Absolute zero-mode tolerance EPS_GAP breaks D1/D3/D4/D16/D19/D20 under rate rescaling — false A11/F4 candidate and negative gap

[marcohost33-maker]

Summary

The zero-mode filter |lambda| > EPS_GAP with absolute EPS_GAP = 1e-10 is duplicated across five modules — diagnostics/spectral.py (D1/D3/D4), diagnostics/lep.py (D16), diagnostics/mpemba.py (D19/D20), diagnostics/nonnormality.py, _zhou.py (D24). A Liouvillian carries rate dimension, so this is the same defect class as #101, but located in the spectral core: D1 feeds D17 gap_rate_consistency, kappa_trans, the F5 reach leg and gap_to_gns_ratio.

Both failure directions were reproduced with running code on main@f137053 (not read — measured).

claim_status: pending.

Direction A — small c: false A11/F4 Mpemba candidate

Amplitude-damped qubit (H = 0, jump sigma_-), rho_0 = |+><+|, pure unit rescale L -> cL:

c	D1 gap	gap/c	D19 c_1	is_mpemba_candidate
1e-10	0.0	0.0	0.0	True
1e-5	5.0e-06	0.5	0.5	False
1	0.5	0.5	0.5	False
1e10	5.0e+09	0.5	0.5	False

At c = 1e-10 every genuine mode falls below the absolute floor, so _slowest_mode() returns None, overlap_c1 returns 0.0, and 0.0 < overlap_threshold = 1e-4 fires the candidate flag. The issue-#68 non-triviality guard cannot correct this: it also depends on _slowest_mode, so it silently reports trivial_overlap = False.

End-to-end verdict flip (diagnose(), same physics, only the rate unit changed):

c=   1e+00  a_class=A12  f_family=none  verdict=NOT_EXCLUDED tier=EXPLORATION   c1=5.000e-01 cand=False
c=   1e-10  a_class=A11  f_family=F4    verdict=CANDIDATE    tier=CONFIRMATION  c1=0.000e+00 cand=True

F4 is the highest-priority rung of the decision ladder, so this over-claim shadows every other mechanism. A textbook amplitude-damped qubit is promoted to a quantum-Mpemba candidate by a change of units alone — the exact over-claim class the library exists to prevent.

Direction B — large c: negative spectral gap

The numerical zero mode scales as ~eps * ||L||, so it eventually exceeds a fixed absolute floor and is counted as a genuine mode. Rabi-driven amplitude-damped qubit (H = 0.7 sigma_x, jump sigma_-, rate 0.4), true gap/c = 0.2:

| c | smallest |lambda| | D1 gap | gap/c |
|---|---|---|---|
| 1 | 1.17e-16 | 2.00e-01 | 2.00e-01 |
| 1e6 | 1.63e-10 | 1.63e-10 | 1.63e-16 |
| 1e10 | 1.57e-06 | -1.43e-06 | -1.43e-16 |
| 1e12 | 5.79e-05 | -5.76e-05 | -5.76e-17 |

A negative gap is not merely inaccurate, it is contract-violating: Delta = -max{Re lambda : lambda != 0} is non-negative for any GKSL generator. Downstream, gap <= 0 unconditionally sets _psr_fires = True in the F5 branch, so this also feeds the gapless blind spot already noted in #101.

Why the existing conformance suite did not catch it

tests/test_scale_conformance.py (#101 slice B) covers the new scale-relative diagnostics D8b/D10b/D13 and the rate_scale helper. The spectral layer (D1/D3/D4), D16, and the Mpemba layer (D19/D20) are outside its scope, so the defect sits in the gap between the suite and the classifier.

