373e-5); the 8 aggregate 11f quantities reproduce (8/8 PASS, max abs difference 0); formal-input SHA-256 matched before and after. Observed time structure around the W peak:

Event	Time
W peak	t = 7.70 (W_peak 3.3852501213e-3)
site7->load current, decline onset	t = 7.5
site7->load current, zero crossing after peak	t ~ 8.15
site7->load current, persistent negative onset	t = 8.16
load passive-energy increase onset	t = 7.70

From the W peak to t=10: load Delta E = -1.86345075e-4, Delta W = -1.05878221e-3, Delta passive = +8.72437140e-4. Final classification mixed_transport_and_load_local_change_with_fallback_diagnostic. Both the transport-side signature (current decline / zero crossing / persistent negative direction) and the load-local signature (W decrease with concurrent passive increase) are logged with status = observed only. No causal claim is made: the current zero crossing is not called a proof of boundary reflection, the current change is not said to cause the W decrease, and the passive-energy increase is not called thermalization or decoherence. Not examined: causality, boundary reflection, group velocity, modes, mutual information, negativity, entanglement, t>10, N>7, other Omega.

Finite-chain (N=3..7) event-structure diagnostics

Two follow-up analyses read only stored / formally adopted trajectories and add no new mechanism claims.

N=3 W-peak branch check. Testing whether the noise-dependent N=7 W-peak candidate generalizes to N=3, only the three stored N=3, Omega=0.2 trajectories (eta = 0, 1.5, 3.0) were read — no new time evolution, smoothing, or fitting. Under fixed extraction rules, the resolved peak sits in different time windows across the branches: eta=0 at t=9.48 (late window), eta=1.5 at t=5.63 and eta=3.0 at t=5.39 (early window). They cannot be aligned as a single time-window peak branch. Verdict peak_branch_correspondence_incomplete (secondary none): neither an early- nor a late-window correspondence is supported on the stored grid, and the incompleteness comes from absent local peaks in the predefined windows, not from grid resolution. The earlier N=7 preregistered W-peak prediction (post-11k experiment 3) is retained as-is; it is not generalized into an N-common law, and no claim is made about eigenmodes, phase transitions, exponential-law validity, or causal noise selection.

Equal-input N=3..7 event comparison. N=3, 4, 5, 6, 7 are compared at equal drive input energy, each using its own matched Omega (N=7 reuses the 11M trajectory; N=8 is not included). Per-chain categorical signatures:

N	parity	t_W_peak_end	current negative onset	signature
3	odd	5.63	not observed	negative_not_observed
4	even	6.04	8.69	W->passive->negative
5	odd	6.62	not observed	negative_not_observed
6	even	7.16	7.72	W->passive->negative
7	odd	7.70	8.16	W->passive->negative

The formal artifact classification is odd_even_alternation_candidate, driven by the continuous side of the precommitment: three quantities (backflow_amount_post_peak, post_peak_W_loss_fraction, Delta_passive_peak_to_t10) show non-zero alternating local differences A_4, A_5, A_6. However the categorical odd-even condition is not met: N=7 (odd) shares the even-group W->passive->negative signature rather than the negative_not_observed signature of N=3 and N=5, so N=7 breaks the odd-group commonality. t_W_peak_end also varies smoothly with chain length rather than alternating. The careful statement is therefore: a simple odd/even rule was not confirmed; only some continuous quantities show local alternating differences. This is not a parity mechanism, a universal parity order parameter, or a statistically significant difference. Numerical quality across N=3..7: 1001 points each, state/solver nonfinite 0, max continuity residual 1.373e-5 (< 5e-4), max trace error 2.220e-15, max energy-ledger residual 5.197e-7; N=7 carries one robust-positivity fallback point (consistent with the reused 11M trajectory).

Post-11k mini-experiments: input dependence on stored trajectories

Building on the stored time series and the tail structure obtained up to Milestone 11k — and keeping mechanism hypotheses separate — three small questions were examined on the existing stored trajectories before any larger computation. They propose no new theory or mechanism. Only experiment 3 ran a new trajectory (a single eta=0.75 run); experiments 1 and 2 reuse stored trajectories with no new time evolution. All results are observations for this finite model and these finite conditions.

• Event order (Case B). Across total-noise inputs eta = 0, 1.5, 3.0, the per-metric event order did not agree (the main difference is the position of the usable event). A simple input-independent common event order is not supported. Stop here for this question.
• Coherence vs W (Case D, not comparable). The three trajectories share no common absolute W range (common lower bound 1.3085e-2 exceeds common upper bound 6.9917e-4), so a raw C(W) comparison at equal W cannot be performed. This is neither support nor non-support of a common C(W) relation; "not comparable" is not a negative result.
• W-peak vs total phase noise (Case A). Using only the existing eta = 0/1.5/3.0 points, an exponential-decay candidate W_peak(eta) = A exp(-k eta) and its prediction were fixed in advance (A, k, thresholds, and eta = 0.75 committed in PRECOMMIT.md). A single new eta = 0.75 trajectory was then run (1001 saved points, numerical checks passed, completed_with_fallback_diagnostic): predicted 9.3038620128203936e-3, observed 9.1509856038800939e-3, relative error 1.643150%, verdict phase_noise_W_peak_exponential_candidate_retained. The result is agreement with one added point evaluated after precommitment, within the pre-specified threshold; the exponential-decay candidate is provisionally retained. This is not establishment of an exponential law. The 3-point R-squared was not used for the verdict; A, k, and thresholds were not changed after seeing eta = 0.75; no second eta and no alternative model were run.

What can be said: for this finite model and these finite conditions, event order is input-dependent (Case B); the stored trajectories are not comparable at equal W (Case D); and one preregistered added point is consistent, within the pre-specified threshold, with a fixed exponential-decay candidate that is provisionally retained (Case A).

Not claimed: no universal law, no new quantum channel, no discovered causal mechanism, no claim that coherence determines W, no proof of an exponential law, no general scaling law. W_time_area remains a state-quantity time-area, not cumulative extracted work; usable fraction and passive energy are not treated as independent performance quantities.

Milestone 12: what the post-peak W decrease looks like (fixed matched N=7)

Milestone 12 keeps the M11M matched N=7 condition fixed (TOTAL_GAMMA=1.5, Omega = 0.18748395731510084, dt=0.0025, t=0..10, 1001 saved points, vacuum initial state) and asks three descriptive questions about the post-peak decrease of the load's extractable work: which component changes (12A), in what temporal order (12B), and how load-chain correlation behaves around it (12C). No matching, re-optimization, root search, fine-dt run, or other N/gamma/Omega was performed.

