MIsCORE–RIUD: Paper II — Connected Multiscale Paths from Residual Morphology to Raw H1/L1 Strain

1. Introduction

Paper I [22] closed a lower-order question: recurrent merger-local post-model morphology can be defined, audited, and carried across a population without first assigning a unique physical cause. Paper II begins exactly at that boundary. It does not reopen the Paper-I discovery problem; it asks whether the already-defined local objects remain isolated or connect into a higher-order structure.

A localized object is still point-like. It has a time, frequency, temporal support scale, detector relation, and versioned observational provenance, but recurrence alone does not show whether neighboring local objects belong to the same evolving structure. If the feature changes scale while it evolves, exact equality of the lower-order cell can fragment one higher-order object into several detections that are locally different but topologically connected.

Paper II therefore asks two serial, non-equivalent questions. First: do the frozen local structures form ordered connected multiscale paths across time, frequency, and temporal scale? Second: once that path is frozen, does the same path population receive support in H1/L1 raw strain without a second discovery search? The first question constructs the higher-order object; the second tests whether that object survives a change of representation.

This shift changes what must remain invariant. Paper I needed exact local coordinates because its scientific question was where recurrent local morphology existed. Paper II instead asks whether those localized objects are connected. Exact width identity is therefore not promoted into a universal invariant: a path may preserve temporal ordering and connectivity while moving through compatible temporal scales.

The second shift concerns validation freedom. During path construction, observed data and matched controls must receive symmetric search freedom. After the path is frozen, however, reopening a free search in raw H1/L1 strain would answer a different question: whether some favorable raw path can be found. Paper II instead closes the coordinates before raw projection so that the raw-strain stage tests survival of the same object. A later dynamics analysis must restore control-native map and path freedom because the scientific object changes again, from path survival to path-generating dynamics.

The Paper-II source archive contains 68 preserved executions. They are not 68 independent hypothesis tests. The archive contains successful results, negative criteria, technical recovery, reruns, hardening, alternative representations, nuisance branches, and computational-reproduction steps. Its role is provenance: to preserve which experiment created, changed, weakened, blocked, or reproduced the path object.

The final claim is intentionally narrower than several intermediate interpretations. The path receives population support under fixed raw-strain projection, survives event-influence and conditioning attacks, and shows a separate phase-directionality result under its own null. At the same time, the strictest eventwise coherence criterion does not pass, simple eventwise source-parameter attribution does not explain the path, and single-detector transient nuisance families remain viable alternatives. Those negative results are part of the endpoint, not exceptions removed from it.

This standalone manuscript is source-audited against the preserved 68-execution Paper-II results ledger and frozen authority-package provenance. Historical cohorts and denominators are kept separate whenever they answer different scientific questions.

1.1. Scientific Handoff: From Local Objects to Connected Paths

Paper II is best read as a sequence of changes in scientific object and evidential constraint.

Evidential constraint	Change, decisive consequence, and surviving claim
Local recurrence does not establish connectivity	Raise the object from node to ordered multiscale path. Exact lower-order cell identity is relaxed where it fragments continuity. Survives: the connected path becomes the target.
Path algorithms can manufacture attractive lines	Audit path construction and scorer geometry; make path-search/model-selection freedom explicit and mirror it in controls. Survives: only the more symmetric path construction.
Residual-map support may be representation-specific	Freeze the path and project it into raw H1/L1 strain. No second raw free search is allowed. Survives: cross-representation fixed-path support.
One event may dominate a population result	Leave one event out at a time. Every omission remains below the frozen threshold. Survives: population support is not single-event driven.
Conditioning or one coordinate convention may create the result	Remove the 60-Hz notch and perform a locked-coordinate ablation. Both variants remain supported.
A nuisance family may mimic the population	Prelock six nuisance families over 62,400 jobs. Four coherent/independent families fail to reproduce baseline; single-detector transients remain weaker alternatives.
Population support does not imply phase directionality	Add a separate selector-aware phase statistic. 0/100,000 conditional phase draws exceed the observed value. Survives: conditional directionality under its own null.
“Detector coherence” can mean different tests	Preserve population/map and strict eventwise criteria separately. Population/map support survives; strict eventwise FDR05 survivors = 0.
A path may simply encode known source parameters	Run the strict physical-attribution audit. Global FDR05 survivors = 0 in direct and visibility-controlled families. Simple eventwise source-parameter attribution is not established.
A frozen path must be reproducible without importing prior outputs	Reacquire and cleanly replay from frozen selector/rule assets. Fixed branch is reproduced without reselection; prior result import is prohibited as input.

1.2. Relation to Paper I and Established Residual, Unmodeled, and Path/Track Analyses

Paper I [22] asked whether recurrent merger-local post-model morphology could be made into a stable scientific object before physical attribution. Paper II does not redefine that object retroactively. It takes the surviving local object as its lower-order input and asks whether local instances are connected. This distinction separates Paper II from ordinary residual consistency testing and from a second unmodeled search. Standard residual tests ask whether post-model data are compatible with expected noise or whether additional coherent power remains after subtraction [2,3,12]. Flexible unmodeled approaches such as BayesWave reconstruct transient morphology [13]. Population residual approaches such as SCoRe ask whether specified residual quantities show source-dependent organization [14,15,16]. Connected tracks and graph-like time-frequency objects are not, by themselves, new ideas. Seedlessclustering methods search spectrograms for extended gravitational-wave tracks, including applications to compact-binary coalescences [17,18,19]. Wavegraph uses graph-constrained time-frequency clustering inside a model-independent transient-search architecture [20]. More generally, chirplet path pursuit treats connected time-frequency trajectories as detection objects in noisy data [21]. Paper II therefore does not claim invention of path finding, graph clustering, or time-frequency track detection. Its distinction is evidential ordering. The path is built from already-audited post-model local objects, the path detector itself is attacked for morphology-inducing bias, the path-search and model-selection freedom granted to the observed construction is mirrored in matched controls, and only after that object is frozen is it projected at fixed coordinates into raw H1/L1 strain. The raw stage is therefore not another track search.

