F5 evidence path: scale-relative D8/D10/D11b/D13 and mechanism-valid calibration

[marcohost33-maker]

Context

Follow-up to issue #97 item 2. The known absolute henrici_eta > 1.0 threshold is not the only rate-unit dependence in the A10/F5 path. Re-audit of main@b1a5681 found that the numerical D10/D11b/D13 estimators also contain absolute rate scales.

claim_status: pending; no classifier change in this issue until calibration and independent physics review.

Verified findings

D8

henrici_eta is rate-dimensioned and transforms as eta(cL)=c eta(L) for c>0.

D10

kreiss_constant() uses fixed grid constants:

• sigma_lo = 1e-3
• sigma_hi >= 1
• omega_max >= 1

The exact continuous-time Kreiss constant is invariant under positive time/rate rescaling, but this dimensional grid is not. A direct reproduction of the current algorithm on A=[[-1,20],[0,-2]] gave materially different estimates under A -> cA (approximately 1.00 at c=1e-10, 9.79 at c=1, 6.77 at c=1e10). This is grid error, not changed physics.

D11b/D12

resolvent_peak_curve() defaults to absolute sigma=1e-3 and omega_max >= 1; the peak and FWHM therefore compare different dimensionless offsets after a unit change.

D13

pseudospectral_radius() uses absolute eps=1e-3 plus absolute 1e-3 span floors. For positive c,

Sigma_eps(cL) = c Sigma_{eps/c}(L).

Thus a fixed absolute epsilon changes the perturbation family under L -> cL. In the same reproduction, the current pseudospectral_radius(cA)/(c*gap) changed from about 7.1e6 at c=1e-10 to 2 at c=1/1e10.

Mechanism mismatch

F5 currently consumes the maximum modulus of the epsilon-pseudospectrum. For slow/phantom relaxation this is not necessarily the relevant extremum: it can be dominated by fast modes far from the origin. The literature ties transient/phantom behaviour more directly to pseudospectral abscissa/intrusion, resolvent growth, localized or ill-conditioned modes, and direct propagator amplification. Henrici alone is generic non-normality evidence, not a proof of F5.

Correct staged solution

Slice A — shared scale and additive diagnostics; no verdict change

1. Define one positive operator rate scale with explicit zero-operator semantics, preferably a documented norm-based scale.
2. Add relative Henrici dep_F(L)/||L||_F while preserving legacy D8.
3. Reparameterise D10/D11b grids in dimensionless coordinates and convert back only for rate-valued outputs.
4. Define D13 using eps_abs = eps_rel * scale(L); record both values and grid/refinement metadata.
5. Add a pseudospectral abscissa or gap-directed intrusion diagnostic; do not silently redefine the existing D13 field.
6. Label coarse-grid results as estimates/lower bounds and provide convergence/refinement metadata.

Slice B — conformance suite

For c in {1e-10,1e-5,1,1e5,1e10}:

• dimensionless diagnostics invariant within declared tolerance;
• rate-valued diagnostics scale by c;
• D14 agrees after t -> t/c;
• zero, normal, gapless-normal and ordinary non-normal negatives remain non-F5;
• dense/sparse and basis-change metamorphic tests.

Slice C — preregistered classifier study

• calibration and holdout sets separated;
• known phantom/Jordan/skin positives;
• normal, detailed-balance, gapless-normal and non-phantom non-normal negatives;
• system-size families to expose Frobenius/dimension bias;
• FPR/TPR/invariance criteria declared before fitting;
• no tuning merely to preserve existing anchors;
• classifier switch only after independent physics review.

Release/claim gate

Until all dimensioned thresholds and absolute grids are removed from the F5 decision path, LiouScope must not claim that A10/F5 is invariant under a change of rate units. The honest interim status is research/candidate evidence with the limitation documented.

Refs #97 item 2.

[marcohost33-maker]

Additional state-of-the-art requirement: norm and perturbation structure

The scale fix is necessary but not sufficient. D8/D10/D11b/D13/D14 currently use the Euclidean/Frobenius geometry of the full vectorised operator space. Their numerical values depend on the chosen inner product/norm, and the extremising vector may be non-Hermitian, traceful or otherwise outside the physical set of density-matrix perturbations.

Required separation:

1. Mathematical non-normality diagnostics — unstructured Hilbert-Schmidt/2-norm quantities may remain, but must be labelled as such.
2. Operational relaxation diagnostics — evaluate state differences on the traceless-Hermitian subspace and report trace-distance or a justified GNS/KMS-weighted norm. CPTP evolution contracts trace distance between physical states, so a full-space 2-norm transient larger than one is not itself an operational increase in distinguishability.
3. Perturbation structure — ordinary pseudospectra allow arbitrary matrix perturbations. A physics claim should additionally study perturbations preserving at least Hermiticity and trace preservation, and ideally the GKSL/locality parameterization. Unstructured and structured results answer different questions and must not be conflated.

