Toy decay identifiability → CMS Open Data 4‑lepton pairing audit

Bottom line

This is a two‑part research record. Stages 1–3 use a toy model to study when the decay history X→Y1Y2→4 particles becomes indistinguishable from a direct four‑body decay.
Stage 4 applies the same closest‑to‑Z pairing heuristic to CMS 2012 Open Data. The central Stage‑4 finding is that on 101 same‑flavour (4e/4mu) simulated events, the closest‑to‑Z rule agrees with the generator ancestry record on 59/101 (58.4%) and disagrees on 42/101 — and the 95% interval on that point estimate still contains chance (0.5).
For identical‑flavour four‑lepton final states, this is agreement with one internal generator bookkeeping record rather than a unique observable history. The real‑data work consists of an exploratory examination of 26 four‑lepton candidates, a Z‑peak validation check, and modestly tighter selection criteria for four candidates in the 120–130 GeV range; these are pipeline checks rather than a Higgs measurement.
A full ancestry audit of all 42 mismatches (§5m) traced every mother/daughter edge and found 0/42 pairing‑label changes and 0/42 ambiguous labels — this validates how the labels were built, not a unique physical pairing, so the 59/101 agreement stands unchanged. A follow‑up official JHUGenMELA interference audit (§5n) of the same fixed 101 events found relative identical‑lepton interference of median +4.54% in the 42 mismatch events versus +10.03% in the 59 matched events, a difference of −5.49 percentage points whose bootstrap interval [−15.80, +6.16] comfortably includes zero — this sample gives no stable evidence that interference explains the mismatches.
Three small detector‑covariance follow‑ups round out this update. A fixed‑three‑event local hybrid‑covariance benchmark (§5o) and a related 91‑complete‑case screen (§5p) found the exact nonlinear boundary distance within 0.4% of the local linear estimate for the two events it could evaluate; a prospective, target‑unseen two‑event pilot (§5q) found a fixed quadratic correction outperformed the linear one for both events (absolute errors 0.0173σ vs 0.306σ for 4e, and 0.000343σ vs 0.0228σ for 4mu). A larger, prespecified fixed‑16 extension (§5r) — eight independent‑file candidates per channel, no NA replacement — then mechanically compared 14 paired events (4e 8, 4mu 6) and found the quadratic correction improved on the linear one for 14/14, with median DeltaE = 0.0794076 and all 14 signed residuals positive. None of these establish a global detector‑covariance distance, a measured mispairing probability, a population‑wide bias, or a unique physical pairing — they remain local, small‑sample sensitivity checks over a deliberately selected near‑distance cohort.

1. Research question

Two nested questions.

1. Toy (Stages 1–3): Under a controllable artificial generator + detector, at what resonance width does the sequential‑decay history stop being recoverable — both as an A/B classification and as a correct 4‑particle pairing — and which pieces of information vanish first?
2. Open Data (Stage 4): When the fixed closest‑to‑Z heuristic is applied to CMS simulated 4‑lepton events, how often does it match the generator ancestry label, and — crucially — is that label even a valid ground truth for identical‑flavour final states?

2. Project timeline

3. Data, fixed ranges, event counts

Fixed physics constants used throughout Stage 4: mZ = 91.2 GeV, score S = ((mZ1−91.2)/10)², softmax T = 1.0, reco–gen DeltaR < 0.1 (truth_summary.json).

4. Method and main fixed conditions

• Toy (1–3): RAMBO 4‑body (A) vs Källén two‑body sequential (B); identical toy detector response; RF/logistic/histogram‑likelihood classifiers; width, missing‑rate, mass‑ratio scans; ablations and permutation importance (predictive dependence, not causal effect).
• Stage 4 pairing: For a 4e/4mu event with 2 positive + 2 negative same‑flavour leptons, exactly 2 valid OSSF complete matchings exist (the third uses a same‑sign pair).
  The fixed rule ranks each matching by how close its mZ1 is to 91.2 GeV. 2e2mu has only 1 OSSF matching and is excluded from the primary agreement calculation, reported separately.
• Generator "truth": a matching is "generator‑record‑correct" only when both reco pairs join leptons carrying the same generator‑Z index and the two pairs come from different Z indices — i.e. it follows generator ancestry, not proximity to 91.2 GeV.
• JHUGenMELA interference audit: for each fixed same-flavour event, P_full is the official JHUGenMELA complete decay matrix element. P_A and P_B are controls formed by retaining the measured four-vectors while temporarily assigning the two forced OSSF pairs different charged-lepton flavours.
  The frozen same-normalization definition is P_no-interference = 0.5(P_A+P_B) and relative interference = (P_full−P_no-interference)/P_no-interference. These controls are not event-level probabilities that A or B is the true pairing.

5. Main results

5a. Toy identifiability (Stages 1–3)

The ability to distinguish the two decay models and the ability to recover the correct pairing degrade at different rates (STAGE3_FINAL_REPORT.md). All values are conditioned on the toy generator and detector; they do not imply the existence of a corresponding physical process or any experimental discovery sensitivity.

5b. Stage 4 — closest‑to‑Z vs generator record (101 events)

The 95% interval includes chance (0.5), so this fixed sample does not establish that the rule performs better than chance. The mismatches are not merely close ties: in all 42 mismatched events, even the generator‑record pairing places the more on‑shell Z candidate more than 20 GeV away from 91.2 GeV (FAILURE_ATLAS_REPORT.md).

5c. Label‑validity audit

The existing saved ROOT stores only mother(0)‑derived Z/H indices — not mother counts, statuses, or generator four‑momenta — so from these files alone the 42 labels cannot be confirmed correct or incorrect. Current exclusive primary classification: D = 42, A = 0, B = 0, C = 0 (D = undetermined with the saved information) (LABEL_VALIDITY_AUDIT.md). A full‑family‑tree canary on the top‑10 highest‑gap failures found 0/10 label changes under all‑mother traversal or Z‑copy collapse; those labels were internally consistent, and the canary's primary class is C = 10 — internally consistent bookkeeping, but no unique physical pairing for identical leptons (TOP10_CANARY_REPORT.md).

5d. Pairing‑rule strength scan

Weight w∈[0.5,1.0] on C(w)=w·d1²+(1−w)·d2² (w=1 = legacy Z1‑only):

Agreement with the generator record is 54/101 at w=0.50 and 59/101 at w=1.00; the scan's maximum was 60/101, but that is a post‑hoc maximum over the same 101 events and is not advertised as a performance improvement (pairing-rule-strength-101_REPORT.md).

5e. Three physically motivated rules

All three agree on 22/101. Agreement with the generator record: on‑shell‑Z 59/101, symmetric propagator 59/101, compact ΔR 43/101. These are explainable fixed heuristics, not the full H→ZZ*→4l matrix element (the symmetric‑propagator rule is propagator‑only: no phase space, couplings, spin correlation, or identical‑particle interference) (physical-pairing-rules-101_REPORT.md).

5f. Two events where the mass-based rules disagree

The on‑shell‑Z and symmetric‑propagator rules agree on 99/101; the two exceptions are the only same‑flavour events where they select different pairings. Both are 4e events near m(4l) ≈ 125 GeV, so two simultaneous ~91 GeV Z candidates are kinematically impossible — the disagreement is the trade‑off between a slightly more on‑shell‑like Z1 and a higher Z2 mass, sometimes with a more compact angular pairing.

• 1:604:181002 is a genuine near_tie: the two Z1 distances differ by only 0.915 GeV and the continuous weight scan flips at w = 0.980285, close to the Z1‑only endpoint. Option B buys 0.915 GeV on Z1 but drops Z2 from 46.95 to 29.13 GeV.
• 1:608:182172 is rule_dependent, not near_tie: the Z1 distances differ by 3.441 GeV and the scan flips at w = 0.763425. The on‑shell‑Z rule takes the closer Z1 (option B); the symmetric‑propagator and compact‑ΔR rules take the higher Z2 and smaller mean within‑pair ΔR (option A).
• The two events split their generator‑record agreement: the symmetric‑propagator rule matches the record for 181002, the on‑shell‑Z rule matches it for 182172. One event each — so they do not favor either mass rule overall. This is a concrete rule‑design boundary illustration, not a measurement: no systematic uncertainty, background tail, matrix element, or interference was computed (two-mass-rule-edge-events_REPORT.md).

5g. Pairing-rule geometry (final summary)

A consolidated view of the rule‑comparison results above — a visualization of prior numbers, not a new analysis, physical law, or performance gain (final-pairing-geometry-101_REPORT.md).
The 4×4 agreement matrix reproduces §5b/§5e (generator record vs on‑shell‑Z / symmetric propagator / compact ΔR = 59 / 59 / 43; the two mass rules agree 99/101; each mass rule agrees with compact ΔR only 23/101). The one new breakdown is how the four rules vote per event:

Because only two valid OSSF pairings exist per event, a four‑way (ABCD) split is impossible. The decision‑boundary panel places each event in Δ(d1²) vs Δ(d2²) coordinates (the difference between the two candidate matchings, not a single selected (mZ1, mZ2)); a weight‑w mass rule flips across the line w·Δd1² + (1−w)·Δd2² = 0, and points are colored by the fixed classification (stable 65, rule‑dependent 34, near‑tie 2). Rule agreement or disagreement here does not mean physical correctness or error, and the generator record remains an internal reference, not an observable rule or a unique physical truth.

5h. Pairing stability, mass degeneracy, and local boundary geometry

All of the following are derived analyses of the same fixed 101 events — no new downloads, no new CMS processing, no independent validation sample. The perturbations are artificial numerical stress tests; their widths are not CMS detector resolution, calibration, or systematic uncertainties, and no per-species resolution model was used.

Fixed one-dimensional perturbation audit. Search bounds were ±20% for a common pT scale and for a single lepton's pT, and ±100 mrad for a single lepton's eta or phi. Within these bounds an on‑shell‑Z flip was found for common pT in 6/101, one‑lepton pT in 5/101, one‑lepton eta in 5/101, and one‑lepton phi in 3/101 events. These are minima inside the specified one‑dimensional families, not global multidimensional distances, and events with no boundary found are not asserted to be infinitely far from one (minimum-observable-perturbation-101_REPORT.md).