It is a cross-representation attack on a previously defined object. The paper consequently separates four questions that neighboring methods may combine or answer under different contracts:

• Can the Paper-I local objects be connected under a frozen multiscale rule without forcing a path in every event?
• Can the path construction survive an audit of the scorer and its own search freedom?
• Does the already-defined path receive support in raw strain without coordinate reselection?
• What physical or instrumental mechanism generates the path? Paper II closes the first three within their recorded roles and leaves the fourth open. The contribution is therefore not “a new way to draw lines in a spectrogram.” It is the promotion of a lower-order residual object into a frozen connected object, followed by a validation contract that changes when the representation changes.

1.3. Central Evidential Claim

Paper II contains two serial scientific acts that must not be conflated:

Act	Object	What establishes it	What does not establish it
Construction	Connected multiscale residual-derived path	Frozen node provenance, multiscale connectivity, scorer-bias audit, matched construction freedom, and path-free counterexamples.	The later raw p-value by itself.
Projection	Support for that already-defined path in raw H1/L1 strain	Fixed coordinates, no raw reselection, population null, LOEO, conditioning/ablation, nuisance and phase roles.	A second raw-domain path discovery.

The central claim is therefore not “a line was found with a small p-value.” The evidential object is serial: a connected path is first defined and hardened under a path-construction contract, then the frozen coordinates are challenged in raw strain under a fixed-projection contract. The raw-strain result is meaningful because coordinate freedom has been closed before the representation change.

For readers accustomed to Gaussian-tail language, the finite-null probabilities can be translated descriptively: p=9.9999×10⁻⁶ corresponds to approximately 4.265σ one-sided, and the worst leave-one-event-out p=5.99994×10⁻⁵ corresponds to approximately 3.85σ one-sided. These conversions do not create a new test, do not imply a Gaussian null, and are not used as universal discovery labels; the empirical p-values and their finite-null resolution remain primary.

This distinction also provides the bridge to the rest of the paper: Section 2 and Section 3 define the higher-order path and close its search freedom; Section 4, Section 5, Section 6 and Section 7 test survival, influence, nuisance alternatives, and conditional phase directionality; Section 8, Section 9 and Section 10 preserve negative boundaries and reproducibility; Section 11, Section 12, Section 13 and Section 14 state the method, interpretation, limitations, and endpoint.

2. From Local Nodes to Connected Multiscale Paths

Paper II first raises the observational object from a local node to an ordered multiscale path. This section defines that promotion before any raw-strain validation is considered.

2.1. Why Exact Point Identity Could Not Remain Universal

Paper I localizes morphology in time, frequency, temporal width, detector lag, and detector relation. Those coordinates are necessary when the object is a local node. They are not automatically the correct invariants when the object becomes a path. Suppose two adjacent local structures occupy nearby time-frequency coordinates but have different widths. If exact width equality is required, they become separate objects even when their ordering and continuity suggest one evolving structure. A lower-order detector can therefore over-segment a higher-order object. Paper II treats this as a representation problem rather than a license to erase provenance. The original node remains frozen. Its lower-order coordinates are not rewritten. What changes is the higher-order equivalence rule: compatible changes in scale may be connected when the path criterion is satisfied. The canonical multiscale width family spans: 10, 20, 30, 40, 50, 60 ms. The scientific principle is: Lower-order precision can become higher-order nuisance. This does not mean that precision is undesirable. It means that a coordinate useful for identifying one object may become an unnecessary identity constraint when the next object is a relation among those objects.

2.2. Path Object

For event e, let a frozen local node be represented schematically as:

x(e,j) = [t(e,j), f(e,j), w(e,j), ell(e,j), m(e,j), s(e,j), q(e,j)]

where t is local time, f is frequency, w is temporal width or scale, ell is detector lag, m is the detector-relation label, s is the version-specific local score, and q is the provenance object.

A path is not defined by equality of all coordinates across nodes. It is an ordered higher-order object:

P(e) = [x(e,1), x(e,2), ..., x(e,n_e)]

Its admissibility is determined by the frozen Paper-II connectivity contract:

• Candidate nodes retain their original lower-order provenance rather than being rewritten after connection.
• Nodes are ordered in merger-local time.
• Neighboring nodes satisfy the frozen edge-continuity rule in time, frequency, and scale.
• Temporal support may move within the locked 10/20/30/40/50/60 ms width family.
• Weak intermediate nodes may carry topological information when they satisfy connectivity rather than being discarded solely by a point-only strength threshold.
• Exact lower-order width or frequency-cell identity is not promoted into a universal path invariant.
• Observed path-search and model-selection freedom is reproduced in the matched construction null.
• Detector, lag, score, source file, and version provenance remain attached to constituent nodes and the path.