12A — local ergotropy decomposition. One new trajectory under the same fixed condition. Using the existing energy-basis convention (H_L = diag(0,1,2)), the load ergotropy is split as W = W_pop + W_coh, with W_pop = W(Delta_H(rho_L)) the diagonal (population) part. At every saved time W_pop = 0 and therefore W = W_coh; over t = 7.70 -> 10.00, Delta W = Delta W_coh = -1.0587822145716654e-3, Delta W_pop = 0, Delta E_load = -1.8634507496082930e-4, Delta E_passive = 8.7243713961083613e-4. The observed signed W decrease is thus described entirely as a W_coh decrease. The pre-registered ratio rule, however, required both components to decrease positively, and L_pop = 0 made the ratio undefined, so the formal classification is component_change_inconclusive. That verdict reflects the pre-committed rule being inapplicable — it does not mean the decomposition failed; the decomposition itself is exact (max |W - (W_pop + W_coh)| = 0). Numerical quality 20/20 PASS, M11M aggregates reproduced 8/8 PASS (max abs difference 0), formal input SHA-256 7/7 unchanged. Population and coherent parts are not independent conserved quantities or separate physical energies.

12B — temporal relation (no new time evolution). Reading only the 12A saved 1001-point series under a pre-registered five-interval persistence rule (M12B_PRECOMMIT.md; no smoothing, fitting, FFT, or extremum interpolation), the saved-grid order is:

Event	Time
site7->load current decline onset	t = 7.50
W_coh peak / persistent decline	t = 7.70 / 7.70
Cl1 (coherence l1-norm) peak / persistent decline	t = 7.73 / 7.73
current zero crossing	t = 8.15
current persistent negative onset	t = 8.16
current persistent positive return	t = 9.68

Formal classification coherence_and_ergotropy_decline_before_negative_current (3/5/7-interval sensitivity all agree). The persistent negative current begins 0.46 later than the W_coh decline onset, so the simple account "the turn to negative current initiated the W_coh decrease" is not supported by the time ordering. This is a statement about ordering only; no causal relation between current and W_coh is established in either direction. Checks 15/15 PASS, W = W_coh max difference 0, input SHA-256 3/3 unchanged. The current integrals are time-integrals of a state-derived quantity, not work, efficiency, loss, dissipation, or heat.

12C — load-chain correlation. One new full trajectory under the same fixed condition, evaluating load-chain mutual information I(C:L) = S_chain + S_load - S_total (natural log, nats), the individual entropies, and purities at all 1001 saved times. The MI global peak and its five-interval persistent decline both fall at t = 7.51, with peak value 4.5985207125390781e-3 nats — 0.19 earlier than the W_coh peak/decline at t=7.70. Formal classification mutual_information_peaks_before_Wcoh_decline, unchanged across the 3/5/7-interval sensitivity grid. Numerical quality 29/29 PASS; existing formal artifacts SHA-256 30/30 unchanged; M12A common quantities reproduced to max abs difference 0. Mutual information is a total-correlation observable including classical and quantum correlation — it is not work, ergotropy, or entanglement. No claim is made that the MI decrease caused the W_coh decrease, or that W was converted into MI.

Milestone 13: short-horizon counterfactual branches and their temporal resolution limit

13A — formal counterfactual branch experiment. Starting from the same stored total-system state as M12C and changing only the generator after the branch time, short branches were propagated on a 0.01 saved grid to t=8.50 from two branch starts (t=7.50 and t=7.70; 728 formal saved rows total):

• B0 baseline continuation (reproduces M12C exactly, max abs difference 0);
• B1 chain dephasing set to 0 after the branch;
• B3 chain-load coupling halved after the branch (g: 0.25 -> 0.125);
• B4 coupling set to 0, a boundary control (max invariant drift 1.9984014443252818e-15).

B3 gives the same formal classification at both branch starts: weaker_coupling_weakens_Wcoh_decline.

Branch start	endpoint W_coh relative difference	W_coh time-area relative difference
7.50	+6.1424321151409798 %	+1.4670925175957812 %
7.70	+6.1471886834895455 %	+2.1027954259932112 %

B1 differs between starts and was not averaged: dephasing_counterfactual_mixed at 7.50, future_dephasing_change_has_small_effect at 7.70. B4 is zero_coupling_boundary_control_pass at both starts. All numerical-quality rows PASS (max trace error 1.776e-15, Hermiticity error 0, no fallback used, drive after branch 0).

The admissible summary is exactly: from the same state, halving the chain-load coupling after the branch produced a consistently weaker W_coh decrease within the finite window, at both branch starts, relative to the baseline that keeps the normal coupling. Explicitly not claimed: that coupling is the sole cause of the W_coh decrease; that weaker coupling is always better; that an optimal coupling was found; that halving the coupling was shown to protect W_coh by some mechanism. The B1 and B3 percentages are not compared head-to-head to rank which knob "matters more". W_coh time-area is a time-integral of a state quantity, not cumulative extracted work.

13B — temporal response decomposition (no new computation). Using only the formal 13A saved CSVs (new trajectories 0, new branches 0, new parameter conditions 0), 13B asks which saved observable first develops a persistent B3-minus-B0 difference. Under the frozen five-interval rule:

• start 7.50: current at t=7.50, then all five other quantities (Cl1, MI, W_coh, load energy, passive energy) together at t=7.51;
• start 7.70: current at t=7.70, then MI / load energy / passive energy / W_coh at t=7.71, then Cl1 at t=7.75.

However the three-interval rule places the main state quantities simultaneous with W_coh, while the five- and seven-interval rules produce a start-time dependence; across the full 3/5/7 x epsilon (x0.5/x1/x2) grid the classification is not stable (Wcoh_and_other_differences_emerge_together vs branch_response_order_depends_on_start_time). The formal classification is therefore temporal_response_inconclusive. This is a resolution limit, not a computational failure, and it is not a negation of the 13A B3 result: the numerical audit is 35/35 PASS with 11/11 SHA-256 unchanged for the formal M13A inputs, and the M13A B3 classifications and relative differences reproduce within 1e-12. The immediate current difference at the branch time is structural — the current expression itself contains the coupling coefficient that was changed (audited as current_B3 = 0.5 * current_B0 at branch start) — and it is not read as the current causing the later W_coh difference. Also not shown: that passive energy converted into coherent ergotropy, or that correlation suppression protected work.

Where this leaves the project. M12 described the component (W = W_coh, 12A), the temporal order (12B), and the load-chain correlation (12C) of the post-peak W_coh decrease. M13A examined the finite-time response to a generator change from an identical state and found, at both branch starts, that halving the coupling weakens the W_coh decrease. M13B could not uniquely resolve the internal response ordering of that branch at the available saved resolution. Coupling sensitivity is therefore a confirmed observation, while the specific physical mechanism and any optimal condition remain undetermined.