Rare Δm tail. With Δm = |mZ1,A − mZ1,B| and relative Δm = Δm / ((mZ1,A + mZ1,B)/2), over the 101 events absolute Δm has minimum 0.882 GeV, median 36.936 GeV, maximum 77.051 GeV.

The sample is entirely M‑125 Higgs signal MC, so this does not show that degeneracy concentrates near 125 GeV, and events outside 118–130 GeV are not an experimental background sideband (delta-m-distribution-101_REPORT.md).

Fixed Δm-bin behavior (counts, not rates alone — the first bins are tiny):

All 5 events with Δm < 5 GeV flipped under the one‑lepton‑pT ±20% audit; none of the 96 with Δm ≥ 5 GeV did. This is structurally expected, because Δm is closely related to the on‑shell‑Z decision margin itself — it is not an independent predictive or discovery result. The 5 GeV split was not preregistered and is not a natural physical threshold; the accompanying tests are exploratory and unadjusted (delta-m-bin-rule-behavior-101_REPORT.md).

Continuous local-pT boundary. Using fractional coordinates u_i = ΔpT_i / pT_i for the four leptons and d_local = g(0) / ‖∇_u g(0)‖₂, where g is the on‑shell score difference between the unselected and selected matching: 79/101 events have both candidate Z1 masses below 91.2 GeV, and 22/101 straddle it. For the 79 both‑below events Δm equals the on‑shell score margin algebraically. Spearman correlations with the local boundary distance:

This strong correlation is mainly structural, not an independent validation: Δm supplies the numerator for the 79 both‑below events, and the remaining spread comes from event‑specific gradient norms. For straddling events Δm is not the exact score margin, and the actual score margin tracks the local distance more directly. For the two closest events, 1:604:181079 gives a linear local pT L2 of 1.748% (nonlinear boundary 1.748%, rank 1/101) and 1:604:181002 gives 1.838% (nonlinear 1.838%, rank 2/101) — linear/nonlinear agreement better than 0.01%. A nonlinear boundary was confirmed for 16 local candidates inside a per‑lepton ±20% box; the remaining distant linear extrapolations must not be read as exact global minima (delta-m-local-pt-boundary-101_REPORT.md).

Four-event multivariate audit — a different coordinate system. The earlier top‑four audit used normalized 12‑dimensional coordinates (pT change ÷ 20%; eta/phi change ÷ 0.1 rad) and let pT and angles move cooperatively. Its "combined audit distance" and the pT/angle components printed in that table therefore belong to the combined pT+angle solution and are not competing values against the later pT‑only L2 distances. In local_boundary_results.csv the pT‑only rows for 181079 and 181002 reproduce the later figures of ≈1.748% and ≈1.838%, while the smaller pT numbers in the top‑four summary (≈1.577% and ≈1.485%) come with simultaneous angle components (local-multivariate-boundaries-top4_REPORT.md).

181002 versus 181079. The two events share the two smallest absolute and relative Δm values among all 101, the two shortest local on‑shell pT boundary distances, both candidates lying below 91.2 GeV, and m4l near 125 GeV. The saved gradient comparison gives a cosine similarity of about 0.845 using same‑index bookkeeping labels; because cross‑event lepton indices are not a permutation‑invariant physical correspondence, this is descriptive similarity only, not evidence that the two events form a physical class (comparison-181002-181079_summary.json).

5i. Real‑data pipeline validation

Input‑quality flag: of the 16 selected electrons in the four candidates, 14 had isolation at the missing‑value sentinel −999 (isolation_available = False); the original isolation < 0.40 cut passes −999, so isolation was effectively inert for these candidates (strict-cuts-125gev_REPORT.md, verified in selected_electrons.csv).

5j. Analytic fractional-pT gradient audit

This subsection explains, in closed form, the local gradient of the on-shell score difference used in §5h, evaluated on all fixed 101 events. It is deliberately narrow in scope:

• fixed 101 events only (79 both-below, 22 straddling);
• fractional-pT coordinates u_i = ΔpT_i / pT_i, with η and φ held fixed;
• piecewise-local analysis with the active Z1 pair held fixed;
• not a detector-resolution, detector-covariance, or global-boundary analysis (analytic-gradient-audit-101_REPORT.md, DERIVATION.md).

Structure. The two competing active Z1 pairings of an event share exactly one lepton. Ordering the four fractional-pT coordinates as (shared, other-only, chosen-only, unused) and writing the two candidate Z1 masses as m_a (the "other" matching) and m_b (the "chosen"/selected matching), with on-shell signs q_k = sign(m_k − m_Z), the massless closed form gives

grad g = 1/2 ( q_a m_a − q_b m_b,  q_a m_a,  −q_b m_b,  0 )

so the gradient norm follows the plus-sign closed form — all three squared terms are added:

||grad g||^2 = 1/4 [ m_a^2 + m_b^2 + (q_a m_a − q_b m_b)^2 ],   q_k = sign(m_k − m_Z).

For the 79 both-below events the shared term is a mass difference (q_a = q_b); for the 22 straddling events it is a mass sum (q_a ≠ q_b). The two unique-lepton components (q_a m_a and −q_b m_b) are always present.

Which masses enter. The primary closed form uses the candidate masses m0_a, m0_b obtained from massless four-vectors. The derivative computed from the stored electron/muon rest masses is kept separately as an independent cross-check against the saved finite difference, not as the primary formula. The massless form is primary for interpretability — only the massless limit collapses to the clean two-mass expression above — not for accuracy: the stored-mass derivative is the more accurate of the two, matching the saved finite difference to ~1e-8, while the massless form departs from it at the ~1e-3 level (see the error tables below).

Fixed scope and structural checks (from summary.json and internal_z1_branch_audit.csv):

Exact stored-mass derivative vs saved finite difference (independent cross-check, 404 components):

Massless analytic approximation vs saved finite difference (404 components):

The stored-mass derivative therefore matches the saved finite difference to ~1e-8 or better, while the massless approximation departs from it at the ~1e-3 level; the difference between the two is the finite stored-mass correction, isolated from the numerical finite-difference residual (error_metrics.csv).

Shared-lepton contribution. The "shared-component fraction" below is the shared coordinate's share of the squared gradient norm, not a fraction of the gradient amplitude (component_contribution_summary.csv, side_class_comparison.csv):

Interpretation (scope-limited): in these fractional-pT local coordinates, for both-below events the shared component enters as a mass difference and largely cancels, whereas for straddling events it enters as a mass sum and adds; this is why the straddling events carry the larger gradient norms in this fixed sample. This is a statement about the two active candidate masses and the shared-lepton topology within the fixed local coordinates only.

Relation to the §5h both-below correlation. In the fully massless both-below model, g(0) = Δm and ‖∇g‖ = ½√(m_a² + m_b² + Δm²), so the local boundary distance reduces exactly to d = 2Δm / √(m_a² + m_b² + Δm²), using massless masses and the massless Δm. This is an exact massless-model reduction; against the saved finite-mass finite-difference distance of §5h it is a numerically validated approximation, not an identity (the saved analysis carries the stored electron/muon masses). It therefore explains the strong both-below Δm–distance relationship structurally, but does not by itself mathematically determine the observed Spearman ρ = 0.9832, because the candidate-mass scale m_a² + m_b² also varies across events.

Worked examples (from target_examples_181079_181002.csv). The gradient vectors below are printed in physical lepton-index order (u0, u1, u2, u3), not the canonical (shared, other-only, chosen-only, unused) order of the formula above; the per-event index mapping is given with each example.

1:604:181002 — massless other/chosen masses 69.92313036889585 / 70.83805787655506 GeV, q_other = q_chosen = −1 (shared term is a difference); indices shared=0, other-only=1, chosen-only=3, unused=2:

• analytic massless gradient (0.45746375382960025, −34.96156518444793, 0, 35.41902893827753)
• closed-form norm 49.769749086799806
• saved finite-difference norm 49.7697452846256

1:604:181079 — massless other/chosen masses 70.89208275962014 / 71.77380860949629 GeV, q_other = q_chosen = −1 (shared term is a difference); indices shared=2, other-only=1, chosen-only=3, unused=0:

• analytic massless gradient (0, −35.44604137981007, 0.4408629249380738, 35.88690430474814)
• closed-form norm 50.44289950226834
• saved finite-difference norm 50.442898286124404

Both examples above are 4e events, where the massless approximation is at its best (their massless-vs-finite-difference vector relative errors are ~8e-8 and ~3e-8). A representative 4μ event shows the typical — and worst — massless departure, because the approximation error is driven by the muon mass:

1:604:180905 — 4μ, both-below; massless other/chosen masses 24.621186397148737 / 88.6468016747457 GeV, q_other = q_chosen = −1 (difference); indices shared=3, other-only=0, chosen-only=1, unused=2:

• analytic massless gradient (−12.310593198574368, 44.32340083737285, 0, 32.012807638798485)
• closed-form / massless norm 56.044040000787994
• exact stored-mass norm 56.043052199775374
• saved finite-difference norm 56.04305220149693
• massless-vs-finite-difference vector relative error 4.16342873822614e-05 (the sample maximum); exact-mass-vs-finite-difference vector relative error 8.9485529764465e-11

So for this event the exact stored-mass derivative reproduces the saved finite difference to ~1e-10, while the massless form differs in the fifth significant figure of the norm — the interpretability / accuracy trade-off made concrete.

Within-candidate Z1 branch audit. Within the fixed local region:

• no event is nondifferentiable at the origin because of an exact internal Z1 tie (0 origin ties);
• no active pair changes in the ±1e-4 coordinate stencil (0 switches);
• the nearest locally estimated internal branch is event 1:608:182271, matching 0, at a local linear fractional-pT L2 distance of 0.009527043578052046.