The exact numerical edge thresholds and implementation-specific branch logic remain authority-bound to the frozen rule files and reproducibility package. The manuscript states the scientific contract without replacing the executable implementation with an invented cosmetic equation.

Paper II relaxes identity, not provenance. A node discovered at one width does not become a different historical node after connection; the higher-order path simply stops requiring every lower-order coordinate to remain invariant while scale changes.

2.3. Path-Construction Bias Became a Scientific Target

An attractive path can be manufactured by the path detector itself. Earlier scorer forms exposed precisely this risk: directional or rising rewards, jump restrictions, saturation behavior, detector dominance, or single-width dominance could favor visually persuasive line-like structures even when the underlying data did not justify them. Those earlier forms were therefore not treated as final evidence. The Paper II lineage audits the path-construction machinery itself, removes or neutralizes morphology-inducing preferences that were not required by the scientific question, and requires the matched null to repeat the same path-search and model selection freedom used on the observed data.

The governing rule is: A path cannot be validated by allowing the observed data to optimize connectivity while forcing controls to evaluate only the observed path. This makes scorer design part of the falsification program rather than a hidden preprocessing choice. The final fixed raw-strain projection occurs only after the higher-order path object has survived this construction stage and has been frozen. Discovery freedom belongs upstream; projection is downstream.

2.4. Corrected 25-Event Atlas: Path Existence Was Not Guaranteed by Construction

Before the final 13-event population was frozen, the path representation was exercised in a corrected fixed 25-event candidate atlas. That stage is useful because it provides a direct answer to a basic construction objection: does the path algorithm simply produce a path whenever it is given an event? It did not. The corrected atlas recorded:

• fixed candidate events: 25;
• events with one or more paths: 16;
• events with zero path: 9;
• multi-pattern events: 14. The scientific verdict for that stage was

FIXED25_FULL_CANDIDATE_PATTERN_ATLAS_CORRECTED_COMPLETE. The existence of nine zero-path events is important. A connectivity algorithm that mechanically turns every candidate into a path would make downstream population support difficult to interpret. The atlas instead shows a mixture of path-bearing and path-free outcomes under the same higher-order construction framework. The multi-pattern count is equally important. Fourteen of the 25 candidates admitted more than one pattern representation. Paper II therefore did not begin from the assumption that one rigid universal sequence already existed. The later family-dependence audit formalized this multiplicity more directly. This atlas is a development-stage object, not the canonical raw-projection denominator. It is retained here because it explains why the eventual 13-event path population is a selected frozen observational object rather than an inevitable output of the path code.

2.5. Canonical Lock: Window, Width Family, and Outcome Independence

The canonical 13-event stage was explicitly design-locked as a fixed internal validation using a merger-relative center window of -0.12 s <= t_center <= +0.03 s. The design lock bound the analysis to the exact frozen path rule inherited from the earlier engine and recorded outcome_driven_change = false. The accompanying rule lock states:

• LOCK_ROLE = FROZEN_BEFORE_PATH_OUTCOME_REPLAY;
• PATH_OUTCOME_USED_TO_SELECT_PANEL = false;
• EVENT_REPLACEMENT_AFTER_OUTCOME = false;
• THRESHOLD_RETUNING_AFTER_OUTCOME = false. This matters because a path population can be made artificially clean if events are replaced after outcomes are known or if the connectivity threshold is retuned until the desired population survives. Those freedoms were explicitly excluded at the canonical lock. The width policy was also frozen before outcome interpretation. All six widths, 10, 20, 30, 40, 50, 60 ms, were retained for path composition and leave-one-width-out diagnostics. The source lock explicitly records no BEST_WIDTH verdict. The multiscale representation was therefore not reduced after the fact to whichever width happened to produce the strongest final statistic. A technically incomplete 14-event long-window acquisition branch illustrates the same no-replacement policy from another angle. One event failed acquisition after repeated H1 fetch timeouts. That stage was recorded as SCIENTIFIC_VERDICT = NOT_TESTED and TECHNICAL_VERDICT = LOCKED_LONG_WINDOW_ACQUISITION_INCOMPLETE, with the next action RETRY_FAILED_EVENTS_NO_REPLACEMENT. The failure was not converted into a scientific null and the event was not silently replaced by an easier case.

2.6. Frozen Path Population

The canonical Paper-II population contains:

• 13 events;
• 172 locked nodes;
• 26 H1/L1 raw-strain files. These denominators define the primary fixed-path population role.

The 13 events are not interpreted as a random sample from all binary-black-hole mergers, and the 172 nodes are not treated as independent astrophysical detections. They are a frozen connected-path population carried forward from the path-construction lineage.

Event	Locked path nodes	Paper-II role
GW150914_095045	7	canonical frozen path population
GW200129_065458	14	canonical frozen path population
GW200224_222234	11	canonical frozen path population
GW230824_033047	16	canonical frozen path population
GW230831_015414	16	canonical frozen path population
GW230904_051013	16	canonical frozen path population
GW231206_233901	16	canonical frozen path population
GW231221_135041	6	canonical frozen path population
GW240629_145256	16	canonical frozen path population
GW240919_061559	10	canonical frozen path population
GW240920_124024	12	canonical frozen path population
GW241006_015333	16	canonical frozen path population
GW241231_054133	16	canonical frozen path population

The path-provenance package preserves event identity, residual family, node count, time range, frequency range, width range, lag mode, source path, and SHA-256 provenance for each frozen path. The standalone manuscript uses those records as the authority for path membership rather than reconstructing membership from later outcome tables.