Consequences for F5 calibration:

• D14 full-space 2-norm amplification is corroborating evidence, not a necessary condition for phantom relaxation; gap/mixing discrepancies can arise through large expansion coefficients while physical trace distance remains contractive.
• Store the norm, subspace projection and perturbation class in diagnostic provenance.
• Include basis/representation invariance only under transformations preserving the chosen norm; arbitrary similarity transforms change pseudospectra and transient norms.
• Add structured-perturbation robustness as a later research slice rather than silently interpreting unstructured epsilon-pseudospectra as GKSL-robustness.

[marcohost33-maker]

Re-audit addendum: numerical estimator labels and fail-closed gapless semantics

Two further corrections are required before any D10/D13/F5 implementation slice.

D10 is not presently a Mitchell-global computation

The module docstring calls the current implementation a “Mitchell 2020 grid search”. Mitchell 2020 establishes globally convergent algorithms and globality certificates for computing the continuous-time Kreiss constant to arbitrary accuracy. LiouScope currently evaluates a finite hand-chosen (sigma, omega) grid. That result is a grid lower-bound estimate, not Mitchell's globally convergent algorithm and not a certified Kreiss constant.

Required change:

• rename/document the legacy value as kreiss_grid_lower_bound or equivalent;
• record grid ranges/resolution and whether the maximiser lies at an edge;
• add refinement convergence;
• only call a result kreiss_constant when a global or explicitly bounded computation is implemented.

Gapless F5 must not auto-fire on the reach leg

The current classifier sets _psr_fires = True whenever gap <= 0, then only gates on absolute Henrici. A vanishing/floored gap is not positive evidence of phantom relaxation; it can represent normal conserved dynamics, a degenerate stationary manifold, a numerical floor, or a critical system without the F5 mechanism.

Required interim rule: gapless or uncertified-gap inputs should abstain / mark F5 undefined unless separate positive evidence establishes the mechanism. They must not be converted into infinite pseudospectral reach automatically.

Controls must include:

• normal zero operator;
• normal pure-dephasing generators with conserved populations;
• semisimple degenerate stationary manifolds;
• defective Jordan-zero positive controls;
• gap-floor numerical adversaries.

This strengthens the positive-evidence burden already specified in #101. D14/F2 geometry is tracked separately in #103.

[marcohost33-maker]

Status 2026-08-08 — Slice A + Slice B implemented in draft PR #104

Slice A (additive; no verdict change):

1. ✅ Shared positive operator rate scale: liouscope.numerics.scale.rate_scale(L) = ||L||_F, documented zero-operator semantics (0.0), fail-closed on non-finite input, unitary-similarity invariant.
2. ✅ Relative Henrici henrici_relative = η_N/||L||_F ∈ [0,1] (D8b), legacy D8 preserved; clip-tolerance fail-closed per the 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 re-audit slice.
3. ✅ D10/D11b grids reparameterised in dimensionless coordinates (kreiss_grid_lower_bound / resolvent_peak_curve_scaled), converted back only for rate-valued outputs. Legacy values byte-identical.
4. ✅ D13 with eps_abs = eps_rel · rate_scale(L); both values plus grid metadata recorded; the existing pseudospectral_radius field is untouched (no silent redefinition).
5. ✅ New gap-directed intrusion diagnostic pseudospectral_abscissa(_rel) via single-sweep numerics.pseudospec.pseudospectrum_extent.
6. ✅ Coarse-grid results labelled as estimates/lower bounds: KreissGridEstimate carries grid ranges/resolution, a tie-tolerant edge-maximizer flag and refinement-convergence metadata; the legacy D10 docstrings no longer claim "Mitchell 2020" (re-audit addendum) — the value is documented as a grid lower bound without globality certificate.

Slice B (conformance suite, tests/test_scale_conformance.py): for c ∈ {1e-10, 1e-5, 1, 1e5, 1e10} — dimensionless diagnostics invariant (rtol 1e-8, measured drift ~1e-13 over 20 decades); rate-valued outputs scale by c; D14 agrees after t → t/c; zero / normal / gapless-normal (pure dephasing, conserved populations) / ordinary non-normal controls behave per the addendum oracles; unitary-basis invariance; NaN/inf fail-closed. 22 new tests; full suite 600 passed, coverage 94.68 %.

Deliberately NOT done (stays open in this issue):

• Slice C — preregistered calibration study, threshold selection, F5 gate switch, and the gapless-abstention rule from the addendum. All of these are classifier-influencing and remain gated on the calibration design + independent physics review. The new values are advisory-only (ADVISORY_EVIDENCE_KEYS, metamorphic non-influence pinned) with claim_status: pending.
• Norm/perturbation-structure separation (traceless-Hermitian subspace, structured perturbations) from the 2026-08-05 comment — that is D14/F2 geometry territory and overlaps D14 transient amplitude: remove steady-projector and norm-geometry confounds #103; not touched here.