Across all 202 candidate matchings the local-linear internal-branch distance spans 0.9527% to 133.44% L2, and taking the nearest branch per event spans 0.9527% to 114.87%. These are local linear estimates, not verified radii of validity, so the large values must not be read as exact branch-free neighborhoods; the 0.9527% minimum is the tightest local region and stays well outside the ±1e-4 stencil. That figure is a local linear estimate of the internal branch distance, not an exact nonlinear branch distance, and it does not assert that the closed form carries across an internal Z1 branch (internal_z1_branch_audit.csv).

5k. Fixed five-event pT-projected uncertainty canary

A partial feasibility result, not a completed detector-aware analysis. This canary ran cmsRun (CMSSW_5_3_32, the recommended analysis release) over 5 preselected EventIDs to answer one narrow question: for the leptons of the saved pairing, is a per-lepton momentum uncertainty actually present in the fixed 2012 AODSIM, and can it be projected onto the on-shell score-difference direction of §5j? It is a pT-projected uncertainty canary — not a full detector-aware metric (detector-aware-canary-5_REPORT.md, summary.json).

Scope and method. The metric is pT-projected only, with η and φ held fixed, zero cross-lepton covariance, and a local linear Gaussian approximation. For each event it forms z = |score margin| / σ_g, where σ_g projects the per-lepton pT uncertainties onto the §5j analytic score-difference gradient, and reads a linear-Gaussian flip probability from that z. That probability is a number inside the local linear Gaussian model — it is not a measured probability of a wrong pairing. Muons used the track pT uncertainty (candidate pT and muonBestTrack pT agreeing to 1e-5); electrons projected the stored candidate-P4 p4Error as a direction-fixed relative pT uncertainty, which is not a full electron covariance.

Availability audit (20 leptons matching the saved selection):

Per-event pT projection (from pt_projected_event_metrics.csv):

The 181002 electron result is provisional: it rests on the candidate-P4 p4Error standing in for a full electron covariance, so its 0.219952 flip number must not be read as a physical mispairing probability. 181079 is not usable / NA.

Why 181079 is NA. Its lepton slot 3 (reco_index = 3) is a candidateP4Kind = 2 (P4_PFLOW_COMBINATION) electron whose stored candidate p4Error = 0. The same electron does carry correctedEcalEnergyError = 0.567356646 GeV and a GSF trackMomentumError = 999, but neither was substituted for the missing candidate-P4 error, so the row was left NA rather than filled with a stand-in.

Independent repository-wide verification of the deltaP omission. Beyond inspecting the two source files cited above, a sparse checkout of the official cms-sw/cmssw repository at tag CMSSW_5_3_10 was used to grep every call site of setDeltaP( across the full RecoParticleFlow package tree (all files under RecoParticleFlow/, ~300+ .cc/.h files), DataFormats/ParticleFlowCandidate, and RecoEgamma/EgammaElectronAlgos — not just the two files quoted in the discussion. Result: exactly one occurrence of setDeltaP( in that entire scope, and it is the inline setter definition itself in PFCandidate.h:257 — zero call sites. The backing member deltaP_ is touched only by the two constructors in PFCandidate.cc (deltaP_(0.)), never reassigned. This confirms, at the scope of the whole relevant module tree rather than the two originally cited files, that PFCandidate::deltaP() cannot be nonzero anywhere in this CMSSW tag's PF-electron path, so every electron reaching GsfElectronAlgo.cc:866-869 with candidateP4Kind() == P4_PFLOW_COMBINATION will deterministically carry p4Error = 0 — not as a property specific to 181079. Scope of this specific verification: tag CMSSW_5_3_10 only (the AODSIM's HLT/RECO production release); the analysis release CMSSW_5_3_32 was checked earlier only for file-identity of the same three files, not re-grepped repository-wide; PF2PAT or other wrapper/customization producers outside the three listed packages were not searched; and this does not establish developer intent, only the absence of any code path writing deltaP_ to a nonzero value.

CMSSW source audit (bounded). The target AODSIM's HLT/RECO was produced with CMSSW_5_3_10; the recommended analysis release is CMSSW_5_3_32, and the relevant PF-electron source files are identical between the two. PFCandidate does provide deltaP / pflowP4Error fields — so the data format is not missing an error field — but deltaP is initialized to 0, the standard RecoParticleFlow PF-electron path does not call setDeltaP(), and PFElectronTranslator copies that 0 into the P4_PFLOW_COMBINATION error. The bounded reading is therefore that, for this electron, the combined-uncertainty field was left unset by the standard PF-electron producer path — this is not a true zero uncertainty, not an extraction mistake, and not a confirmed CMSSW defect. The official combined error cannot be taken from this AOD's p4Error; an independent approximation from the ECAL error and GSF covariance could be built later, but that would be an approximation, not a recovery of the official value.

Reproducibility. cmsRun processed exactly 5/5 fixed events (wall time 1:29.34, maximum RSS 390,864 kB). Independent postprocessing reproduced REPORT.md, summary.json, and both CSV outputs with 0 SHA-256 mismatches, and no additional CMS event processing was performed during the check (REPRODUCIBILITY_CHECK.txt, runtime_environment.txt).

The five-event canary above is retained as its historical scope; the audits below later generalized the same availability question to all 101 fixed events, to an independent file, and to a pooled covariate-overlap diagnostic (§5l).

5l. Uncertainty-availability audits and covariate-overlap diagnostics

These four audits extend the §5k five-event feasibility canary to the full fixed 101 events, to an independent AODSIM file, and to a pooled covariate-overlap diagnostic. They remain availability and overlap diagnostics — no z or flip number here is a full detector-aware distance or a measured probability of a wrong pairing.

Fixed 101-event uncertainty-availability audit

All 404 selected leptons of the fixed 101 events were audited for stored per-lepton uncertainty availability (uncertainty-availability-audit-101_REPORT.md, summary.json):

• 42 4e events = 168 selected electrons; 59 4mu events = 236 selected muons (404 leptons total).
• candidateP4Kind=2 && p4Error==0.0: 10/168 electrons, affecting 10/42 4e events; each affected event contains exactly one such electron.
• The other 158 electrons are kind=1 with a positive active candidate-P4 error.
• All 10 kind=2 electrons have ecalDrivenSeed=false, trackerDrivenSeed=true; kind=2 with nonzero p4Error: 0; kind=2 with ecalDrivenSeed=true: 0.
• 236/236 muons passed the prescribed best-track / finite-symmetric-positive-definite covariance checks (covariance symmetry max abs 0, global minimum eigenvalue 3.6596e-10, candidate/best-track pT relative difference max 4.96525e-08).
• Events with all four official saved uncertainties usable: 91/101 = 32 4e + 59 4mu.
• Affected EventIDs: 1:604:180965, 1:604:181079, 1:604:181104, 1:604:181111, 1:608:182217, 1:608:182259, 1:608:182266, 1:608:182324, 1:611:183165, 1:616:184565.
• "Usable" here means usable for the specified pT-only projection, not complete detector-uncertainty modelling. Seed and preselection flags were recorded as producer/reconstruction path diagnostics only — not interpreted as a physical cause and not tied to a low-pT dependence. The known five-event canary electron 1:604:181079 (reco_index = 3) reproduced as kind=2 / error=0.

This generalizes the §5k single-electron observation: the earlier canary happened to hit one such electron; the full audit finds 10/168.

Channel-separated pT projection and sensitivity analysis

The official complete-case projection uses the 91 events with official saved uncertainties, reported per channel and never pooled (pt-projection-sensitivity-101_REPORT.md, summary.json):

• The 10 missing-error 4e events are not silently imputed. They are replaced only under four explicit alternative models — ecal_relative, gsf_relative, inverse_variance_independent (independence proxy), and quadrature_conservative (conservative proxy) — each touching only the 10 missing electrons; the other 394 leptons are unchanged.
• These are sensitivity models, not recovery of the official combined error. Across the four models the 10 affected events' z spans 0.136 to 36.84 with a maximum linear-Gaussian flip of 0.446; the model choice changes the z < 1 class for 3 events and the z < 2 class for 2.
• Complete-case results are not unconditional results for all 101 events. The independent-file sample was not included in this pT projection. The four usable five-event canary z values were reproduced to 1e-5; 181079 is excluded from the official complete case.

Independent-file replication

A pre-fixed different AODSIM file from the same CERN H→ZZ MC dataset, first 1000 raw events only, with the same truth matching and the same 4e/4mu selection (independent-availability-replication_REPORT.md, summary.json):

• Eligible same-flavour events: 121 (62 4e = 248 electrons; 59 4mu). Selected EventID overlap with the original fixed 101: 0.
• kind=2 && p4Error==0.0: 6/248 electrons, affecting 6/62 4e events; each affected event again contains exactly one such electron. kind=2 && nonzero p4Error: 0; kind=2 && ecalDrivenSeed=true: 0.
• Rates with Wilson 95% intervals — electron level: original 10/168 = 5.9524% [3.2650%, 10.6091%] vs independent 6/248 = 2.4194% [1.1134%, 5.1768%]; affected-4e-event level: original 10/42 = 23.8095% [13.4810%, 38.5275%] vs independent 6/62 = 9.6774% [4.5112%, 19.5488%].
• This is a same-dataset / different-file replication, not validation across another physics process, detector condition, dataset, or real collision data.

Pseudoreplication correction

• The electron-level and event-level counts are not independent evidence: in both samples there is exactly one kind=2 / error-0 electron per affected event.
• The primary descriptive unit for affected-event frequency is the 4e event: 10/42 versus 6/62.
• The electron-level Fisher exact two-sided p = 0.073965 and the event-level p = 0.0582609 are not two independent confirmations; the electron-level test is not a primary inferential result, and even the event-level test is only a limited reference because file-level clustering/heterogeneity is not modelled. Failure to reject equality is not evidence that the two rates are equal.