Milestone 14: coupling-control design and switching-work accounting

Milestone 14 asks what it costs, within the model's Hamiltonian ledger, to make the M13A coupling change — and keeps that ledger strictly separate from the state ledger. Fixed protocol names used from here on:

Protocol	Definition	Corresponds to
C0	g = 0.25 throughout	M13A B0
C1	g: 0.25 -> 0.125 at t=7.50, held to 8.50	M13A B3, start 7.50
C2	g: 0.25 -> 0.125 at t=7.70, held to 8.50	M13A B3, start 7.70

No g=0, additional coupling value, switch-back, ramp, feedback, or multiple switch is admitted.

14A — design only (no formal control trajectory). Milestone 14A is a design and audit Milestone: it ran no formal control trajectory, no coupling sweep, no optimization, and no ramp. It audited the implemented interaction term (H_CL(g) = g V_CL, V_CL = sigma_plus_N b + sigma_minus_N b_dagger, exactly linear in g), confirmed the drive envelope is exactly zero for t>tau=3.2 so that post-switch protocol differences are not drive-input differences, fixed the ideal-quench convention W_switch_on_system = Tr[rho(t_s)(H_after - H_before)] (positive = supplied by the external controller to the modeled system; state unchanged across the quench), and fixed three separate ledgers — state quantities, switching work, and post-switch energy — which are not summed or subtracted into any net quantity. Its no-propagation unit checks are 11/11 PASS. The reuse audit found that the M13A artifacts do not persist full density matrices or Tr[rho V_CL], so the verdict was single_prefix_recomputation_required: re-run one baseline prefix only. Final design verdict coupling_control_experiment_ready, recommended plan Plan B (minimal_state_recomputation_for_switching_accounting). The state-level trade-off preview quoted there is explicitly not a formal control verdict, because the switching work was still missing.

14 Plan B — formal switching-work accounting (executed). Exactly one baseline prefix was propagated from t=0 to t=7.70, full density matrices were saved at t=7.50 and t=7.70, and the ideal same-state switch was evaluated at both. New post-switch branches, coupling values, sweeps, ramps, extraction, optimization, fine-dt runs, and t>8.50 propagation: 0. The existing M13A C0/C1/C2 post-switch state trajectories were reused unchanged.

Protocol	switch time	W_switch_on_system	W_switch_out
C1	7.50	1.6980822985791084e-15	-1.6980822985791084e-15
C2	7.70	8.8624029002963674e-16	-8.8624029002963674e-16

Both ideal-quench energy jumps fall below the fixed 1e-12 audit tolerance, so the correct description is numerically zero at this tolerance (status numerically_zero_within_1e-12_audit_tolerance). A numerically zero Hamiltonian-jump work does not mean a real controller is free: the model has no actuator Hamiltonian, finite ramp, bandwidth, or device-cost ledger. Reusing the M13A state values, the formal composite classification for both C1 and C2 is Wcoh_improves_with_load_energy_tradeoff + switching_accounting_available — endpoint W_coh and W_coh area each improve by at least 1%, while endpoint load-energy retention stays below the fixed 99% threshold (C1: W_coh endpoint +6.142%, area +1.467%, retention 97.159%; C2: W_coh endpoint +6.147%, area +2.103%, retention 98.961%). Numerical checks 22/22 PASS; M13A B0 quantities reproduce at both switch times to max abs difference 3.839e-13; source/formal input hashes 15/15 unchanged. The reported switching-work ratios are scale diagnostics, not efficiencies, and the switching ledger is not converted into a profit, a "net protected W_coh", or a control-efficiency claim.

Milestone 15A: end-point extraction accounting after coupling control

Milestone 15A propagates only C0 and C2 from 7.70 to 8.50 and performs one ideal local extraction at the single fixed evaluation time t = 8.50. The formal comparison is C2 - C0 only. Verdict state_improvement_preserved_after_extraction, mandatory numerical failures 0.

Quantity at t=8.50	C0	C2	C2 - C0
W_coh before extraction	3.0626774684866464e-3	3.2509460312412416e-3	1.8826856275459521e-4
load energy before extraction	6.601877630500e-3	6.533284638378e-3	-6.8592992121339208e-5
W_load_gross	3.0626774684866503e-3	3.2509460312412546e-3	1.8826856275460432e-4
W_operation_out_total	3.062677468536e-3	3.250946031265e-3	1.8826856272917414e-4
interaction jump	-4.957529045397e-14	-2.416320556056e-14	2.5412084893412970e-14
load ergotropy after extraction	0	0	0

The gross-work difference reproduces the pre-extraction W_coh difference with residual 9.1072982488782372e-18 (ratio 1.000000000000). Because the interaction-jump difference is only 2.5412084893412970e-14, the bare-load and total-Hamiltonian accountings give the same sign for the comparison. Numerical and provenance checks 40/40 PASS, including trace, Hermiticity, positivity, post-extraction zero ergotropy, M13A endpoint reuse, M14 state SHA-256, and switching-accounting reuse.

The admissible conclusion is exactly: at t=8.50, a single ideal local extraction on C2 yielded a larger gross extracted work than the baseline C0; the C2-C0 gross-work difference agreed with the pre-extraction W_coh difference to numerical precision; and including the interaction term in a total-Hamiltonian accounting did not change the sign of the comparison. In short, the state-level improvement was preserved through this one ideal local extraction.

Ledgers stay separate. W_coh, load energy, switching work, W_load_gross, and W_operation_out_total are not summed into any single "benefit". The M14 Plan B C2 switching work (8.8624029002963674e-16) is carried over for reference only and is not combined with either extraction-work column. Not performed in 15A: new coupling conditions, sweeps, ramps, optimization, feedback, measurement, multiple extractions, other N/gamma/Omega, or any propagation beyond t=8.50. Nothing here is optimal control, a free increase of work, an efficiency improvement, a confirmed net benefit, a zero controller cost, an experimental-hardware advantage, or a quantum advantage.