Figure 1. A. Frozen connected multiscale paths for the first seven canonical events. Each line connects successive nodes from the archived per-event observed_path_nodes.csv asset in frozen path order; marker size encodes temporal width. No interpolation, smoothing, or synthetic trajectory construction is used. B. Frozen connected multiscale paths for the remaining six canonical events. Axes and marker encoding match Figure 1A. Together, Figure 1A–1B display all 13 canonical paths and all 172 frozen nodes used by the Paper-II path population.

Figure 1. A. Frozen connected multiscale paths for the first seven canonical events. Each line connects successive nodes from the archived per-event observed_path_nodes.csv asset in frozen path order; marker size encodes temporal width. No interpolation, smoothing, or synthetic trajectory construction is used. B. Frozen connected multiscale paths for the remaining six canonical events. Axes and marker encoding match Figure 1A. Together, Figure 1A–1B display all 13 canonical paths and all 172 frozen nodes used by the Paper-II path population.

2.7. Canonical Population Versus Archaeology Cohorts

Paper-II archaeology also contains wider 16-event roles. Those roles must not be collapsed into the 13-event primary population. In particular, the family-dependence/effective-multiplicity audit spans 16 events and reports 55 event-family tests, 55 exact unique top paths, Jaccard80 near-duplicate structure in four events, and median effective family count 2.951545. A separate strict physical-attribution role also spans 16 events. These wider roles answer different questions: how many path-family alternatives were effectively present, and whether simple eventwise source parameters explain path features. They do not alter the canonical raw-projection denominator of 13 events / 172 nodes / 26 detector files. This denominator separation is retained throughout the paper.

3. Why Raw Strain Is Not Searched Again

With the path now frozen, the analysis changes role from construction to representation testing. The raw-strain stage therefore inherits coordinates rather than reopening them.

3.1. The Fixed-Projection Question

After the residual-derived path has been frozen, raw strain is not a new discovery field in Paper II. The question is not: Can a strong path be found somewhere in raw H1/L1 data?

The question is: Does the already-defined path receive support at its frozen coordinates in raw H1/L1 data? These questions require different freedom. A free raw search could choose new times, frequencies, widths, detector relations, or path geometry after seeing the raw data. A positive result would then mix the original residual discovery with a second raw-domain discovery and would be difficult to interpret as support for the same object. Paper II therefore uses fixed projection.

Statistical form of the distinction. Let θ̂ₑ denote the residual-derived path coordinates frozen before raw evaluation. Paper II tests T_raw(Dₑ; θ̂ₑ), not max_θ T_raw(Dₑ; θ). The maximized quantity would be a new raw-domain search and would require a new search-aware null with the same optimization freedom. Holding θ̂ₑ fixed therefore does not make the validation easier by fiat; it makes the hypothesis narrower and prevents raw data from choosing a second favorable coordinate system after the residual path is known.

3.2. Fixed Coordinates, Frozen Roles

Let P_e denote the frozen residual-derived path for event e. Let R_{e,d}(P_e) denote the version-locked raw-strain response evaluated at the path coordinates for detector d. The manuscript does not substitute a simplified analytic kernel for the authoritative executable implementation. The scientific contract is more important than a cosmetic closed form:

• the path coordinates are inherited from the frozen path object;
• raw H1/L1 data are conditioned under the recorded preprocessing contract;
• the raw representation does not select new path coordinates;
• event contributions are aggregated under a prespecified equal-weight population rule;
• controls are generated under the corresponding fixed-projection null;
• invalid or unavailable records remain explicitly invalid or unavailable rather than being converted to zero. For each canonical event, the fixed raw role uses frozen 4-s, 4096-Hz H1 and L1 strain files, giving 26 detector files. The locked evaluation band is 40–400 Hz. The primary preprocessing configuration is WHITEN_BP40_400_NOTCH60 with equal event weighting; prespecified variants include band-passed/no-notch processing, visibility weighting, and a locked-coordinate-drop ablation. Public O4 4096-Hz products are downsampled from 16384 Hz and retain the documented calibration, data-quality, and anti-aliasing limitations of the public release [5,6,7,8,9,10]. No coordinate is moved to improve a raw-strain result. The raw projection authority also records 52/52 event-variant rows completed, corresponding to 13 events across four frozen raw-processing roles, with 400 controls per event/variant used in the event-level support layer and 100,000 valid global null draws for the primary population statistic. These counts belong to different levels of the calculation and are not multiplied into one synthetic sample size. The primary population statistic under this contract is 1.171221.

3.3. Why the Freedom Rule Changes at the Next-Order Question

Paper II asks whether the already-found path survives a new representation, so its coordinates are frozen. A later dynamics question would ask whether the process that generates and organizes paths is recurrent. In that different test, controls would need to construct their own maps, paths, trajectories, and family labels; forcing observed coordinates into controls would merely test coordinate rarity again. The apparent contrast disappears once the scientific object is stated correctly.

4. Canonical Fixed-Path Population Result

The fixed-projection contract now permits a direct population test: the question is whether the frozen path population receives support at the same coordinates in raw H1/L1 strain.