Pooled 416-electron covariate-overlap diagnostic

Pooled descriptive sample of 416 electrons = kind=1 400 + kind=2 16 (168 original + 248 independent; 158 + 242 kind=1, 10 + 6 kind=2) (overlap-diagnostic-416_REPORT.md, summary.json, continuous_overlap_summary.csv):

• Closest available kind=1 comparison subgroup: kind=1 && ecalDrivenSeed=true && trackerDrivenSeed=true, n = 370 (146 original + 224 independent).
• Complete categorical separation remains: all 16 kind=2 electrons are (ecalDrivenSeed=false, trackerDrivenSeed=true), while all kind=1 are ECAL-driven; there are zero kind=1 electrons in the same seed cell as kind=2. A kind=1 calibration applied to kind=2 is therefore category-level extrapolation even where the continuous covariates overlap.
• Against the closest 370-electron subgroup, kind=2 central-95%-range inclusion is: candidate pT 14/16, |eta| 16/16, relative ECAL error 16/16, relative track-pT error 14/16, candidate-pT/track-pT ratio 10/16, and event pT rank 16/16.
• Full observed range vs central 95%: for candidate-pT/track-pT ratio all 16/16 kind=2 values lie inside the full kind=1 observed range, while only 10/16 lie inside its central 95% range. The former is weak and outlier-sensitive; the latter means six points lie outside the central region, not necessarily outside all observed kind=1 support.
• Observed descriptive shifts (not called statistically significant): |eta| median 1.323 versus 0.732, relative track-pT error median 0.255 versus 0.187 — a higher descriptive median was observed on the kind=2 side.
• Event 1:608:182324 is the largest multivariate nearest-neighbour outlier: candidate-pT/track-pT ratio 5.155 and nearest distance 4.425 IQR units. This single event does not explain all six central-95%-range exclusions.

The six exclusions relative to the pooled central 95% range of the closest 370-electron kind=1 subgroup (0.7644 to 2.0707) are listed below. Four lie below the interval and two lie above it, so the result is not produced by the single largest outlier alone.

This completes the planned descriptive follow-up for the pooled 416-electron overlap diagnostic. No additional cmsRun, calibration fit, or reconstruction-mechanism claim is implied by this table. - electron_track_momentum_error and electron_combined_p4_error are sentinel 999 for 16/16 kind=2 electrons — a missing-value sentinel, not a large physical uncertainty. The finite covariance-derived track_pt_error is a different field. The candidate-P4 error is exactly 0 for 16/16 kind=2 electrons and is not interpreted as a true zero uncertainty. - This analysis diagnoses observed covariate overlap only; it does not verify equality of reconstruction-error mechanisms or transportability of a kind=1 error model. Any future proxy must be treated as an explicit cross-seed-path sensitivity model, not a recovery of the official value.

5m. Full ancestry audit of all 42 mismatch EventIDs

The fixed top-10 canary method was extended to the complete, prespecified set of 42/42 events where the closest-to-Z rule disagreed with the legacy generator pairing (ANCESTRY_AUDIT_42_REPORT.md, summary_42.json, event_label_comparison.csv). The same single fixed AODSIM file was used, with only the 42 EventIDs selected by eventsToProcess.

• Every event produced exactly four legacy reco-gen matches: 42 events / 168 matched leptons.
• The audit saved 28,143 genParticle rows, 64,840 mother/daughter relation rows, and one detailed family tree for each of the 42 events.
• Graph integrity checks found 0 unresolved relative indices, 0 mother edges without reciprocal daughter edges, and 0 daughter edges without reciprocal mother edges.
• Pairing changes from legacy mother(0) labels to all-mother nearest-Z labels: 0/42.
• Pairing changes after collapsing connected Z status-copy components: 0/42.
• Events with an ambiguous all-mother or Z-copy-collapsed pairing: 0/42.
• Z-copy components were present in 42/42 events, but did not change any event's pairing label.
• The fixed classification is therefore A=0, B=0, C=42, D=0: no event is promoted to a confirmed physical rule error; no label-construction problem or unresolved graph remains; all 42 retain the conceptual limitation that generator bookkeeping is not a uniquely observable physical history.
• The single prespecified recount added 0 agreements, so the original result remains 59/101 (58.4%) agreement and 42/101 disagreement.

The 10-event benchmark completed in 1:57.61 with maximum RSS 362,032 kB. The full 42-event cmsRun completed in 2:26.31 with maximum RSS 365,260 kB. The final artifact check found 0 SHA-256 mismatches; an independent postprocessing rerun reproduced all 10 core artifacts exactly; and the full run's top-10 subset reproduced all eight corresponding canary tables exactly. No reliable XRootD byte counter was present in the log.

This closes the generator-label construction audit for the fixed 42 mismatches. At this stage it did not determine which identical-lepton pairing was physically preferred; the separate fixed-101 matrix-element/interference audit is reported next.

5n. Official JHUGenMELA interference audit of all fixed 101 events

The same stored four-vectors were evaluated for all 101/101 same-flavour events with official JHUGenMELA v7.5.6, ROOT 6.30.02, and a fixed 8 TeV decay-ME configuration (REPORT.md, results_101.csv, group_summary.csv, group_comparison.csv). No additional cmsRun was needed: the frozen input contains 404 rows = 101 events × 4 reconstructed leptons.

The complete same-flavour ME was compared with the same-normalization no-interference control 0.5(P_A+P_B), where P_A and P_B force the two valid OSSF pairings by assigning their pairs different charged-lepton flavours without changing the measured four-vectors. The resulting relative interference is a signed ME comparison, not an A/B truth probability.

The prespecified comparison was the median relative-interference difference, mismatch minus match. Pooled over channels it was −5.49 percentage points, with a deterministic 20,000-resample conditional bootstrap interval of [−15.80, +6.16] points. The 4e estimate was +5.68 points ([−24.22, +25.28]); the 4mu estimate was −6.68 points ([−17.64, +0.89]). All intervals include zero, and the channel point estimates have opposite signs. The fixed sample therefore gives no stable evidence that mismatch events have stronger interference.

Validation was exact within the saved calculation: two independent executions agreed in all 23 non-timing fields for all 101 events; all 101 passed the 16 structured OSSF input-order checks with maximum ME difference 0; the corrected one-event benchmark 1:604:181002 reproduced +25.0719358298%; and all complete/control MEs were finite and positive. Nominal JHUGen InterfOn and InterfOff remained exactly equal in all 101 events, so their difference was not used. The reproducible bundle includes the C++ source, scripts, fixed input, Dockerfile, JHUGen distribution, environment record, two raw result tables, logs, and SHA-256 manifest.

This audit tests association within the fixed selected 101 events. It does not establish causality, choose the physically "true" A/B pairing, include detector/theory/background systematics, or validate the result on an independent event sample.

5o. Fixed-three-event local hybrid-covariance benchmark

This follow-up moved beyond the earlier pT-only projection for exactly three preselected events while keeping electron and muon uncertainty handling separate. It used local coordinates (ln pT, eta, phi), recomputed four-vectors and active closest-to-Z branches under variation, and compared the local linear projection with an exact nonlinear boundary calculation. It did not run an additional cmsRun or extend to the other 98 events.

For 4mu, each best-track 5×5 covariance was transformed to the available within-muon 3×3 covariance. For 4e, the active candidate-P4 relative scale error was combined with the GSF-track (eta,phi) 2×2 block. The unavailable candidate-scale/GSF-angle cross terms and all cross-lepton covariances were set to zero. The covariance was frozen at the central point. This is therefore a hybrid covariance, not a complete electron or event-level detector covariance.

All 8/8 selected electrons passed the fixed candidate-P4 versus GSF centre-direction check; the maximum DeltaR was 0.000365077. The two mandatory saved pT-only checks reproduced with absolute differences 4.46e-11 (4e 181002) and 1.87e-11 (4mu 182137). The missing candidate-P4 error in 1:604:181079 slot 3 was not replaced by ECAL, GSF, an average, or another proxy.

The first fixed-three-event run is retained as an immutable failure record. Its final recorded two-pass job used 1,480 exact-score evaluations, but the session cumulative estimate was 3,700, exceeding the authorized ≈1,494 cap because an aborted attempt and a report-regeneration calculation were also run. It is labelled partial_score_cap_violation; the 4mu SLSQP result remained NA there after the numerical finite-difference Jacobian consumed its 300-evaluation optimizer cap. A proposed cap-400 retry was prepared later but never executed and consumed zero exact-score evaluations.

The 4mu result was completed in a separate adaptive test with a newly derived exact stored-mass, piecewise-active 12D analytic constraint Jacobian. This was not the earlier §5j pT-only analytic gradient reused as a 12D formula. The new derivatives included ln pT, η, and φ and were checked against central finite differences at the origin, half-ray, saved-ray, and accepted endpoint, plus six fixed directional derivatives. The maximum component discrepancy was 2.24e-9 GeV/unit, the maximum relative L2 discrepancy was 5.36e-11, and the maximum directional absolute discrepancy in the debug checks was 4.58e-9; active pairs did not switch.

With the analytic Jacobian, the formal 4mu SLSQP terminated in 3 iterations / 7 optimizer score evaluations, with final g = -2.84e-14 GeV and relative KKT residual 8.28e-14. Formal and reproduction outputs matched. The new test used 99 debug + 121 formal + 121 reproduction = 341/528 authorized exact-score evaluations and 1.164/120 s. The accepted distance differs from the predecessor's last unaccepted point by only about 3.3e-12 σ, so this closes the numerical termination issue rather than finding a materially different boundary.

This is explicitly a numerical-completion test after the preceding failure, not an independently prespecified confirmation from the beginning. Both accepted values are local hybrid-covariance sensitivities. They are not complete detector-covariance distances, global shortest distances, pairing-error probabilities, or evidence that either pairing is the unique physical history.