C2 extension to t=10: long-window C0 comparison

This supplementary comparison asks whether the C2-minus-C0 bare-load ergotropy difference observed at t=8.50 is an artifact of that window's endpoint. It is not a new Milestone. The formally saved M14 Plan B full-system state at t=7.70 was propagated once and continuously with C2 (g=0.125) to t=10.00 — no extraction at t=8.50, no switch-back, no rematching, and no split formal trajectory. Fixed conditions: N=7, TOTAL_GAMMA=1.5, per-site gamma 1.5/7, Omega=0.18748395731510084, restart time 7.70, final time 10.00, internal dt=0.0025, saved interval 0.01, 920 internal steps, 231 saved points, drive exactly 0. The C0 long-window baseline is the M12A CSV, fixed by an explicit artifact audit. Primary classification C2_advantage_positive_at_t10.

t	W_C0	W_C2	Delta W	relative difference
8.50	3.062677468487e-3	3.250946031241e-3	+1.882685627546e-4	+6.147189%
9.68	2.417441223901e-3	2.999355919229e-3	+5.819146953273e-4	+24.071514%
10.00	2.326467906728e-3	2.972492411029e-3	+6.460245043013e-4	+27.768468%

After t=8.50 the difference increased in all 150 saved intervals (decrease count 0, near-flat count 0), never reached near-zero, and never turned negative; its global minimum is at t=8.50 and its global maximum at t=10.00, with no interior local extrema. This is a statement about the C2-C0 difference, not about C2 itself: C2 formed no new W peak after t=8.50, its in-window maximum W remains at t=7.70 (3.3852501212993710e-3, the same value as C0's maximum), and the difference grows because C0 declines faster — t=10 retention is 87.807172% for C2 versus 68.723664% for C0. Over 7.70..10.00 the ergotropy time-areas are A_C0 = 6.5655097929267292e-3, A_C2 = 7.2916111984482872e-3, A_Delta = 7.2610140552155800e-4, relative difference +11.059330%; these are time-areas of the instantaneous bare-load ergotropy and are not cumulative extracted work. At t=10 the accompanying energy differences are Delta E_load = +2.106309466093e-5 and Delta E_passive = -6.249614096404e-4, with energy-identity residual 0. The site7-to-load currents are recorded as observations under the existing definition only and are not used to explain the ergotropy difference.

Audit: mandatory numerical checks 28/28 PASS, existing-artifact reproduction checks 15/15 PASS, SHA audit PASS. The C0 short/long observable residual is 0, the C2 t=7.70..8.50 reproduction residual is 0, and the M15A t=8.50 values reproduce to a maximum residual of 9.107e-18. Eigensolver fallback count 0; formal runtime 682.479 s.

Scope: this is a finite-condition, saved-grid comparison from one propagation of one stored state against one fixed baseline. No extraction was performed in this extension — the only formal extraction in the project remains the single ideal local extraction at t=8.50 in M15A. It does not establish general weak-coupling or C2 superiority, an optimal coupling, switch time, or extraction time, any causal mechanism, any conversion of passive energy into ergotropy, any net benefit combining switching and extraction work, repeated or cyclic operation, steady-state output, behavior beyond t=10, other N/gamma/Omega/g, efficiency improvement, or quantum advantage. Where the word advantage is used here, it means only the sign of the bare-load ergotropy difference under these fixed finite conditions.

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Current main results

Representative values only; all are directly-observed results for this model, these conditions, this finite time, and this implementation — not general laws. Full tables live in the reports and CSVs.

• M5c (N=3, conditional load-energy match, t=10): ergotropy A/B = 6.373 (A 5.2798e-2 / B 8.2846e-3); load-energy relative difference 4.001e-5. Omega and total input energy not matched.
• M9a/9b (N=7, fixed-per-site, reference): noise-free W_max 2.2436e-2 (t=7.71); all-site-noisy (gamma=0.5) W_max 4.0437e-4 (t=7.65); fixed-per-site W_max N3 3.0302e-3 > N5 1.0968e-3 > N7 4.0437e-4 (noise-free "N7 > N5" does not survive here).
• M9c validation (N=7, fixed-total, total gamma=1.5 — source of truth): positivity 1001/1001, solver_failure 0; SymmetricEigen 999 success / 2 failure; Schur fallback 2/2; max difference vs existing 9c trajectory 0; N=7 W_max 0.0038080717406769921 (t=7.70); fixed-total W_max N3 3.0302e-3, N5 3.4876e-3, N7 3.8081e-3; N7/N5 W_max ratio 1.0918825139, within the descriptive "10% band" (not a statistically significant difference, not distance-only causation).
• M11g (N=3 vs N=7, equal drive input energy — distinct from the fixed-total M9c run above): matched at Omega=0.18748395731510084 (relative input mismatch 2.8157e-6); N=7 W_max 3.3853e-3 at t=7.70 vs N=3 3.0302e-3 at t=5.63 (ratio ~ 1.11717); N=7 higher/later/narrower peak but smaller W time-area.
• M11L (matched N=7, dt-halving of the same trajectory): Omega fixed at 0.18748395731510084; dt halved 0.0025 -> 0.00125. Matching preserved (relative mismatch 2.8157045e-6, input change ~3.39e-12 relative); scalar 23/23 PASS, trajectory 10/10 PASS, W-peak shape PASS (peak t=7.70, W_max 3.3853e-3 both grids); solver_failure = 0, one positivity fallback point. matched_condition_dt_halving_passed_with_fallback_diagnostic.
• M11M (matched N=7, site-resolved single-trajectory diagnostic): W peak t=7.70; site7->load energy current zero crossing after peak ~ t=8.15, persistent negative onset t=8.16; load passive-increase onset t=7.70. Peak->t10: load Delta E = -1.86345075e-4, Delta W = -1.05878221e-3, Delta passive = +8.72437140e-4. mixed_transport_and_load_local_change_with_fallback_diagnostic (transport-side and load-local signatures both observed, no causal claim).
• N=3 W-peak branch check (stored N=3, Omega=0.2): resolved peaks at eta=0 t=9.48, eta=1.5 t=5.63, eta=3.0 t=5.39 — not alignable as one time-window branch; peak_branch_correspondence_incomplete.
• Equal-input N=3..7 event comparison: signatures N3 negative_not_observed, N4 W->passive->negative, N5 negative_not_observed, N6 W->passive->negative, N7 W->passive->negative. Formal odd_even_alternation_candidate, but the categorical odd/even condition is not met (N=7 breaks the odd-group commonality); only some continuous quantities alternate. Not a parity mechanism, universal rule, or statistically significant difference. N=8 excluded.
• M12A (matched N=7, load ergotropy decomposition): W_pop = 0 at every saved time, so W = W_coh; over t=7.70->10.00 Delta W = Delta W_coh = -1.0587822146e-3, Delta W_pop = 0, Delta E_load = -1.8634507496e-4, Delta E_passive = 8.7243713961e-4. Formal component_change_inconclusive (the pre-committed both-components ratio rule was inapplicable, not a failed decomposition); 20/20 PASS, M11M aggregates 8/8 PASS.
• M12B (same series, no new evolution): current decline onset t=7.50; W_coh peak/decline t=7.70; Cl1 peak/decline t=7.73; current zero crossing t=8.15; persistent negative onset t=8.16; persistent positive return t=9.68. coherence_and_ergotropy_decline_before_negative_current; the persistent negative current is 0.46 later than the W_coh decline onset.
• M12C (same condition, one new trajectory): load-chain mutual information peaks and begins its persistent decline at t=7.51, peak 4.5985207125e-3 nats — 0.19 earlier than the W_coh peak. mutual_information_peaks_before_Wcoh_decline; 29/29 PASS, 30/30 SHA-256 unchanged. MI is a total-correlation observable, not work or ergotropy.
• M13A (short-horizon counterfactual branches from the M12C state): halving the chain-load coupling after the branch (B3) gives weaker_coupling_weakens_Wcoh_decline at both starts — endpoint W_coh relative difference +6.1424321151% (start 7.50) and +6.1471886835% (start 7.70); W_coh time-area relative difference +1.4670925176% and +2.1027954260%. B1 start-dependent (dephasing_counterfactual_mixed / future_dephasing_change_has_small_effect); B4 zero_coupling_boundary_control_pass, max invariant drift 1.998e-15.
• M13B (13A CSVs only, no new computation): the B3-B0 persistent-difference ordering is not stable across the 3/5/7 x epsilon sensitivity grid — temporal_response_inconclusive; audit 35/35 PASS, 11/11 SHA-256 unchanged, M13A values reproduced within 1e-12. A resolution limit, not a failure, and not a negation of the M13A B3 result.
• M14A (design only): no formal control trajectory, sweep, ramp, or optimization was run. Fixed the protocol set C0/C1/C2, the ideal-quench convention W_switch_on_system = Tr[rho(t_s)(H_after-H_before)], and three separate ledgers; unit checks 11/11 PASS. Verdict coupling_control_experiment_ready, recommended Plan B.
• M14 Plan B (executed): one baseline prefix to t=7.70; ideal same-state switch evaluated at both switch times. W_switch_on_system = 1.6980822985791084e-15 (C1) and 8.8624029002963674e-16 (C2) — numerically zero within the 1e-12 audit tolerance, which does not imply a real controller is free. Both protocols: Wcoh_improves_with_load_energy_tradeoff + switching_accounting_available (C2 endpoint W_coh relative difference +6.147%, endpoint load-energy retention 98.961%). Checks 22/22 PASS; hashes 15/15 unchanged.
• M15A (executed, C2-C0 at t=8.50 only): one ideal local extraction. W_coh before 3.0626774684866464e-3 (C0) vs 3.2509460312412416e-3 (C2); Delta W_coh = 1.8826856275459521e-4; W_load_gross 3.0626774684866503e-3 vs 3.2509460312412546e-3, Delta = 1.8826856275460432e-4; Delta W_operation_out_total = 1.8826856272917414e-4; Delta interaction jump 2.5412084893412970e-14; Delta load energy before -6.8592992121339208e-5; gross-work-vs-W_coh residual 9.1072982488782372e-18; post-extraction load ergotropy 0 for both. Checks 40/40 PASS. Verdict state_improvement_preserved_after_extraction. The ledgers are not summed into a net benefit.
• C2 extension to t=10 (supplementary, no new Milestone number, no extraction): propagating the saved t=7.70 state once with C2 to t=10.00 and comparing against the fixed M12A C0 long-window baseline, the C2-C0 bare-load ergotropy difference is +1.882685627546e-4 (+6.147189%) at t=8.50, +5.819146953273e-4 (+24.071514%) at t=9.68, and +6.460245043013e-4 (+27.768468%) at t=10.00, increasing in all 150 saved intervals after t=8.50 with no near-zero or negative crossing. C2 formed no new W peak after t=8.50 — the difference grew because C0 declined faster (t=10 retention 87.807172% vs 68.723664%). Ergotropy time-area over 7.70..10.00: A_Delta = 7.2610140552155800e-4 (+11.059330%), a time-area of instantaneous ergotropy, not cumulative extracted work. Classification C2_advantage_positive_at_t10; 28/28 and 15/15 checks PASS.