Correct fix (canonical, no free parameter)

Unlike the #101 F5 threshold — where the replacement value cannot be derived without a calibration study — this correction has a canonical form and needs no calibration:

1. One shared zero-mode tolerance derived from the spectrum itself. The functions receive only eigenvalues, not L, so the natural scale is the spectral radius max|lambda|: homogeneous of degree one, unitary-similarity invariant, and consistent with numerics.scale.rate_scale.
2. tol = rtol * max|lambda| with rtol = 1e-10. At max|lambda| ~ O(1) this reproduces the historical absolute floor, so anchor behaviour is expected to be unchanged (must be verified byte-identically, not assumed).
3. Preserve an atol opt-in for the legacy absolute floor, mirroring the resolution of Deep-review follow-ups 2026-07-12: methodology-gated findings (D7b formula, F5 Henrici scale-dependence, AR(1) correction, input-hash framing, steady-state tolerance semantics) #97 item 5 in fix: scale-relative steady-state tolerance, dense + sparse guard (#97 item 5) #99.
4. Clamp the gap at zero with a fail-closed rationale, so a round-off-dominated spectrum can never report a negative gap.
5. Extend the conformance suite to D1/D3/D4/D16/D19/D20 over c in {1e-10, 1e-5, 1, 1e5, 1e10}, including the end-to-end verdict invariance that this issue's repro exercises.

This is a numerical-correctness fix in the fail-closed direction, not a threshold change: no classifier threshold is re-tuned, and the F5/A10 gate switch remains gated on #101 slice C.

Refs #101 (same defect class, disjoint location), #102 (the false A11 path is exactly the priority-shadowing failure mode tracked there).

[marcohost33-maker]

Fix implementiert in PR #107 (472c4b2) — beide Richtungen behoben, Anker unverändert

Die kanonische Korrektur ist umgesetzt: numerics.scale.spectral_zero_tolerance() leitet die Schwelle aus dem Spektrum selbst ab (ZERO_MODE_RTOL * max|lambda|, Spektralradius — homogen Grad 1, unitär-ähnlichkeitsinvariant, konsistent mit rate_scale). Alle sechs Call-Sites sind umgestellt: spectral.py (D1/D3/D4 + has_complex_pairs), lep.py (D16 — sowohl Koaleszenz-Clamp als auch Kandidaten-Fenster), mpemba.py (D19/D20), nonnormality.py (D9 Petermann), _zhou.py (D24).

Nachgemessen, nicht behauptet — dieselben Systeme wie im Issue:

Grösse	vorher	nachher
gap/c, c ∈ {1e-10 … 1e10}	0.0 bei c=1e-10, sonst 0.5	0.5 konstant
gap/c, Rabi-System, c bis 1e12	1.6e-16 / −1.4e-16 / −5.8e-17	0.2 konstant, nie negativ
D19 c_1 bei c=1e-10	0.0 → cand=True	0.5 → cand=False
End-to-end Verdict bei c=1e-10	A11/F4 CANDIDATE	A12/none NOT_EXCLUDED (= c=1)

Anker-Beweis: Bei Einheitsskala reproduziert die relative Schwelle den historischen absoluten Floor, daher bleiben alle 21 Anker-Regressionen und die volle Suite unverändert grün — nur reskalierte Generatoren ändern sich. Das war die tragende Annahme des Fixes und ist geprüft, nicht unterstellt.

Zwei bewusste Design-Entscheidungen:

1. Kein Clamp bei Null in D1. Nach dem skalenrelativen Filter ist ein positives Re(lambda) kein Rundungsartefakt mehr, sondern ein echt instabiler (Nicht-GKSL-)Modus. Ein Clamp würde ein stilles Versagen gegen ein anderes tauschen. Der negative Gap bleibt für diesen Fall sichtbar und ist als Kontrakt gepinnt (test_unstable_generator_still_reports_a_negative_gap).
2. Legacy-Floor als atol=-Opt-in auf jeder betroffenen Funktion erhalten, analog zu fix: scale-relative steady-state tolerance, dense + sparse guard (#97 item 5) #99 für die Steady-State-Toleranz. Ein Diskriminierungstest belegt, dass das Opt-in den alten Defekt tatsächlich wiederherstellt — ohne ihn würde auch ein No-op-„Fix" die Invarianz-Assertions bestehen.

Neue Konformanzsuite tests/test_zero_mode_scale.py (33 Tests, c ∈ {1e-10 … 1e12}) schliesst die Lücke, durch die der Defekt fiel: die #101-Suite deckt die skalenrelativen Nachfolger D8b/D10b/D13 ab, aber nicht die Schicht, die entscheidet, welche Moden der stationäre Zustand sind.