5p. Fixed 91-complete-case hybrid-linear audit

The subsequent fixed audit covered all 91 complete cases (4e 32, 4mu 59). Under the unchanged hybrid definition and event-level QC, 83 passed and 8 failed: 4e had 24 PASS / 8 FAIL, while 4mu had 59 PASS / 0 FAIL. All eight failures were candidate-P4/GSF hybrid-compatibility-gate failures; they are not physics failures. Thus, 91 complete cases and 83 hybrid-QC-usable cases are distinct denominators.

The preregistered operational screen, QC PASS and hybrid d_lin <= 3, selected exactly two events: 1:604:181002 (4e) and 1:608:182137 (4mu). They are the same two events already used in the local nonlinear pilot, not an independent nonlinear validation. Their linear distances differed from the existing nonlinear results by less than 0.4% (0.3989% and 0.3708%, respectively). The other 89 complete cases produced zero new local candidates; no additional nonlinear calculation was run.

The d_lin <= 3 cutoff is an operational screening threshold for limiting nonlinear follow-up far from the centre of a frozen local hybrid covariance. It is not a statistical three-sigma statement, a theoretical guarantee of linear accuracy, or a calibration from Delta-m, the pT-only distance, or the +/-20% perturbation study. These results remain local hybrid-covariance sensitivities subject to the omitted scale-angle and cross-lepton correlations and do not establish a global distance, probability, unique physical pairing, or independent confirmation.

5q. Prospective two-event quadratic exact-ray prediction audit

This pilot tested whether a fixed second-order correction predicted the exact boundary better than the local linear distance on two target-unseen events. Candidate selection was completed before target generation from the independent-file audit: among 115 complete cases / 96 event-QC PASS cases, the minimum hybrid d_lin event was fixed separately for 4e (1:120:35923) and 4mu (1:120:35961). A target-exposure inventory found 31 candidate-related records, 0 candidate-specific numerical exact-ray exposures, and 0 scan errors. Here, “target-unseen” means this saved-artifact exposure test passed; it is not a claim about inaccessible human memory or information outside the audited files.

The computation was separated into three sealed stages:

1. Stage P (prediction): computed and saved the linear and quadratic predictions without reading or generating the exact targets. It used 20/20 event-score evaluations and 40/40 pairing-score calls; evaluation 21 was not started. The deterministic repeat matched exactly, target semantic parse count was 0, and wall time was 0.246 s.
2. Stage T (target): generated only the two exact ray targets, without semantically parsing the prediction artifact. It used 174/204 event-score evaluations and 348/408 pairing-score calls; evaluation 205 was not started. Both events were TARGETED, the deterministic repeat matched exactly, prediction semantic parse count was 0, and wall time was 0.739 s.
3. Stage C (comparison): read each sealed numerical artifact once and performed no exact scoring, ROOT/truth read, or scoring-kernel import. The deterministic repeat matched exactly and wall time was 0.0226 s.

Thus the quadratic prediction had smaller absolute error than the linear approximation for 2/2 fixed events, and it overpredicted both exact ray targets. The sealed Stage-P prediction JSON has SHA-256 EFF53FB41D60537E75670413DE9DEE25035FB12A20AEE0223ED446AA2AD5A19E; the Stage-T target JSON, D96432E5EA11F26E1A4120D359989A6FD8C675DDEA0BB2A675A6916390AD4FA8; and the Stage-C comparison JSON, 91BF6A0781033AFF2B9BCFD567DFDF48D21242A309B2F4CBF38DF95968C1D17F.

The computational-integrity checks passed, but scientific accuracy is a separate question. With only one event per channel, no population error rate, coverage, calibration, channel comparison, or generalization performance can be estimated. These remain fixed-ray, frozen local hybrid-covariance sensitivities, not global shortest detector-covariance distances or measured pairing-error probabilities.

5r. Fixed-16 prospective quadratic exact-ray extension

The extension was fixed before its exact targets were generated. It retained the eight smallest eligible d_lin events per channel after excluding the previously known target and the two pilot events. The fixed ranges were 7.1660–13.6458 σ for 4e and 11.0305–20.0789 σ for 4mu. All 16 EventIDs and their order were retained throughout; an event-level NA was never removed, replaced, or used to redefine the denominator.

The computation again used three sealed stages:

1. Stage P (prediction): used 160/160 event-score evaluations and 320/320 pairing-score calls. It produced 15 PREDICTED rows and one NA (1:121:36059, selected-pair mass-minus-MZ sign changed from the origin). The two deterministic passes matched exactly, target semantic parsing was zero, and wall time was 0.855 s.
2. Stage T (target): used 1228/1632 event-score evaluations and 2456/3264 pairing-score calls. It produced 14 TARGETED rows and two NAs (1:122:36334 and 1:121:36059, both active-pair switches from the origin). The two 614-evaluation passes matched exactly, prediction semantic parsing was zero, and wall time was 3.755 s.
3. Stage C (comparison): mechanically formed the primary set Stage-P PREDICTED ∩ Stage-T TARGETED, preserving all 16 formal rows. The overlap of the Stage-P and Stage-T NA sets left 14 paired events and 2 formal NA rows. Stage C parsed each sealed JSON once, performed the in-memory comparison twice with exact agreement, made zero score calls, and completed in 0.128 s.

The primary event-level quantities were fixed as

E_lin  = abs(d_lin  - d_ray) / d_ray
E_quad = abs(d_quad - d_ray) / d_ray
DeltaE = E_lin - E_quad
signed quadratic residual = (d_quad - d_ray) / d_ray

The sealed prediction, target, and comparison JSON SHA-256 values are respectively C70EBE2D8C8F091592CDE19F3CC00CE7D4C67174BE76A1BB31DF8952A658BD1E, 827D97C0F90555BC7479B514A11C00A9A269BDCBAAD206EF054D4BEDE4289E98, and 1B16667EB45549218F2F9CF1582CD3FC4D4F357DF6E9279252494FBCC46210CC. The Stage-C artifact manifest contains 13 entries with zero mismatches and has SHA-256 4A568B2B2394EF2BE46D80B20F6C9A538F28A1AEC8C98EE927F8B6A60607A423.

This prospectively fixed cohort gives substantially stronger evidence than the two-event pilot that the quadratic correction improves the fixed-ray approximation in the selected near-distance independent-file range. It still does not establish population accuracy, coverage, calibration, a global shortest detector-covariance distance, or transport to another process, detector condition, dataset, or real data. The 14/14 positive residuals are a reproducible cohort pattern and a hypothesis for external validation; they are not by themselves proof of a population-wide overprediction bias.

6. Figures

7. Corrections and terminology updates

Where documents report different values, this README uses the result from the most recent audit and records the change as "earlier wording → post‑audit wording."

• "Physical accuracy" → "agreement with the generator ancestry record." 59/101 (58.4%) is agreement with generator bookkeeping, not a physical correctness rate. For 4e/4mu, generator history is one internal bookkeeping choice, not the unique observable past.
• "The rule beats chance" → undetermined. The 95% CI on 58.4% contains 0.5.
• "The 42 failures are the rule's real errors" → still unsupported after the full audit. The later §5m run saved and checked the ancestry graph, status, and four‑momenta for all 42 events: classes B = 0 and D = 0, while A remains 0 and C = 42. The label‑construction mechanism is now traced, but generator bookkeeping is still not a uniquely observable physical pairing.
• Scan maximum 60/101 → not a performance claim. It is a post‑hoc maximum over the same events.
• Stage 3 derived‑table bugs (a KeyError on a mis‑assumed manifest column; inclusion of events with <4 objects) were fixed and re‑reconciled to the original seed means (max diff 4.1e‑5) (STAGE3_FINAL_REPORT.md).
• Stage 4 scope deviation logged: a validation check approved for a 1‑file Higgs‑MC transfer actually connected to 4‑file metadata; event arrays/quick analysis were not run and the overrun was recorded (stage4_cms_open_data_README.md).
• "Checked in two cited files" → "confirmed absent across the full relevant module tree." The §5k claim that no CMSSW code path calls setDeltaP() was initially argued from two quoted files and was flagged as unverified ("found none" vs. "there is none"). An independent grep of the entire RecoParticleFlow, DataFormats/ParticleFlowCandidate, and RecoEgamma/EgammaElectronAlgos trees at tag CMSSW_5_3_10 found the same result: setDeltaP( occurs exactly once (the setter's own declaration), with zero call sites anywhere in that scope. See §5k for the bounded scope of this check (single tag, three packages, no PF2PAT/wrapper producers searched).
• "Final two-pass job stayed below its cap" → cumulative session violated the cap. The first §5o benchmark's saved two-pass job used 1,480 evaluations, but including the aborted calculation and a report-regeneration rerun gives an estimated cumulative 3,700. The record is not retroactively called compliant. The later 4mu analytic-Jacobian calculation is reported separately as an adaptive completion test, not an independent confirmation.