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Directly validated findings

• The equal-input match of N=7 to N=3 at Omega=0.18748395731510084 (relative mismatch 2.8157e-6), with passing numerical-quality checks (M11f).
• Peak values/timing N=3 vs N=7 and the two W crossings; N=7 higher/later/narrower peak, smaller 0->10 W time-area (M11g).
• The 9c fixed-total positivity accounting (1001/1001, fallback 2/2, solver_failure 0) and the max-0 agreement with the existing 9c trajectory (M9c validation).
• Milestone 10 fixed-total metric rankings under both TOTAL_GAMMA = 1.5 and 3.0 (M10 Final).
• Numerical-quality checks (per-run) and, for 11j/11k, SHA-256 input-integrity checks; the 11i checks are design/audit checks.
• dt-halving (0.0025 -> 0.00125) of the same matched N=7 trajectory with Omega fixed: matching preserved and scalar 23/23 / trajectory 10/10 / W-peak shape all PASS, solver_failure = 0 (M11L).
• Site-resolved single-trajectory diagnostic for the matched N=7 run: per-site populations, bond currents, site7->load energy current, nearest-neighbor coherences, and load E/W/passive on 1001 times, with the W-peak-relative time ordering above; continuity/unit checks PASS, 11f aggregates reproduced 8/8 (M11M).
• N=3 stored-trajectory W-peak branch windows (eta=0 t=9.48; eta=1.5 t=5.63; eta=3.0 t=5.39) and their non-correspondence as a single time-window branch (peak_branch_correspondence_incomplete).
• Equal-input N=3..7 per-chain categorical signatures and numerical-quality checks; the categorical odd/even condition is not met (N=7 breaks the odd-group commonality).
• For the matched N=7 condition, that the load's diagonal (population) ergotropy is exactly 0 at every saved time, so W = W_coh, with an exact decomposition residual of 0 and the signed peak->t10 changes above (M12A).
• The saved-grid event ordering current decline (7.50) -> W_coh decline (7.70) -> Cl1 decline (7.73) -> current zero crossing (8.15) -> persistent negative current (8.16) -> persistent positive return (9.68), stable across 3/5/7-interval rules (M12B).
• The load-chain mutual information peak/persistent-decline time t=7.51 and peak value 4.5985207125e-3 nats, 0.19 earlier than the W_coh peak, stable across 3/5/7-interval rules (M12C).
• That, from an identical stored state, halving the chain-load coupling after the branch weakens the W_coh decrease within the finite window at both branch starts, with the endpoint and time-area relative differences above; plus the B0 reproduction (max abs difference 0) and the B4 zero-coupling boundary control (max invariant drift 1.998e-15) (M13A).
• The numerical/SHA-256 audits of the re-analysis stage: 35/35 checks PASS, 11/11 formal M13A inputs unchanged, M13A B3 values reproduced within 1e-12 (M13B).
• That the implemented interaction term is exactly linear in g and that the drive is exactly zero at t=7.50, 7.70, and 8.50, so post-switch protocol differences are not drive-input differences; unit checks 11/11 PASS with zero residuals (M14A, no time evolution).
• The ideal-quench Hamiltonian-jump work for both coupling-down protocols, 1.6980822985791084e-15 (C1) and 8.8624029002963674e-16 (C2), each below the 1e-12 audit tolerance, with the state exactly unchanged across the evaluated switch and M13A B0 quantities reproduced to 3.839e-13; checks 22/22 PASS (M14 Plan B).
• That at t=8.50 a single ideal local extraction gives W_load_gross larger for C2 than for C0 by 1.8826856275460432e-4, agreeing with the pre-extraction W_coh difference to a residual of 9.1072982488782372e-18, with the total-Hamiltonian accounting giving the same sign and post-extraction load ergotropy exactly 0 in both protocols; checks 40/40 PASS (M15A).
• That, extending C2 from the formally saved t=7.70 state to t=10 and comparing it with the fixed C0 baseline on the common saved grid, the C2-C0 bare-load ergotropy difference is still positive at t=10 and increased across every saved interval after t=8.50, under these fixed finite conditions. This concerns the difference between the two protocols, not growth of C2's own ergotropy, which formed no new peak after t=8.50; checks 28/28 and 15/15 PASS (C2 extension, no extraction performed).