4.1. Primary Equal-Weight Stack

The frozen population statistic is T_obs = 1.171221. The empirical null contains 100,000 valid draws, with k=0 observed-or-more-extreme exceedances. Using the plus-one rule, p(+1)=(k+1)/(N+1)=1/100001=9.9999×10⁻⁶. As a descriptive one-sided Gaussian-tail translation, this is approximately 4.265σ. The empirical probability remains the primary result: the σ value does not imply that the null is Gaussian, does not increase the finite-null resolution, and is not assigned to individual events. This probability belongs to the specific fixed-path population null; it is not a catalog-wide false-alarm probability or a prevalence estimate.

4.2. Event-Level Evidence is Subordinate to the Population Contract

Paper II does not require every event to be individually significant under an eventwise test. The primary object is the population of frozen paths under the prespecified aggregate statistic. This is important because a population object can be supported by a distributed pattern that no single event reproduces at the same strength. Converting the Paper-II claim into “13 individually detected paths” would misstate the statistical role. The relevant influence test is therefore leave-one-event-out rather than a requirement that all 13 eventwise p-values cross one threshold.

Unit-of-inference guard. The equal-weight population statistic is event-weighted. The 172 nodes define the frozen geometry inside the 13 event paths; they are not treated as 172 independent astrophysical trials. This prevents node-rich events from inflating the effective sample size and keeps the primary claim at the population-of-events level.

5. Population Robustness: Leave-One-Event-Out, Conditioning, and Ablation

5.1. Leave-One-Event-Out

Each event is omitted in turn and the frozen population calculation is repeated. Every leave-one-event-out omission remains supported under the frozen threshold. The largest empirical probability is p_LOEO,max = 5.99994×10⁻⁵, corresponding descriptively to approximately 3.846σ on a one-sided Gaussian tail. Thus no single event can be removed and make the population result disappear. This does not imply equal contribution across events; it establishes robustness to single-event omission under the recorded population statistic.

5.2. No-Notch Replay

A replay without the 60-Hz notch gives: p = 9.999×10⁻⁴ (≈3.090σ one-sided, descriptive). The result weakens numerically relative to the canonical fixed-path test but remains supported. The path population therefore does not depend entirely on that specific notch operation. The no-notch replay is a conditioning robustness test. It is not an independent discovery experiment and is not pooled with the primary p-value.

5.3. Locked-Coordinate Ablation

A locked-coordinate ablation gives: p = 1.99998×10⁻⁵ (≈4.107σ one-sided, descriptive). This branch tests sensitivity to the declared coordinate structure without reopening a free raw search. Its purpose is to ask whether the population evidence collapses when a predeclared coordinate component is removed or altered under the frozen ablation design. Again, this is a robustness layer, not a second independent discovery probability.

5.4. What These Variants Jointly Establish

The primary result survives deletion of any single event, a no-notch conditioning variant, and a locked-coordinate ablation. These tests constrain different explanations and remain non-poolable.

5.5. Finite-Null Calibration and why σ is Secondary

The empirical p-values remain the inferential quantities. For readers who use Gaussian-tail language as a familiar scale, this revision reports the one-sided descriptive transform z = Φ⁻¹(1−p). The transform changes neither the empirical null nor its finite resolution. In particular, 0 exceedances in 100,000 draws means p(+1)=1/100001, not p=0 and not infinite σ. The values below are not pooled and do not represent independent discoveries.

Role	Empirical p(+1)	Descriptive one-sided z	Interpretive role
Primary fixed-path population	9.9999×10⁻⁶	4.265σ	Primary fixed-coordinate population null
Worst LOEO omission	5.99994×10⁻⁵	3.846σ	Single-event influence robustness
No-60-Hz-notch replay	9.999×10⁻⁴	3.090σ	Conditioning robustness
Locked-coordinate ablation	1.99998×10⁻⁵	4.107σ	Coordinate-ablation robustness

6. Prelocked Nuisance Calibration

Population support alone does not identify cause. The next layer therefore asks which predeclared nuisance families can reproduce the same fixed-path response without changing the path.

6.1. Why Nuisance Testing Must Be Prelocked

A path statistic that separates real data from a generic null can still be reproduced by specific artifact families. Paper II therefore includes an explicit nuisance-calibration program rather than treating a small empirical p-value as causal identification. The nuisance program was prelocked at 62,400 jobs. The baseline nuisance-comparison population statistic was: T_nuisance,baseline = 4.468201, with empirical p = 9.9999×10⁻⁶. The prelocked nuisance families were:

• common-mode linear chirp;
• common-mode same phase;
• common-mode opposite phase;
• independent detector ringing;
• H1-only transient;
• L1-only transient. The design used prelocked amplitude levels rather than selecting a favorable artifact strength after observing the result.

6.2. Calibration Accounting and Parity Closure

The final nuisance/population rerun was not a single opaque batch. The authority record resolves the 62,400 scorer jobs into a reproducibility accounting layer:

• passing shards: 104;
• basis-amplitude cells: 24;
• locked leave-one-event-out calibration rows: 312;
• in-memory scorer jobs passed: 62,400;
• map-summary parity passes: 62,400;
• exact-adapter parity passes: 62,400;
• failed jobs: 0;
• imported prior-pass jobs from the superseded B13B result layer: 0. The purpose of reporting these counts is not to inflate the statistical denominator. They are execution-integrity counts. The empirical population probability remains defined by its own null contract. The 62,400-job accounting instead shows that the nuisance/calibration layer was rerun under a single frozen scor-ing contract without silently importing completed positive outcomes as if they were newly executed jobs. The later evidence-freeze stage records FULL62400_SCORER_AND_POPULATION_PASS = true and freezes the raw and calibration panels together. The source package contains separate tables for the 24cell calibration panel, 312-row LOEO calibration panel, 13-event integrated evidence panel, nuisancefamily summary, raw global variants, raw global LOEO, raw-input manifest, and residual-path provenance.