8. Reproduction and main scripts

# Stages 1–3 (artificial)
python -m pip install -r requirements.txt
python -m unittest discover -s tests -v
python run_all.py --quick        # 2,500 events/hyp
python run_all.py                # full: 50,000 events/hyp
python run_stage2.py             # → outputs_stage2/
python run_stage3.py             # → outputs/stage3/

• Stage 4T truth: run the 100‑event benchmark first, review ETA, then the fixed 1,000‑event config; postprocess with truth_postprocess.py. The runner refuses any CMSSW_VERSION other than CMSSW_5_3_32 (truth_validation_README.md).
• Failure atlas: build_failure_atlas.py regenerates the 101‑event derived analysis.
• Ancestry validation check: CMSSW TruthPairingAnalyzer.cc + all‑mother/Z‑copy audit on 10 fixed EventIDs.
• Analytic gradient audit: run_analytic_gradient_audit.py is a derived, read‑only audit that pins the SHA‑256 of its three fixed input files, checks the 101 / 79 / 22 event counts, and verifies the baseline pairing selection before comparing the closed‑form gradient against the saved finite difference and the exact stored‑mass derivative (analytic-gradient-audit-101/run_analytic_gradient_audit.py).
• Detector-aware pT-projected validation check (§5k): run_detector_canary_5.sh drives cmsRun over the 5 fixed EventIDs (CMSSW_5_3_32, slc6_amd64_gcc472); preflight_canary.py and postprocess_canary.py build the diagnostics and projected-metric CSVs. cmsRun processed 5/5 events; an independent rerun reproduced the four derived files (REPORT.md, summary.json, and both CSVs) with 0 SHA-256 mismatches (detector-aware-canary-5/REPRODUCIBILITY_CHECK.txt).
• Fixed-101 availability audit (§5l): run_uncertainty_availability_audit_101.sh drives cmsRun (CMSSW_5_3_32) over the fixed 101 events; prepare_audit.py / postprocess_audit.py build the 404-lepton availability table (wall 1:41.52, max RSS 391,948 kB). The channel-separated pT projection and the four alternative-error sensitivity models (§5l) postprocess that table together with analytic_vs_finite_difference_101.csv; flip probabilities are model numbers, not measured mispairing probabilities.
• Independent-file replication (§5l): the same analyzer and selection over the first 1000 raw events of a different AODSIM file from the same H→ZZ dataset (wall 2:20.74, max RSS 407,356 kB); the pooled 416-electron overlap diagnostic postprocesses the original 168-electron and independent 248-electron tables (overlap-diagnostic-416/).
• Fixed-three-event hybrid-covariance benchmark (§5o): run_benchmark.py records the first three-event calculation and its score-cap violation. The separate analytic-Jacobian completion test uses compute_analytic_retry.py, a persistent score_ledger.jsonl, and write-once debug/formal/ reproduction JSON. render_report.py reads those stored files only and cannot rerun the event score. The benchmark and completion-test folders are local Codex outputs and are not bundled in this README snapshot.
• Fixed-16 quadratic extension (§5r): the local Codex folders fixed16-quadratic-ray-audit-stage-p-v2, fixed16-quadratic-ray-audit-stage-t-v2, and fixed16-quadratic-ray-audit-stage-c-v1 hold the separately sealed prediction, target, and comparison records. Stage C is a pure JSON comparison with zero score calls; its 13-entry artifact closure has zero mismatches. These folders are not bundled in this README snapshot.
• Every artifact folder carries summary.json / manifest.json / artifact_sha256.txt for input/output auditing. The fixed real‑data candidate file SHA‑256 is D9F9830F8CBC45AC8A4A60A19D6164070303A4F90EC93A570D568D680C96462D.

9. Suggested artifact layout

Relative links above assume this consolidated tree (rename your dated Codex folders to match):

.
├── project/
│   ├── README.md                     # Stage 1 (= artificial_four_particle_decays_README.md)
│   ├── STAGE2_FINAL_REPORT.md
│   └── STAGE3_FINAL_REPORT.md
├── stage4_cms_open_data/
│   ├── README.md                     # Stage 4 pre-flight
│   └── truth_validation/
│       ├── README.md
│       └── outputs/fixed_1000/truth_summary.json
├── failure_atlas_101/                # FAILURE_ATLAS_REPORT.md, events_101.csv
├── label_validity_audit_101/         # LABEL_VALIDITY_AUDIT.md
├── top10_label_canary/               # TOP10_CANARY_REPORT.md
├── cms-real-data-small/              # REPORT.md, summary.json, m4l_data_vs_mc.png
├── z-peak-canary/                    # REPORT.md, summary.json, z_peak_ee_mumu.png
├── strict-cuts-125gev/               # REPORT.md, selected_electrons.csv
├── pairing-rule-strength-101/        # REPORT.md, event_classification_101.csv, *.png
├── physical-pairing-rules-101/       # REPORT.md, rule_agreement_counts.csv, rule_definitions.csv
├── two-mass-rule-edge-events/        # REPORT.md, two_events.csv
├── final-pairing-geometry-101/       # REPORT.md, final_pairing_geometry.png
├── minimum-observable-perturbation-101/  # REPORT.md, event_minimum_perturbations_101.csv, *.png
├── fragile-pairing-events-audit/     # fragile_event_rankings.csv, top10_pairing_candidate_values.csv
├── local-multivariate-boundaries-top4/   # REPORT.md, local_boundary_results.csv
├── comparison-181002-181079/         # summary.json, event_level_comparison.csv
├── delta-m-distribution-101/         # REPORT.md, event_delta_m_101.csv, *.png
├── delta-m-bin-rule-behavior-101/    # REPORT.md, delta_m_bin_rule_behavior.csv, *.png
├── delta-m-local-pt-boundary-101/    # REPORT.md, event_local_pt_boundary_101.csv, structural_correlations.csv, *.png
└── analytic-gradient-audit-101/
    ├── REPORT.md
    ├── DERIVATION.md
    ├── analytic_vs_finite_difference_101.csv
    ├── error_metrics.csv
    ├── side_class_comparison.csv
    ├── component_contribution_summary.csv
    ├── internal_z1_branch_audit.csv
    ├── target_examples_181079_181002.csv
    ├── analytic_gradient_error_distributions.png
    ├── analytic_gradient_error_distributions.pdf
    ├── run_analytic_gradient_audit.py
    ├── summary.json
    └── artifact_sha256.txt
├── detector-aware-canary-5/
│   ├── REPORT.md
│   ├── summary.json
│   ├── selected_lepton_uncertainty_diagnostics.csv
│   ├── pt_projected_event_metrics.csv
│   ├── REPRODUCIBILITY_CHECK.txt
│   ├── runtime_environment.txt
│   ├── configs/detector_canary_5_cfg.py
│   ├── run_detector_canary_5.sh
│   ├── preflight_canary.py
│   ├── postprocess_canary.py
│   └── cmssw/Stage4/DetectorAwareCanary5/plugins/   # DetectorAwareCanary5Analyzer.cc, BuildFile.xml
├── uncertainty-availability-audit-101/
│   ├── REPORT.md
│   ├── summary.json
│   ├── selected_lepton_availability_404.csv
│   └── source/    # fixed_events_101.txt, prepare_audit.py, postprocess_audit.py,
│                  # uncertainty_availability_audit_101_cfg.py, run_uncertainty_availability_audit_101.sh,
│                  # DetectorAwareCanary5Analyzer.cc, BuildFile.xml
├── pt-projection-sensitivity-101/
│   ├── REPORT.md
│   └── summary.json
├── independent-availability-replication/
│   ├── REPORT.md
│   └── summary.json
├── overlap-diagnostic-416/           # REPORT.md, summary.json, overlap tables and figures
├── ancestry-audit-42/                # report, graph tables, event comparison, hashes and logs
├── mela-interference-101/
│   ├── REPORT.md
│   ├── input_101x4.csv
│   ├── results_101.csv
│   ├── results_101_run2.csv
│   ├── group_summary.csv
│   ├── group_comparison.csv
│   ├── mela_interference_101.cpp
│   ├── prepare_input.py
│   ├── aggregate_results.py
│   ├── compare_runs.py
│   ├── run_all.ps1
│   ├── Dockerfile
│   ├── JHUGenerator.v7.5.6.tar.gz
│   ├── validation.txt
│   ├── reproducibility.txt
│   ├── environment.txt
│   ├── source_provenance.txt
│   └── artifact_sha256.txt
├── detector-aware-3event-benchmark/  # immutable first attempt; partial_score_cap_violation
├── muon-182137-analytic-jacobian-completion-test/
│   ├── PLAN.md
│   ├── REPORT.md
│   ├── compute_summary.json
│   ├── debug_result.json
│   ├── formal_result.json
│   ├── reproduction_result.json
│   ├── score_ledger.jsonl
│   ├── compute_analytic_retry.py
│   ├── render_report.py
│   └── artifact_sha256.txt
├── fixed16-quadratic-ray-audit-stage-p-v2/  # sealed fixed-16 predictions
├── fixed16-quadratic-ray-audit-stage-t-v2/  # sealed fixed-16 exact-ray targets
└── fixed16-quadratic-ray-audit-stage-c-v1/  # fixed paired comparison and integrity closure