Descriptive model support

• Single amplitude/shift/scale transform, normalized RMSE ~ 0.05485 (M11h).
• Asymmetric rise/fall transform, normalized RMSE ~ 0.03899, asymmetric_time_scaling_partially_supported (M11j).
• Post-peak two-stage exponential, normalized RMSE ~ 0.00810, two_stage_tail_decay_supported; late-tail concentration ~ 59.6% in t=9->10, late_tail_structure_remains (M11k).

These are descriptive shape diagnostics, not evidence of any physical mechanism.

Counterfactual branch response (M13A). Not a curve fit but the same category of evidence: a descriptive, finite-window comparison between branches that share an identical state and differ only in the generator applied after the branch time. The B3 (coupling halved) branch shows a weaker W_coh decrease than the B0 baseline at both starts (endpoint W_coh relative difference +6.142% / +6.147%; time-area relative difference +1.467% / +2.103%), with the B4 zero-coupling boundary control passing. This establishes a sensitivity of the observed W_coh decline to the chain-load coupling within this window — it is not a mechanism, not a protection scheme, not an optimum, and the B1 (dephasing) and B3 (coupling) percentages are not ranked against each other. M13B, which re-analyzed only these saved values, could not uniquely order the internal response (temporal_response_inconclusive); that is a limit of the saved temporal resolution and does not withdraw the B3 result above.

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What has not been confirmed

• Quantum advantage/supremacy or superiority over a classical method.
• Universal scaling laws, exponential/power-law decay, thermodynamic limits, N->infinity behavior.
• Any general advantage of longer chains, or universal laws of noise position/protection strength.
• Any physical mechanism behind the observed shapes — group velocity, boundary reflection, mode beating, entanglement/correlation fronts, or any causal relationship — and that transform/decay fits correspond to specific physical processes.
• Control cost of extraction, finite-time switch-off/extraction, repeated cycles, continuous supply, steady output, long-time stability.
• dt-halving convergence beyond the single matched N=7 trajectory checked in 11L (arbitrary dt, other matched conditions, a fine-dt-specific matching root); matching-root uniqueness/monotonicity.
• Equal-input results for TOTAL_GAMMA=3.0, for other Omega roots, for N>7, or for t>10; N=8 is not included in the formal chain-length comparison; other external solver crates (out of 9c-validation scope). (Equal-input N=4/5/6 now exist only as finite-chain event-structure diagnostics, not as a scaling law.)
• Any parity mechanism, universal odd/even effect, parity order parameter, or statistically significant parity difference; the N=3..7 odd_even_alternation_candidate is a descriptive finite-size candidate only, and N=7 breaks the categorical odd-group commonality.
• Any causal reading of the 11M site-resolved diagnostic: that the current change causes the W decrease, that the current zero crossing / reversal proves boundary reflection, or that the passive-energy increase proves thermalization or decoherence.
• The internal temporal ordering of the weaker-coupling branch response: under the 3/5/7-interval x epsilon sensitivity grid the B3-B0 persistent-difference order was not stable, so the order in which current, load energy, passive energy, Cl1, MI, and W_coh respond could not be uniquely determined at the current saved interval and persistence rules (M13B, temporal_response_inconclusive). The immediate current difference at the branch time is structural (the current contains the changed coupling coefficient) and is not evidence of a response order.
• The physical mechanism behind any of the M12/M13 observations. Coupling sensitivity is observed (M13A), but no mechanism is identified: it is not shown that coupling is the sole cause of the W_coh decrease, that weaker coupling is generally better, that an optimal coupling exists or was found, that passive energy converted into coherent ergotropy, that correlation suppression protected work, or that the mutual-information decrease caused (or was converted into) the W_coh decrease.
• Generalization of any M12/M13 result beyond this single fixed matched N=7 condition, these two branch start times, the finite window (branch t<=8.50, trajectory t<=10), and this saved resolution; within M12/M13 themselves no extraction, recovery, control, or repeated-cycle operation was performed (the later C2 extension to t=10 is a separate supplementary check and likewise performed no extraction).
• Anything beyond the ideal, instantaneous idealizations used in M14/M15A. Specifically not confirmed: a finite-time switch or ramp (the modeled quench has mathematically zero duration, no controller Hamiltonian and no bandwidth limit); the implementation cost of any actuator or controller (a numerically zero Hamiltonian-jump work is not a zero device cost); the implementation cost of the extraction unitary; any net benefit combining switching work and extraction work (the ledgers are deliberately kept separate and are never summed); repeated extraction or cycle operation; general long-time stability, steady state, or behavior beyond t=10 — note that the formal windows and the single extraction time of M13A and M15A themselves remain t=8.50, while a separate supplementary check has propagated C2 to t=10, so saved-grid comparisons at t=9.68 and t=10 have been carried out (without extraction); parameter generalization to other coupling values, sweeps, N, gamma, Omega, branch times, or conditions beyond those fixed here; optimal control of any kind (no optimum was searched for or found, and weaker coupling is not shown to be generally better); any experimental advantage on real hardware; and any quantum advantage.
• Fair baselines against classical wave/stochastic models; novelty/priority vs the literature (no literature review done).

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Reproduction and verification

Complete reproduction packages are stored as Milestone-specific ZIP archives in the repository's zip/ directory. Source code, reports, CSV outputs, audit files, and any saved states required by a given stage are contained in the corresponding archive; they are not all exposed as individual files at the GitHub repository root.