Release gate honoured: docs now state explicitly that the A10/F5 verdict path is not yet rate-unit invariant (docs/explanation/layers-and-taxonomy.md, README status disclaimer).

---

Generated by Claude Code

[marcohost33-maker]

Unabhängige Gegenprüfung von PR #104 (Vero, 2026-08-08) — Slice A + B verifiziert, #104 gemergt

Nicht die Tests des PR nachgelesen, sondern die Behauptungen mit eigenem Code
nachgemessen — am Beispiel A=[[-1,20],[0,-2]] aus diesem Issue, über
c ∈ {1e-10, 1e-5, 1, 1e5, 1e10}.

Der dokumentierte Defekt reproduziert exakt. D10 legacy: 1.00 → 1.10 → 9.79 → 6.77 → 6.77 (das Issue nannte 1.00 / 9.79 / 6.77). D13 legacy
pseudospectral_radius(cA)/(c·gap): 7.07e6 → 72.1 → 2 → 2 → 2 (Issue:
~7.1e6 → 2). Die Befunde waren also nicht theoretisch, sondern messbar.

Die Korrektur greift.

Grösse	Spannweite über 20 Grössenordnungen
henrici_relative (D8b)	1.1e-16
kreiss_grid_lower_bound (D10b)	1.8e-15
pseudospectral_radius_rel	2.8e-17
pseudospectral_abscissa_rel	6.9e-18
D10 legacy (unverändert erhalten)	8.79

rate_scale: homogen Grad 1 (rel. Abweichung ≤ 1.5e-16), rate_scale(0) = 0.0,
unitär-ähnlichkeitsinvariant auf 12 Stellen, fail-closed bei nicht-finiter
Eingabe (ValueError). Die ratenbehafteten Legacy-Felder zeigen weiterhin den
alten Defekt — das ist gewollt (additiv, keine Verdikt-Änderung) und sollte bei
Slice C nicht versehentlich als „behoben" gelesen werden.

Diskriminierung der Konformitätssuite belegt. rate_scale konstant 1.0 →
9 Tests rot. scale = rate_scale(L_super) in nonnormality.py auf 1.0 →
D10b-Invarianz bricht (Spannweite 9.03 statt 1.8e-15), Suite meldet
test_kreiss_scaled_invariant_under_rate_rescale und
test_kreiss_scaled_zero_operator_is_exactly_one rot. Restore → 22/22 grün.

Eigener Prüffehler, offengelegt: mein erster Bruchversuch ersetzte
rate_scale(A) — dieses Muster steht nur in Docstrings, die echten Aufrufe
lauten rate_scale(A, name="A") bzw. rate_scale(L_super). Der Test lief
grün und ich hätte beinahe eine Lücke in der Suite gemeldet, die es nicht
gibt. Erst der korrekt gesetzte Bruch war aussagekräftig.

Volle Suite auf dem PR-Zweig: 599 passed. Der eine Fehlschlag
(test_version_single_source) ist ein Artefakt meiner Umgebung — site-packages
trug liouscope 0.5.0, mein PYTHONPATH zeigte auf den Worktree. CI grün.

Offen bleibt Slice C (präregistrierte Kalibrierstudie, Kalibrier-/Holdout-
Trennung, FPR/TPR vorab deklariert) und der F5-Gate-Wechsel — beides
ausdrücklich hinter unabhängiger Physik-Review, siehe #102. claim_status
bleibt pending.

[marcohost33-maker]

State audit at main@9892f25 (2026-08-21) — slice A/B landed, slice C untouched

Re-checked against the code on the default branch, not against the PR description.

Landed (PR #104, merged 2026-08-08, merge commit 5054274). Slice A and slice B are on main: src/liouscope/numerics/scale.py exists, and henrici_relative (D8b), kreiss_scaled / kreiss_grid_lower_bound (D10b), pseudospectral_radius_rel (D13) and pseudospectral_abscissa_rel are computed and serialised (src/liouscope/diagnostics/classification.py:308-311), with the rate-unit conformance suite in tests/test_scale_conformance.py.

Still open — the F5 decision path is unchanged. All four scale-relative quantities sit in ADVISORY_EVIDENCE_KEYS (classification.py:84-100) and drive no verdict, exactly as this issue requires until slice C. The rung that actually decides A10/F5 still reads:

• classification.py:376-378 — the reach leg compares the absolute pseudospectral_radius (D13 with absolute eps=1e-3) against gap;
• classification.py:388 — the gate is ev["henrici_eta"] > 1.0, the rate-dimensioned absolute threshold this issue opened on;
• classification.py:387 — _psr_fires = True unconditionally when gap <= 0, i.e. the gapless auto-fire from the 2026-08-05 re-audit addendum is still in place; the required abstain / F5 undefined semantics are not implemented.