10. Limitations

• Stages 1–3 are artificial: constant matrix element or propagator‑only amplitude, isotropic angles, no spin/interference/background/pileup. Results condition on this toy generator and detector and do not imply the existence of a corresponding physical process or experimental discovery sensitivity.
• An official decay-only JHUGenMELA ME/interference comparison is now available for the fixed 101 same-flavour events (§5n). It does not prove that interference causes a mismatch, assign an A/B pairing probability, include detector/theory/background systematics, or identify the correct pairing.
• Generator ancestry is MC bookkeeping. For identical‑flavour 4‑lepton final states it can be a unique internal record but is not guaranteed to be the unique physically observable past.
• The Z‑peak canary guarantees only the basic chain — charge/momentum → invariant mass — not 4‑electron quality or pairing correctness.
• The three fixed rules are explainable heuristics, not a full H→ZZ*→4l matrix element.
• The strength‑scan maximum (60/101) is a post‑hoc maximum over the same 101 events, not an improvement.
• The four 120–130 GeV real‑data candidates are not Higgs events; background estimation and formal electron ID are missing, and 14/16 selected electrons had isolation at the −999 sentinel.
• 2e2mu is descriptive only (single OSSF matching) and excluded from the primary agreement calculation.
• The perturbation widths in §5h are artificial numerical stress‑test coordinates, not CMS detector resolution, electron/muon calibration, or systematic uncertainties; electrons and muons received the same dimensionless protocol with no per‑species resolution model.
• The minimum flip perturbations are minima within the specified one‑dimensional families (±20% pT, ±100 mrad angle), not global multidimensional minimum distances. Events with no boundary found inside those bounds are not claimed to have none.
• The all‑101 boundary distances are local linear values; only 16 local candidates had a nonlinear boundary confirmed inside a per‑lepton ±20% box. Distant linear extrapolations are not exact global minimum distances.
• The Δm‑bin tests are exploratory, unadjusted, and rest on small bins (n = 2, 1, 2, 6, 90); the 5 GeV split was chosen post hoc and is not a natural physical threshold.
• The strong Δm–boundary correlation is largely algebraic: for the 79 both‑below events Δm is the score‑margin numerator, so this is not independent validation.
• The 181002/181079 gradient cosine (~0.845) uses same‑index bookkeeping labels and is not a permutation‑invariant physical correspondence.
• The two‑mass reduction of §5j is specific to the fractional‑pT metric. In absolute‑pT coordinates or under a detector covariance, per‑lepton kinematics re‑enter through the metric/Jacobian, so the closed form does not transfer unchanged.
• The §5j result is piecewise local and assumes the active Z1 pair remains fixed; it is not a global minimum‑boundary result.
• The §5j result is not a CMS‑resolution or detector‑instability result, and no independent event sample was tested — all figures come from the same fixed 101 events.
• The pT-projected uncertainty canary (§5k) processed only five preselected events; it is a partial feasibility result, not a detector-aware analysis.
• Its metric is pT-projection only, with η and φ held fixed.
• It assumes zero cross-lepton covariance.
• It uses a local linear Gaussian approximation, so its flip probability is not a measured probability of a wrong pairing.
• The electron candidate-P4 p4Error it projects is not a full electron covariance.
• The PF-combination (candidate-P4) error was missing (stored as 0) for one selected electron (1:604:181079) in the five-event canary (§5k). The later fixed-101 audit (§5l) found this is not unique: 10/168 electrons (10/42 4e events) carry candidateP4Kind=2 && p4Error==0, each affected event holding exactly one such electron.
• No complete detector-covariance distance or official PF-electron combined-error model was built. Section 5o forms local within-lepton 3×3 hybrid blocks and includes η/φ, but the electron scale-angle cross terms and every cross-lepton covariance are unavailable and set to zero. The §5l channel-separated projection evaluates four alternative-error sensitivity models for the 10 missing electrons, but these are explicit sensitivity proxies, not a recovery or reconstruction of the official combined uncertainty.
• Independent-sample scope (§5l): a same-dataset / different-file replication now exists (6/248 electrons, 6/62 4e events), but broader validation across another physics process, detector condition, dataset, or real collision data still does not. The electron-level and event-level Fisher tests are not two independent confirmations, and failure to reject equality is not evidence of equal rates.
• The §5l pT projection is complete-case on 91/101 events and is reported per channel (4e 32 / 4mu 59), never pooled; it is not an unconditional result for all 101 events.
• The §5l overlap diagnostic shows complete categorical seed separation (all kind=2 are tracker-driven-only, all kind=1 are ECAL-driven), so a kind=1 error model applied to kind=2 is category-level extrapolation even where continuous covariates overlap; it diagnoses observed overlap, not equality of reconstruction-error mechanisms or transportability of a kind=1 model.
• The sentinel value 999 on electron_track_momentum_error / electron_combined_p4_error (16/16 kind=2) is a missing-value flag, not a large physical uncertainty; the candidate-P4 p4Error == 0 is not interpreted as a true zero uncertainty or a confirmed CMSSW defect.
• The §5o benchmark contains only three preselected events from the same fixed AODSIM sample. Its covariance is frozen at the central point, its nonlinear calculation is local and single-start, and it neither proves a global shortest boundary nor validates any ranking on an independent sample.
• The first §5o run exceeded its authorized cumulative score budget (estimated 3,700 vs ≈1,494) and is retained as partial_score_cap_violation. The later 4mu analytic-Jacobian result is an adaptive numerical-completion test after that failure, not an independently prespecified confirmation.
• The §5o analytic Jacobian is exact only for the stored-mass four-vector parameterization inside a piece with fixed active pairs. Finite-difference and branch checks passed for this event, but that does not turn the result into a global or coordinate-independent detector geometry.
• The prospective §5q audit contains exactly two fixed events, one per channel. Its 2/2 improvement is descriptive and cannot estimate population accuracy, uncertainty, calibration, or generalization. The events come from a different AODSIM file but the same H→ZZ MC dataset; this is not external validation across a process, detector condition, dataset, or real data. The exposure audit establishes absence of candidate-specific numerical exact-ray values in the scanned artifacts, not absence from human memory or every possible external source. The exact target is along the fixed ray under a frozen local hybrid covariance, not a global shortest boundary. No preregistered population-level success threshold was tested.
• The fixed-16 §5r extension is larger and prospectively sealed, but it is still a deliberately selected near-distance cohort: the eight smallest eligible d_lin events in each channel after fixed exclusions, not a random or population-representative sample. Its 14 paired rows cannot supply population coverage, a calibrated error distribution, or transport beyond the same H→ZZ MC dataset. The 14/14 positive signed residuals are an observed cohort pattern, not proof of a universal overprediction bias. The two NA rows remain part of the formal fixed denominator and were not replaced. As in §5q, the targets are exact only along the fixed rays under a frozen local hybrid covariance; they are not global shortest boundaries in a complete detector covariance.

11. Next steps and current interpretation limits

Scientifically valuable next:

1. The full 42-event generator ancestry audit is now complete (§5m). All 42 EventIDs were traced; no all-mother or Z-copy-collapsed label changed, no ambiguity remained, and the one prespecified recount left the bookkeeping agreement at 59/101. Treat this step as done.
2. The fixed-101 official JHUGenMELA interference audit is now complete (§5n). It found broad overlap between mismatch and match events and no stable evidence of stronger interference in the mismatch group. Treat the fixed-sample calculation as done. A future extension would require an independently fixed event sample and a separately specified physical observable; do not reinterpret P_A or P_B as probabilities that one pairing is the true history.
3. Independent seeds and confidence intervals for the toy scans; parent‑boost, spin correlation, background and generator‑systematics extensions.
4. The analytic local‑boundary denominator is now explained (§5j) — treat that step as done. Within the fractional‑pT metric, ‖∇_u g‖ reduces to the two active candidate masses and the shared‑lepton topology, matched to the saved finite difference and to an exact stored‑mass derivative. A complete detector‑covariance distance remains future work and should not be built from a uniform 1–2% smearing. Section 5o now supplies only a fixed-three-event local hybrid-covariance benchmark. Future work should, in order: - first verify which per‑electron and per‑muon uncertainty or covariance quantities are actually available in the fixed 2012 AODSIM representation; - treat electrons and muons separately; - distinguish a diagonal resolution approximation from a covariance‑aware metric; - validate on an independently fixed sample; - do not call a resolution‑weighted distance a measured pairing‑error probability. A first feasibility step is now done (§5k): a fixed five-event cmsRun canary confirmed the AODSIM carries per-lepton track covariance (20/20 finite, symmetric, positive) and formed a pT-projected uncertainty for 4/5 preselected events. The availability side is now audited more fully (§5l): across the fixed 101 events 236/236 muons and 158/168 electrons carry usable stored uncertainties (91/101 events fully available), the 10 missing electrons were characterized and replicated at 6/248 in an independent file, and a pooled 416-electron overlap diagnostic showed complete categorical seed separation between the two electron kinds. Still not done: an official PF-electron combined-error model (only explicit sensitivity proxies exist, not a recovered official value), a complete covariance-aware distance with the unavailable scale-angle and cross-lepton terms, and validation beyond this one H→ZZ MC dataset — the independent file is same-dataset/different-file, not another process, detector condition, dataset, or real data. The §5o result adds η/φ and available within-lepton covariance for three events, but remains local, hybrid, and adaptively completed. Every §5k–§5l z and flip number remains a local-linear-Gaussian model value, and every §5o distance remains a local sensitivity rather than a measured pairing-error probability. Any cross-seed-path proxy must be treated as an explicit sensitivity model rather than a recovery of the official uncertainty.
5. The two-event pilot (§5q) and fixed-16 extension (§5r) are now complete. The extension found 14/14 improvements under the mechanically fixed paired rule and retained both NA rows without substitution. Treat this as strong fixed-cohort evidence for the quadratic fixed-ray approximation, not as population calibration. The next informative test is a newly sealed sample from another AODSIM file and preferably another production condition or dataset, using the same metrics and no target-informed replacement. It should test both quadratic improvement and whether the 14/14 positive residual direction reproduces. Do not add events to either completed cohort after seeing its targets.

Do NOT over‑interpret now: do not report 58.4% as physical accuracy; do not call the 42 mismatches confirmed rule errors; do not treat the scan's 60/101 as an improvement; do not call the 120–130 GeV real‑data candidates Higgs events; do not call any of Stages 1–3 experimental sensitivity, discovery significance, or a Higgs re‑discovery.