Download and extract the relevant archive, then run the commands below from the extracted directory that contains its Cargo.toml. A Rust toolchain is required (Cargo.toml: edition 2021; deps nalgebra, num-complex, thiserror; dev-dep approx). Archive contents vary by Milestone, so a binary or data file listed below may be present only in the package for that stage.

cargo fmt --all -- --check
cargo test --release --offline
cargo build --release --offline
# opt-in 24-dimensional smoke test:
cargo test --release full_24d_short_time_smoke_test -- --ignored --nocapture

Representative milestone binaries:

cargo run --release --offline --bin time_dependent_sanity          # M5a
cargo run --release --offline --bin dephasing_kernel_benchmark     # M8c
cargo run --release --offline --bin n7_noise_free_full             # M9a
cargo run --release --offline --bin n7_all_site_noisy_full         # M9b
cargo run --release --offline --bin fixed_total_noise_comparison   # M9c
cargo run --release --offline --bin n7_t002_eigen_diagnostic       # M9c diagnostic
cargo run --release --offline --bin n7_fixed_total_validation      # M9c validation (source of truth)

The N=7 full runs take on the order of tens of minutes (e.g. 9a ~ 2953s, 9b ~ 2899s, 9c validation ~ 2703s). Per-milestone reports record cargo fmt PASS, the analysis binary PASS, and release-test counts that grow across the line (e.g. M2 26 tests; M5a 47 passed / 0 failed / 1 ignored; M10b 101, M10c 104, M10 Final 107; M11c 110, M11d 113, M11e 116, M11f and M11h-M11k 119, each 0 failed / 1 ignored). The 9c-validation runtime checks are 47/47 PASS and the robust eigenvalue diagnostic's required checks are 10/10 PASS. 11j and 11k additionally verify their inputs by SHA-256 before and after analysis.

Later stages report their own runtimes and audits: the M14A design checks needed no time evolution (3.455 s after build, 11/11 PASS); M14 Plan B propagated a single 3080-step baseline prefix to t=7.70 in 2177.4 s (36.29 min; construction 5.2 s, prefix 2169.8 s, two-state diagnostics/write 2.4 s) with 22/22 checks PASS and 15/15 hashes unchanged; M15A reports 40/40 numerical and provenance checks PASS. The two saved 384x384 states from Plan B are stored as row-major little-endian complex-f64 binaries with QWNRHO1 metadata (2,359,320 bytes each), individually hashed in m14_plan_b_state_files.csv. The supplementary C2 extension to t=10 reports a formal runtime of 682.479 s (11.37 min) with 28/28 mandatory numerical checks PASS, 15/15 existing-artifact reproduction checks PASS, SHA audit PASS, and eigensolver fallback count 0; it saves two unextracted C2 states (c2_pre_extraction_state_t850.bin, c2_pre_extraction_state_t1000.bin) in the same QWNRHO1 format, each with a recorded SHA-256.

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Repository and archive layout

The public GitHub repository is organized as a small entry point plus Milestone-specific reproduction archives:

README.md                 # integrated project overview
LICENSE                   # MIT license
reports/                  # optional browser-readable report copies, where provided
zip/                      # Milestone-specific complete reproduction packages
  *.zip

The reports/ directory is a convenience for reading selected reports in the browser. The reproduction packages in zip/ are the self-contained records used for downloading and reproducing a stage. A report may therefore appear both as a browser-readable copy and inside its corresponding ZIP.

After extracting the relevant ZIP, a typical Rust reproduction package has the following internal layout. Not every archive contains every later-stage file or binary:

src/
  operators.rs            # operators (M1)
  partial_trace.rs        # partial trace (M1)
  ergotropy.rs            # local ergotropy (M1)
  matrix.rs               # ComplexMatrix / C64
  error.rs                # PhysicsError
  liouvillian.rs          # column-major vectorization / superoperators (M2)
  propagator.rs           # dense matrix-exponential propagation (M2)
  diagnostics.rs          # state diagnostics / signed-power accounting (M3 / M3.1)
  time_dependent.rs       # time-dependent RK4 propagator (M5a)
  coherent_drive.rs       # coherent drive (M5b/5c line)
  coherent_drive_matching.rs
  matching.rs / protocol.rs / experiment.rs
  load_extraction.rs      # ideal local-unitary extraction (M6a)
  dephasing_kernel.rs     # exact dephasing kernel (M8c)
  bin/
    time_dependent_sanity.rs        # M5a
    local_noise_placement.rs        # M7a
    local_noise_damage_analysis.rs  # M7b
    ideal_partial_protection.rs     # M7c
    partial_end_protection.rs       # M7d
    chain_length_reachability.rs    # M8a
    n7_feasibility_probe.rs         # M8b
    dephasing_kernel_benchmark.rs   # M8c
    n7_noise_free_full.rs           # M9a
    n7_all_site_noisy_full.rs       # M9b
    fixed_total_noise_comparison.rs # M9c
    n7_t002_eigen_diagnostic.rs     # M9c diagnostic
    n7_fixed_total_validation.rs    # M9c validation (final source of truth)
    matched_n7_dt_halving_validation.rs    # M11L
    matched_n7_site_resolved_transport.rs  # M11M
tests/
  full_24d_short_time.rs            # M2.1 (ignored smoke test)
MILESTONE_*.md                      # per-stage reports
*.csv                               # per-stage outputs

This tree describes archive contents, not the GitHub repository root. The src/lib.rs module declarations are based on the actual files; the bin/ mapping and the Milestone 10-11 binaries are organized from the "generated files" sections of the respective reports. For a complete list of individual CSVs, see the end of each MILESTONE report in the relevant ZIP.

---

Detailed reports and data

The filenames below refer to files inside the relevant Milestone reproduction ZIP. Some reports may also be mirrored under reports/ for browser reading. Each report lists its own full CSV set.

Minimal pointers for the load-bearing results:

• Foundations & central comparison: MILESTONE_4_RESULT.md, MILESTONE_5A/5B/5C, MILESTONE_6A_REPORT.md; Milestone_4-6a_研究結果ノート.pdf summarizes 4-6a.
• Noise location / protection: MILESTONE_7A/7B/7C/7D_REPORT.md.
• Chain length / feasibility: MILESTONE_8A/8B/8C_REPORT.md.
• N=7 full runs: MILESTONE_9A/9B_REPORT.md.
• 9c (source of truth + records): MILESTONE_9C_VALIDATION.md (final), MILESTONE_9C_DIAGNOSTIC.md, MILESTONE_9C_REPORT.md; key CSVs n7_fixed_total_validation_summary.csv, ..._checks.csv, ..._trajectory_comparison.csv, fixed_total_noise_final_comparison.csv, robust_eigen_diagnostic_unit_checks.csv.
• Fixed-total / XGamma: MILESTONE_10A/10B/10C_REPORT.md, MILESTONE_10_FINAL_REPORT.md.
• Equal-input N=3 vs N=7: MILESTONE_11C_PRECHECK_REPORT.md, MILESTONE_11D/11E_REPORT.md, MILESTONE_11F/11G/11H_REPORT.md, MILESTONE_11I_MOUNTAIN_MECHANISM_DESIGN.md, MILESTONE_11J_ASYMMETRIC_TRANSFORM_REPORT.md, MILESTONE_11K_POST_PEAK_DECAY_REPORT.md; key CSVs input_matching_interpolated_trial_summary.csv, equal_input_timeseries_shape_summary.csv, equal_input_peak_widths.csv, equal_input_curve_transform_models.csv, equal_input_asymmetric_transform_models.csv, equal_input_post_peak_decay_models.csv.
• Matched N=7 dt-halving (11L): MILESTONE_11L_MATCHED_N7_DT_HALVING_REPORT.md (scalar/trajectory comparison CSVs referenced therein).
• Matched N=7 site-resolved transport (11M): MILESTONE_11M_SITE_RESOLVED_TRANSPORT_REPORT.md; key CSV matched_n7_site_resolved_events.csv.
• N=3 W-peak branch check: MILESTONE_N3_W_PEAK_BRANCH_REPORT.md; key CSV n3_W_peak_branch_comparison.csv.
• Equal-input N=3..7 event comparison: MILESTONE_N3_7_EVENT_COMPARISON_REPORT.md (provided as FINAL_REPORT.md); key CSVs event_summary.csv, odd_even_temporal_comparison.csv, numerical_checks.csv.
• Post-peak ergotropy decomposition (12A): MILESTONE_12A_LOCAL_ERGOTROPY_DECOMPOSITION_REPORT.md (fixed-condition trajectory and decomposition/quality CSVs listed therein).
• Temporal relation (12B): MILESTONE_12B_TEMPORAL_RELATION_REPORT.md and M12B_PRECOMMIT.md (event, interval-change, and normalized-comparison CSVs listed therein).
• Load-chain correlation (12C): MILESTONE_12C_LOAD_CHAIN_CORRELATION_REPORT.md; key CSV m12c_interval_changes.csv.
• Counterfactual branch experiment (13A): MILESTONE_13A_FORMAL_SHORT_HORIZON_COUNTERFACTUAL_BRANCH_REPORT.md; key CSV m13a_formal_branch_timeseries.csv (728 formal rows).
• Branch temporal response (13B): MILESTONE_13B_WEAKER_COUPLING_TEMPORAL_RESPONSE_REPORT.md and M13B_PRECOMMIT.md; key CSVs m13b_input_audit.csv, m13b_maximum_differences.csv, m13b_interval_summary.csv, m13b_current_event_comparison.csv, m13b_energy_decomposition.csv, m13b_normalized_delta_timeseries.csv, m13b_start_time_comparison.csv.
• Coupling-control design (14A, design only): MILESTONE_14A_COUPLING_CONTROL_AND_SWITCHING_WORK_DESIGN.md (no formal control trajectory; the 11-row no-propagation unit-check CSV is described therein).
• Switching-work accounting (14 Plan B): MILESTONE_14_PLAN_B_SWITCHING_ACCOUNTING_REPORT.md; key CSVs m14_plan_b_switching_work.csv, m14_plan_b_classification.csv, m14_plan_b_scale_diagnostics.csv, m14_plan_b_state_files.csv.
• End-point extraction accounting (15A): MILESTONE_15A_ENDPOINT_EXTRACTION_ACCOUNTING.md; key CSVs endpoint_extraction_comparison.csv, gross_work.csv, energy_ledgers.csv, numerical_checks.csv.
• C2 extension to t=10 (supplementary, no new Milestone number): MILESTONE_C2_EXTENSION_TO_T10_COMPARISON.md and C2_EXTENSION_INPUT_AUDIT.md; key CSVs c0_c2_extended_summary.csv, c0_c2_extended_comparison_timeseries.csv, c0_c2_extended_time_area_summary.csv, c0_c2_extended_difference_events.csv, c2_extension_numerical_checks.csv, c2_extension_input_manifest.csv.

Reading conventions for the CSVs: W_time_area/E_time_area are time-areas of state quantities (not cumulative work or input); W/Ein is not an overall efficiency; usable_fraction is ergotropy over load energy; XGamma is a diagnostic, not a loss/efficiency. W_coh/W_pop are a descriptive energy-basis split of the load ergotropy, not independent conserved quantities; mutual information is a total-correlation observable in nats, not work, ergotropy, or entanglement; W_coh time-area is a time-integral of a state quantity, not cumulative extracted work. Switching work (W_switch_on_system/W_switch_out) is an ideal instantaneous Hamiltonian-jump ledger, not an actuator or controller cost; W_load_gross and W_operation_out_total are single-shot ideal-extraction ledgers. The state, switching, and extraction ledgers are reported separately and are never added or subtracted into a net benefit.

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Items requiring confirmation

No numerical or judgment contradictions were found between the two source READMEs when merging; Milestone 10 appeared in both and has been unified. The following points could not be resolved from the provided materials and are left open rather than guessed:

• Whole-suite Cargo test count "90/90". The 9c-validation runtime checks are 47/47 PASS and the robust eigenvalue diagnostic's required checks are 10/10 PASS, but a total of "90/90" referenced elsewhere for the entire Cargo test suite cannot be confirmed from the materials and is not adopted here.
• Milestone 11g release-test count. The 11g report records formatting and analysis execution as PASS but does not state a numerical release-test count.
• Two distinct N=7 W_max values are not a contradiction. 3.8081e-3 (M9c fixed-total, Omega=0.2) and 3.3853e-3 (M11g equal-input, Omega~0.18748) belong to different trajectories and conditions; readers should not conflate them. Noted here to avoid misreading.
• M14/M15A binary names and repository paths. The 14A, 14 Plan B, and 15A reports identify their source and output paths on the author's working machine but do not state repository-relative binary names, so no entries were added to the bin/ map in Repository layout rather than guessing.
• The M14A unit-check CSV and the M15A numerical_checks.csv are distinct artifacts. 14A records 11/11 no-propagation unit-check rows; M15A's numerical_checks.csv holds the 40/40 extraction-stage checks. An earlier-milestone file of the same generic name exists for the N=3..7 event comparison; the three should not be conflated when collecting CSVs.
• A numerically zero switching work is not a cost statement. W_switch_on_system values of order 1e-15/1e-16 are below the 1e-12 audit tolerance for the modeled Hamiltonian jump only. No actuator, ramp, bandwidth, or device-cost model exists in the project, so no controller-cost conclusion can be drawn from them.

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Citation and license

License: MIT (see Cargo.toml / LICENSE). When citing, please reference this repository and the specific MILESTONE_*.md report(s) underlying the result used, since each report defines the exact conditions and reservations for its numbers.