The same three properties hold on the head of the open PR #107 (4a8ae9c), where the rung was refactored into a declarative spec and the limitation is documented in the condition strings rather than fixed — which is the correct scope for that PR.

Therefore the release/claim gate of this issue remains active, and the issue is correctly open. What is left is unambiguous:

1. slice C — preregistered calibration/holdout design, control families, FPR/TPR criteria declared before fitting;
2. the gapless abstention rule from the addendum;
3. the norm / perturbation-structure separation from the 2026-08-05 comment (traceless-Hermitian subspace, structured perturbations), which overlaps D14 transient amplitude: remove steady-projector and norm-geometry confounds #103;
4. the classifier switch itself, gated on independent physics review together with Classifier semantics: replace heuristic confidence with calibrated, hypothesis-wise evidence #102.

No code or classifier change is proposed here; this comment only pins the current state so the remaining scope is not re-derived.

[marcohost33-maker]

Re-check at main@81371dd (2026-09-07) — audit of 2026-08-25 still holds, plus external anchors for slice C

Verified against the default branch via the contents API, not against PR bodies or earlier comments.

Nothing in the F5 decision path has moved since the last audit. The most recent commit touching the classifier is 5af088c (2026-08-09); 9892f25, c18a48d, 386041b, 81371dd are license/CI/docs only. The three defects named in this issue and its 2026-08-05 addendum are verbatim present on main:

• src/liouscope/diagnostics/classification.py:388 — the gate is still the rate-dimensioned absolute ev["henrici_eta"] > 1.0, i.e. the exact D8 scale dependence this issue opened on.
• src/liouscope/diagnostics/classification.py:376-377 — the reach leg still divides the absolute pseudospectral_radius by gap. That field is produced with a hard-coded absolute eps = 1.0e-3 (src/liouscope/diagnostics/resolvent.py:134, :144, :158); the scale-relative pseudospectral_radius_rel exists at :202 but is advisory-only.
• src/liouscope/diagnostics/classification.py:387 — _psr_fires = True unconditionally when gap <= 0. The abstain / F5 undefined semantics required by the re-audit addendum are not implemented.

Slice A/B (PR #104, merged into main 2026-08-08, merge commit 5054274) remain correctly landed and correctly advisory (classification.py:84-100). This issue is therefore correctly open: the remaining work is slice C, which is classifier-influencing and gated on the calibration design plus independent physics review — not a leftover.

External anchors for the slice C design (searched 2026-09-07)

1. Kreiss constant labelling. Mitchell, Computing the Kreiss Constant of a Matrix, SIAM J. Matrix Anal. Appl. (final version 2020-12-22, arXiv:1907.06537, https://epubs.siam.org/doi/10.1137/19M1275127) gives the first globally convergent algorithms to arbitrary accuracy, with O(n^6) standard cost and interpolation-based globality certificates (arXiv:1910.01069). This confirms the addendum: a fixed (sigma, omega) grid is a lower-bound estimate, and kreiss_grid_lower_bound is the honest name until a certified computation exists.
2. Structured vs. unstructured perturbations. Scaling of pseudospectra in exponentially sensitive lattices, Phys. Rev. Research 7, L032043 (2025, arXiv:2409.12036, https://doi.org/10.1103/PhysRevResearch.7.L032043) separates complex from structured pseudospectra and shows they scale differently with system size. This is direct support for the 2026-08-05 norm/perturbation-structure comment: unstructured epsilon-pseudospectra must not be read as GKSL robustness, and the system-size families in slice C need the structured variant to avoid a dimension artefact.
3. Threshold selection. A tailored fit that doesn't fit all: the problem of threshold overfitting in diagnostic studies (PubMed 40966727, 2025, https://pubmed.ncbi.nlm.nih.gov/40966727/) documents that deriving and evaluating a threshold on the same data inflates sensitivity/specificity and is routinely misfiled as low risk of bias; TRIPOD-style external validation with calibration slope/intercept is the reporting standard. The slice C requirement "calibration and holdout separated, criteria declared before fitting, no tuning merely to preserve existing anchors" matches this and should stay non-negotiable.
4. Slice B methodology. The rate-rescaling suite is textbook metamorphic testing against the oracle problem; current practice measures a metamorphic relation's worth by mutation adequacy (semantic mutation metrics for MR adequacy in scientific computing, IEEE TSE 2025, arXiv:2605.17437). The suite's discrimination was already shown by mutation on 2026-08-08 — worth keeping that as the acceptance standard for the slice C controls too.

No source file was changed by this check.