6.3. Nuisance Outcomes

The tested common-mode linear chirp, same-phase common mode, opposite-phase common mode, and independent detector-ringing families did not reproduce the baseline population behavior. The H1-only and L1-only transient families remained significant under the scorer but were weaker than the baseline across the prelocked amplitude levels. The correct interpretation is therefore not “detector artifacts have been excluded.” The correct interpretation is that several predeclared coherent and independent nuisance families fail to reproduce the baseline path population, while weaker single-detector transient families remain surviving alternatives. This boundary matters. A nuisance program is valuable only if it is permitted to leave nuisance explanations alive.

Nuisance family	Frozen Paper-II outcome		Claim consequence
Common-mode linear chirp	baseline not reproduced		constrained within tested amplitudes
Common-mode same phase		baseline not reproduced		constrained within tested amplitudes
Common-mode opposite phase		baseline not reproduced		constrained within tested amplitudes
Independent detector ringing		baseline not reproduced		constrained within tested amplitudes
H1-only transient		weaker surviving alternative		remains scientifically live
L1-only transient		weaker surviving alternative		remains scientifically live

6.4. No Omnibus Artifact p-Value

The nuisance families test different mechanisms and amplitude structures. Paper II does not combine them into one synthetic “all systematics excluded” number. The nuisance result constrains a specified artifact space and records what survives within it.

7. Selector-Aware Conditional Complex Phase

The nuisance analysis constrains several amplitude/morphology alternatives but does not answer whether the selected path also carries directional complex-phase structure. That question is tested separately under its own selector-aware null.

7.1. A separate Scientific Question

The fixed-path population result asks whether the frozen path receives support in raw strain. It does not establish whether cross-event phase information has directional organization. Paper II therefore adds a separate selector-aware complex-phase statistic under a different null. The observed equal-event values are:

• zero-direction real component = 0.3671572579932161;
• imaginary component = 0.020037823041199183;
• resultant length = 0.3677036394289937;
• mean phase = 0.0545214997095115 rad.

The direct frozen-real-data stage also retained two diagnostics rather than reporting only the primary equal-event number. A pooled-node complex average had real component 0.319976381974908, imaginary component -0.01116049149998053, and resultant length 0.320170956821935. A mode-flip diagnostic produced real component -0.3199763819749081, imaginary component 0.011160491499980635, and the same resultant length 0.3201709568219351. The preserved transformation reverses the complex direction while retaining the magnitude, providing an implementation-level directional diagnostic rather than an additional independent discovery statistic. The direct on-source phase stage used all 172 locked nodes. Event-level node counts were 7, 14, 11, 16, 16, 16, 16, 6, 16, 10, 12, 16, and 16 in the canonical event order listed above. No path coordinates were reselected for this calculation.

7.2. Selector Rerun on Real Off-Source Controls

Real H1/L1 off-source controls were rerun through the exact selector at 200 controls per event. For 13 events, this produced 2,600 selector executions. The frozen observed path is reproduced for all 13 events under the selector-aware equivalence contract: 8 events by exact coordinate match and 5 by eventwise constant-tau-frame equivalence. Across the equivalence cases, the recorded maximum absolute shift is 0.032 s, and the maximum residual numerical mismatch is 1.2143 x 10^-17 s. This step matters because the phase statistic is evaluated only after the selector has defined the relevant path-level material. The off-source controls therefore pass through the same selector machinery rather than receiving a simplified downstream treatment.

7.3. Observed Phase Geometry and Conditional Randomization

The direct equal-event complex statistic is C_eq = 0.3671572579932161 + 0.020037823041199183 i, with resultant length 0.3677036394289937 and mean phase 0.0545214997095115 rad. A separate pooled-node diagnostic gives Re = 0.319976381974908, Im = -0.01116049149998053, and resultant length = 0.320170956821935. A mode-flip diagnostic reverses the complex direction while preserving the resultant length, as expected for that transformation. These are diagnostics of the frozen phase construction; they are not additional independent detection probabilities.

The primary conditional phase-randomization null contains 100,000 draws. Exceedances are 0/100,000, giving p(+1)=9.9999×10⁻⁶ and a descriptive one-sided Gaussian-tail equivalent of approximately 4.265σ. The observed statistic also exceeded the largest primary-null realization, 0.34150643201659386; the primary-null 99th percentile was 0.16333814077083206. These values describe the frozen null distribution and are not additional p-values. A separate joint path-existence randomization also gives 0/100,000 with the same plus-one empirical value. The phase result remains conditional on the selector-aware contract and must not be relabeled as an empirical off-source detector-phase false-alarm probability.

7.4. Explicit Missing Layer

Paper II does not contain a complete empirical off-source detector-phase null in the sense of reconstructing the entire complex-phase statistic directly from an empirical off-source phase distribution. The real off-source data rerun the selector. The final conditional phase statistic is evaluated by its own randomization contract. The supported statement is therefore: Selector-aware conditional phase directionality is supported under the specified randomization null. The prohibited statement is that an empirical off-source detector-phase distribution has been completely rejected.

8. Detector Coherence Has Two Non-Equivalent Answers

8.1. Population/Map-Level Coherence

At the population and map level, detector-resolved regeneration supported coherence across the frozen path population. This evidence contributes to the interpretation that the path is not merely a collection of unrelated single-detector local maxima.