12. Fact / interpretation / unverified separation

• Measured facts (in the artifacts): 59/101 agreement and its exact CI; 42 mismatches with their gaps; stable 65 / rule‑dependent 34 / near‑tie 2, 35 flip‑point events; three‑rule agreement 99/99/23/22; the two mass‑based rules differ on exactly 2/101 (1:604:181002, 1:608:182172), of which only 181002 is in the fixed near_tie class; the four-rule vote split is 4‑0 = 12, 3‑1 = 56, 2‑2 = 33 (a four-way split is impossible with two pairings per event); the fixed-sample Δm distribution (min 0.882, median 36.936, max 77.051 GeV) with 181079 and 181002 ranked 1st and 2nd; the Δm-bin counts 2/1/2/6/90 and their flip counts; the verified local boundary distances 1.748% and 1.838% with linear/nonlinear agreement better than 0.01%; the Spearman correlations 0.8388 / 0.8996 / 0.9832 / 0.8532; 26 real candidates, 4 in 120–130 GeV; Z‑peak fit centers 91.25 / 90.79 GeV; 0/4 survive strict cuts; 14/16 electrons at −999; and, for the analytic gradient audit (§5j), the closed‑form/analytic‑component norm agreement (max difference 1.42e‑14), the exact stored‑mass derivative vs saved finite‑difference agreement (component RMS 2.64e‑09, max 1.92e‑08), the 79 / 22 difference‑ vs sum‑relation split with its shared‑component squared‑norm fractions (mean 0.1248 / 0.6316), and the branch‑audit counts (0 origin ties, 0 active‑pair switches in the ±1e‑4 stencil). For the pT-projected uncertainty canary (§5k): 5/5 fixed events processed, 20 selected leptons audited, 20/20 track covariances finite/symmetric/positive, 19/20 usable pT projections and 4/5 events fully usable, the per-event z and linear-Gaussian flip numbers in the §5k table, and the one NA electron (1:604:181079 slot 3, candidate-P4 p4Error = 0) reproduced with 0 SHA-256 mismatches; and, as an independent repository-wide check (not limited to the two files originally cited), a full grep of RecoParticleFlow, DataFormats/ParticleFlowCandidate, and RecoEgamma/EgammaElectronAlgos at tag CMSSW_5_3_10 found exactly one occurrence of setDeltaP( — the setter's own declaration — and zero call sites, bounded to that single tag and those three packages. For the §5l availability audits: 404 selected leptons audited (168 e + 236 μ), candidateP4Kind=2 && p4Error==0 in 10/168 electrons (10/42 4e events, one per event), 158/168 electrons kind=1 with positive active error, 236/236 muons usable, 91/101 events fully available; the channel-separated complete-case z summaries (4e n=32 median 18.189 max 74.517; 4mu n=59 median 35.4552 max 111.123) and the four-model affected-event z span 0.136–36.84 (max flip 0.446); the independent-file counts 6/248 electrons and 6/62 4e events with 0 EventID overlap and their Wilson intervals; and, in the pooled 416-electron overlap diagnostic, kind=1 400 / kind=2 16, the 370-electron dual-seed subgroup (146 + 224), complete seed-cell separation, the central-95% inclusion counts (candidate pT 14/16, |eta| 16/16, relative ECAL error 16/16, relative track-pT error 14/16, ratio 10/16, event pT rank 16/16), the descriptive median shifts (|eta| 1.323 vs 0.732; relative track-pT error 0.255 vs 0.187), and the 1:608:182324 nearest-neighbour outlier (ratio 5.155, 4.425 IQR units). For the full 42-event ancestry audit (§5m): 42/42 events and 168 matched leptons processed; 28,143 genParticle rows and 64,840 relation rows saved; 0 unresolved indices and 0 reciprocal-edge failures; 0/42 all-mother changes, 0/42 Z-copy-collapsed changes, 0/42 ambiguous labels; A/B/C/D = 0/0/42/0; and the one recount remained 59/101. Runtime was 2:26.31 with maximum RSS 365,260 kB; artifact hashes, independent postprocessing, and the embedded top-10 reproduction all had zero mismatches. For the fixed-101 JHUGenMELA audit (§5n): 101/101 events and 404 input lepton rows processed; mismatch/match relative-interference medians +4.54% / +10.03%; pooled median difference −5.49 points with fixed bootstrap interval [−15.80, +6.16]; 101/101 structured-order checks passed with maximum ME change 0; the 181002 benchmark reproduced +25.0719358298%; and two independent runs had 0 non-timing-field mismatches. For the fixed-three-event local hybrid-covariance benchmark (§5o): all 8 electron candidate/GSF centre-direction checks passed (maximum DeltaR 0.000365077); the two saved pT-only values reproduced to absolute differences 4.46e-11 and 1.87e-11; 181002 gave linear/ray/iterative distances 0.772266030 / 0.775368015 / 0.775346555 σ; 181079 remained NA without imputation; and 182137 gave 1.607406407 / 1.613524556 / 1.613366841 σ. The initial three-event record retains its estimated 3,700-vs-≈1,494 cumulative score-cap violation. In the separate 4mu completion test, the new 12D analytic Jacobian agreed with finite differences to maximum component difference 2.24e-9 GeV/unit and maximum relative L2 difference 5.36e-11; formal and reproduction results matched, no active pair switched, and the ledger recorded 341/528 exact-score evaluations. For the prospective two-event audit (§5q): candidate-specific numerical exact-ray exposure was 0 in 31 candidate-related artifact records; Stage P used 20 event-score evaluations, Stage T used 174, and Stage C used 0; the exact ray targets were 4.762146960 σ (4e) and 4.487082139 σ (4mu); the quadratic absolute errors were 0.017346340 σ and 0.000342525 σ, smaller than the corresponding linear errors 0.306491862 σ and 0.022768396 σ; both quadratic predictions were overestimates; and all three deterministic-repeat and stage-isolation checks passed. For the fixed-16 extension (§5r): Stage P used 160 event-score evaluations and produced 15 predictions / 1 NA; Stage T used 1228 evaluations and produced 14 targets / 2 NAs; their overlapping NA sets yielded 14 mechanically paired rows. Stage C used zero score calls and found IMPROVED / TIED / WORSE = 14 / 0 / 0, median DeltaE = 0.0794076141, linear relative errors 0.00868198–0.158321, quadratic relative errors 0.000403737–0.0194545, and positive signed quadratic residuals for 14/14 paired events. Both computational passes and all Stage P/T/C integrity closures passed; the Stage-C 13-entry artifact manifest had zero mismatches.
• Interpretation: the 42 mismatches concentrate where ancestry and closest‑to‑Z strongly conflict; classification and pairing information are distinct in the toy study; Δm is a compact indicator of an event's proximity to the on‑shell mass‑rule decision boundary; and, within fractional‑pT local coordinates, the local‑boundary denominator is explained by the two active candidate masses and their shared‑lepton topology (§5j); and the fixed AODSIM does carry per-lepton track covariance, while for one PF-combination electron (181079) the candidate-P4 combined-uncertainty field was left unset by the standard producer path — a bounded reading of the CMSSW source, not a confirmed defect (§5k). The §5l audits generalize this: across the fixed 101 events the same candidateP4Kind=2 && p4Error==0 pattern appears in 10/168 electrons and replicates in an independent file; the pattern coincides perfectly with the tracker-driven-only seed category, which is read as a producer/reconstruction-path diagnostic, not as a physical cause, a seed-status cause, or a low-pT/bremsstrahlung mechanism (no such mechanism test was performed). The kind=2 electrons overlap kind=1 on most continuous covariates but sit in a disjoint seed category, so a kind=1 error model would be category-level extrapolation. The §5m audit shows that the 42 disagreements are not explained by using only mother(0), ignoring Z status-copy collapse, or forcing an ambiguous ancestry label. They remain disagreements with an internally traced generator bookkeeping convention, not confirmed physical pairing errors. The §5n audit further shows that relative identical-lepton interference is broadly distributed in both groups and is not stably larger in the mismatch events; this is evidence against that specific fixed-sample association, not proof of absence or a causal conclusion. The §5o comparison shows that, for the two accepted benchmark events, exact local boundary iteration changed the hybrid-covariance distance by less than 0.4% from the local linear value. For 182137, the analytic Jacobian completed the termination at essentially the predecessor's last numerical point; this is a numerical-resolution result, not evidence for a new physical boundary or detector effect. The §5q two-event result shows that the fixed quadratic correction improved the fixed-ray approximation for those two audited events. The §5r result extends that observation to all 14 paired members of a prospectively fixed near-distance cohort in both channels. This supports the quadratic correction within the audited cohort and range, while the non-random selection and same-dataset scope still prevent a population-wide accuracy or bias claim.
• Structural qualification: much of the Δm–boundary correlation is algebraically expected rather than discovered — for the 79 both‑below events Δm is the score‑margin numerator, and the flip behavior across the Δm bins follows from the same relation. The 5 GeV bin split was post hoc, the tests are unadjusted, and the smallest bins hold 2, 1, and 2 events.
• Unverified / undetermined in this artifact set: whether the rule truly beats chance; whether identical-lepton interference causally produces any mismatch or favors a uniquely physical A/B pairing; detector‑level instability (the perturbations are artificial, not resolution models); physical pairing correctness; generalization of the §5n ME comparison to any independent sample; background, detector-systematic, and theory-systematic behavior; the real XRootD byte totals for several jobs (no reliable counter in the logs); and — for the analytic gradient audit specifically — detector‑level instability, any detector‑aware ranking, the global (non‑local) boundary distance, generalization to an independently fixed sample, and physical pairing correctness. The §5j closed form is a coordinate‑dependent statement inside the fractional‑pT metric with the active Z1 pair fixed, not a coordinate‑independent geometry. For the §5k canary specifically: a full detector-aware / covariance-aware distance (η/φ and cross-lepton terms), generalization beyond the five preselected events, and any physical mispairing probability remain unverified — the §5k flip numbers are local-linear-Gaussian values, and the official PF-combination electron error is not recoverable from this AOD's p4Error (an ECAL+GSF approximation is possible but was not built). For the §5l audits specifically: whether the kind=1 and kind=2 reconstruction-error mechanisms are equal, whether a kind=1 error model is transportable to kind=2, and whether the affected rate generalizes beyond this one H→ZZ MC dataset all remain unverified — the independent file is same-dataset/different-file, the two Fisher p-values are not independent confirmations, and failure to reject equal rates is not evidence of equality. The alternative-error models are sensitivity proxies, not the official combined uncertainty, and every §5l flip number is a local-linear-Gaussian model value, not a measured physical mispairing probability. For §5o specifically, the complete electron covariance, scale-angle and cross-lepton correlations, covariance evolution away from the centre, global shortest boundary, independent-sample generalization, physical pairing correctness, and any measured mispairing probability remain unverified. The adaptive analytic-Jacobian completion does not supply an independent confirmation of the 4mu distance. For §5q specifically, population accuracy, uncertainty or coverage of the quadratic predictor, performance on additional channels or event regions, validation outside the same H→ZZ MC dataset, a global shortest boundary, and a complete detector-covariance interpretation all remain unverified. For §5r specifically, population coverage and calibration, behavior beyond the selected d_lin range, reproduction of the all-positive residual pattern in a new sealed sample, validation outside the same H→ZZ MC dataset, a global shortest boundary, and a complete detector-covariance interpretation all remain unverified.