Nicht angefasst: der F5/A10-Gate-Wechsel und die absolute henrici_eta > 1.0-Schwelle bleiben gemäss #101 Slice C hinter der präregistrierten Kalibrationsstudie. Dieser Fix ändert keine Klassifikator-Schwelle — er stellt nur sicher, dass die Eingaben der Schwellen einheiteninvariant sind.

CITATION.cff vermerkt #108 im Pending for the next cut-Block ausdrücklich als ergebnisändernde Korrektur (nicht als blosse Ergänzung), damit der nächste Release-Cut das nicht verharmlost.

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Generated by Claude Code

[marcohost33-maker]

Status check against the default branch — this issue stays open.

The fix described above lives in PR #107, which is still open (not merged). Verified on main (9892f25):

• src/liouscope/_consts.py:26 still defines the absolute EPS_GAP: Final[float] = 1.0e-10.
• src/liouscope/diagnostics/spectral.py still uses it as the default zero-mode tolerance at lines 44, 106 (liouvillian_gap), 118 (oscillating_mode_gap), 130 (spectral_spread) and 155 (has_complex_pairs).
• numerics/scale.py on main exposes only rate_scale — there is no spectral_zero_tolerance.
• tests/test_zero_mode_scale.py does not exist on main.

So both reported directions (false A11/F4 candidate at small c, negative gap at large c) are still reachable from a release cut of the default branch.

Note also that PR #114 was merged into the PR #107 branch, not into main, so that work is stacked behind the same open PR.

PR #107 currently reports all required checks green and is mergeable, but its branch is behind main (main advanced with #120 after the PR's last update). Closing #108 requires the merge into main; until then the defect is unfixed on the default branch.

[marcohost33-maker]

Status re-check against the default branch (2026-09-07) — still open, and the blocker changed.

Re-verified on main (81371dd, 2026-09-01) via the contents API, not from memory:

• src/liouscope/_consts.py:26 — EPS_GAP: Final[float] = 1.0e-10 unchanged (absolute).
• src/liouscope/diagnostics/spectral.py — still the default zero-mode tolerance at lines 44, 106, 118, 130 and in the has_complex_pairs test at 155.
• src/liouscope/diagnostics/lep.py:21 (D16) and diagnostics/mpemba.py:43, 65, 111, 159 (D19/D20) likewise.
• src/liouscope/numerics/scale.py exposes only rate_scale (line 44); no spectral_zero_tolerance.
• tests/test_zero_mode_scale.py → 404 on main.

So both reported directions remain reachable from a release cut of the default branch.

New since the 2026-08-25 check: PR #107 (claude/liouscope-repo-analysis-xgztfd → main) is still OPEN, but mergeable has flipped from clean to CONFLICTING / mergeStateStatus: DIRTY — main advanced past the branch (#120, #133). Landing this fix now needs a conflict resolution and a re-run of the checks, i.e. it is gated on the GitHub Actions budget freeze, not on any open design question.

One methodological note for whoever resolves it, from the external standard for this exact problem: the accepted scale-invariant form of a numerical-zero threshold is tol = max(m, n) * eps * ||A|| — LAPACK's deflation uses 8 * eps * max(|d|, |z|), NumPy/MATLAB's matrix_rank uses max(m, n) * eps(||A||_2) (numpy.linalg.matrix_rank, LAPACK slaed2). The PR's ZERO_MODE_RTOL * max|lambda| matches the scale-relative part, which is the correctness-bearing half. The dimension factor max(m, n) is the part it drops: with a fixed rtol = 1e-10 the threshold does not grow with the superoperator dimension d^2, so for large systems the accumulated round-off of a genuine zero mode can again cross a threshold that no longer accounts for problem size — the same failure shape as Direction B, only reached via dimension rather than via c. Worth pinning explicitly (either adopt the factor or record why a fixed rtol is preferred here), since #113 already reports a dynamic-range ceiling in this same filter.

No action taken beyond this comment: merging is out of scope under the freeze.