8.2. Stricter Eventwise Locked-Survivor Criterion

A stricter preregistered eventwise coherence gate asked a different question. All 13/13 events were technically evaluated. The result was FDR05 survivors = 0. That criterion is negative. Paper II therefore preserves both statements: population/map-level detector-coherence evidence is present, and the stricter locked-survivor eventwise preregistered coherence criterion is not supported. The later population evidence does not repair or overwrite the failed eventwise criterion.

8.3. Technical Repair Did Not Convert a Failed Criterion Into a PASS

The strict eventwise coherence branch is especially useful for distinguishing technical recovery from scientific outcome. An intermediate run ended with five technical failures and the verdict SURVIVOR_PATH_DETECTOR_COHERENCE_PARTIAL_TECHNICAL_FAILURE. The recorded next action was to resolve only the failed events. After the technical failures were repaired, the same locked criterion completed on all 13/13 events with 0 technical failures. The resulting scientific answer was still negative: 0 FDR05 survivors against the preregistered requirement. Thus the chronology is: technical incompleteness -> outcome-blind repair of failed cases -> complete execution -> scientific null. That sequence is preserved because it demonstrates why TECHNICAL and SCIENTIFIC_NULL cannot be merged into one status. Repairing the computation did not repair the hypothesis.

8.4. Why the Distinction Matters

If these two levels were collapsed, Paper II could be made to look artificially cleaner than the source record. The correct interpretation is more informative. The path receives population-level cross-detector support, but the evidence is not strong enough to declare eventwise coherence under the strictest preserved criterion. That is a scientific boundary, not a technical defect.

9. Family Multiplicity and Physical-Attribution Boundary

9.1. Path Multiplicity

The family-dependence/effective-multiplicity audit is a separate 16-event archaeology role, not the 13-event canonical raw-projection denominator. It contains 55 event-family tests and 55 exact unique top paths. No event has an exact duplicate top-family path across the tested family set; near-duplicate Jaccard80 structure appears in four events. The median effective family count is 2.9515451975560807. The same audit records 41 event-family FDR05 survivors and 13 event-level max-T FDR05 survivors within its own declared role. Those counts are not imported into the primary fixed-path p-value and do not create a second 16-event Paper-II population claim. Their purpose here is structural: the path search contains genuine family multiplicity rather than one rigid exact sequence copied across events. The common Paper-II object is therefore higher-order connectivity under the frozen path rule, while exact family identity remains heterogeneous.

9.2. Strict Physical-Attribution Audit

A separate strict physical-attribution role spans 16 events. The audit contains:

• 192 physical source tables;
• 65 recovered variables;
• 58 strict testable features;
• 580 direct tests;
• 560 visibility-controlled tests. The direct global FDR05 survivor count is 0. The visibility-controlled global FDR05 survivor count is also 0. Simple eventwise source-parameter attribution therefore does not close the path explanation. This is not evidence that source physics is irrelevant. It means that the tested direct low-order mappings from available source parameters to path features do not survive the stated multiplicity-controlled audit.

9.3. Why Physical Failure Does Not Erase the Path

Paper II separates the observational object from its cause.

A frozen path can receive strong cross-representation support while a simple source-parameter explanation fails. The physical-attribution null therefore reduces interpretation, not the existence of the already-supported observational path. This follows the same logic used in Paper I: a failed interpretation is removed at the interpretation layer unless the observational object itself fails under the corresponding test.

10. Uniform Reacquisition and Clean Computational Reproduction

10.1. Uniform GWOSC Reacquisition

A separate uniform GWOSC reacquisition reobtained the fixed 13-event / 172-node branch without reselection and reproduced the strong waveform-robustness result. The historical data path had been assembled through five acquisition branches; the reacquisition reduced this to one uniform GWOSC acquisition branch. Under the frozen path object, the recorded comparison gave:

• maximum absolute event-mean log-ratio change = 0.0;
• median absolute node log-ratio change = 0.0;
• maximum absolute node log-ratio change = 0.0. This exact numerical parity under a uniform reacquisition is stronger than merely stating that the qualitative conclusion looked similar. It shows that the fixed result was not numerically dependent on the historical mixture of five acquisition branches. This stage strengthens provenance. It does not create a new blind discovery because the target branch and its role were already known. The proper term is reacquisition or frozen replay.

10.2. Clean End-to-End Reproduction

The Paper-II clean-reproduction lineage restored the frozen selector source and frozen rule source, rebuilt the required assets, and executed a fresh end-to-end path from the frozen selection machinery to fresh outputs. The clean DAG explicitly separated source assets from prior frozen result tables. Prior PASS/final-statistic CSVs were not allowed to function as hidden numerical inputs to the reproduction. This matters because a nominal “reproduction” can accidentally become a filecopy operation if historical outputs are imported upstream. The clean replay therefore tests computational reconstruction under the frozen contract rather than mere agreement with a stored number.

10.3. Reproducibility Levels

Paper II distinguishes:

• static verification: source tables, roles, denominators, and locked outputs agree;
• local executable replay: a clean computational path reconstructs the frozen calculation using restored source/rule assets;
• external independent scientific reproduction: a separate person or group independently reconstructs the scientific result. Paper II strengthens the first two. The third remains outside the internal evidence record.

10.4. Later Canonical-XP Parity Recovery: A Stopped Reproducibility Branch

A later recovery experiment attempted a stricter canonical-XP parity reconstruction. Its outcome is deliberately kept separate from the successful uniform reacquisition and clean Paper-II replay. The six reconstructed XP cases all passed the declared numerical-tolerance check, but none achieved byte equality:

• numerical-tolerance pass: 6/6;
• byte-equality pass: 0/6;
• canonical-XP parity pass: 0/6. The stage therefore recorded SCIENTIFIC_CANONICAL_XP_PARITY_FAIL, set SCIENTIFIC_VERDICT = NOT_RUN, and stopped before a new scientific test. It did not invalidate the previously frozen Paper-II evidence because it asked a later, stricter provenance/parity question with its own gate. It also did not contribute any new positive scientific result. This branch is retained because reproducibility evidence should be allowed to fail at the layer at which it was tested. A numerical-tolerance match is not silently relabeled as byte-identical provenance.

11. Methods Architecture

11.1. Frozen Scientific Objects

Paper II uses three related but non-interchangeable scientific objects:

• local frozen nodes inherited from the lower-order morphology stage;
• the connected multiscale residual-derived path;
• the fixed raw-strain projection of that already-defined path.

11.2. Search Freedom by Stage

Stage	Question	Freedom
Local object discovery	Where is the morphology?	Open relevant axes; controls receive matched freedom.
Path construction	Are nodes connected?	Relax exact cell identity; path/model selection is mirrored in controls.
Paper-II raw projection	Does the frozen path survive a raw representation?	No second raw free search.
Next-order dynamics analysis	Is the path-generating process recurrent?	Controls must build their own maps, paths, trajectories, and families.

11.3. Multiscale Continuity

The canonical path representation allows the temporal-support scale to move through the width family W = {10, 20, 30, 40, 50, 60} ms. A valid path must satisfy the frozen ordering/connectivity rules. Exact equality of all lower-order node coordinates is not required.

11.4. Fixed Raw Projection

Once P_e is frozen, raw strain receives only the fixed coordinates associated with P_e. No event can move its path to a more favorable raw coordinate after inspection. For the canonical role, each event contributes frozen 4-s, 4096-Hz H1 and L1 strain, evaluated in the locked 40–400 Hz band. The primary conditioning is WHITEN_BP40_400_NOTCH60, and the population uses equal event weights. Variants are predeclared robustness roles, not coordinate-discovery opportunities. The raw response is aggregated under the frozen equal-weight population contract. The primary empirical tail uses the plus-one rule: p(+1) = (k + 1)/(N + 1). For k=0 and N=100000, p(+1) = 9.9999 x 10^-6.

11.5. Influence Audit

The population statistic is recomputed after omitting each event once. A population claim is weakened if a single event controls the result. All Paper-II LOEO omissions remain supported under the frozen threshold.

11.6. Conditioning and Ablation

Conditioning variants and coordinate ablations are fixed-role robustness tests. They are not treated as new discoveries and their probabilities are not combined with the primary population probability.

11.7. Nuisance Calibration

Nuisance families are specified before evaluating the final comparison. The nuisance program records both failures to reproduce the baseline and nuisance families that remain viable. A surviving nuisance alternative remains surviving.

11.8. Phase Layer

The selector-aware phase statistic has a separate denominator and null. The phase result is therefore not multiplied, combined, or Fisher-pooled with the fixed-path population p-value.

11.9. Multiplicity and Physical Attribution

The family-dependence/effective-multiplicity audit and the physical-attribution audit both use separate 16-event roles. Neither denominator is substituted for the canonical 13-event raw-projection population. The strict physical-attribution audit applies a global FDR05 criterion across its declared test family. Zero global survivors are retained as a negative result. No post-hoc physical feature is promoted because it looks attractive before correction.

11.10. No-Replacement and Fail-Closed Rules

Several development branches encountered missing or incomplete technical assets. Paper II preserves a simple rule: a failed acquisition or missing detector file does not become a negative scientific observation, and an unavailable event is not replaced after the outcome is known merely to restore the intended count. The locked 14-event long-window acquisition is the clearest example. One event failed after repeated H1 fetch timeouts. The branch remained NOT_TESTED scientifically and explicitly requested a retry of the failed event with no replacement. A separate rollback-replay branch likewise recorded a missing H1 long-strain asset for one event as a technical problem rather than a physical null. These branches do not enter the primary p-value. Their relevance is procedural: the canonical population was not cleaned by turning technical inconvenience into favorable scientific selection.

11.11. Status Taxonomy

Paper II preserves distinct operational states: SUPPORTED within a frozen role, NULL under a valid locked test, BLOCKED interpretation, TECHNICAL failure, UNAVAILABLE data or implementation, REPLAY/reacquisition, and HOLD at the surviving claim boundary. A technical failure is not a scientific null, and a negative eventwise criterion is not overwritten by a positive population criterion.

12. Discussion

The evidence layers can now be read in order: object construction, fixed-coordinate representation survival, influence robustness, nuisance calibration, conditional phase directionality, and explicit negative boundaries. Their strength comes from this ordering rather than from pooling p-values.

12.1. The Paper-II Discovery is a Path, Not a Stronger Point

A shallow interpretation of Paper II would describe it as Paper I with a smaller p-value. That would miss the scientific change. Paper I established local recurrent objects. Paper II asks whether those objects can be raised into connected multiscale structure. The path is therefore a new observational object. Its defining information is not merely the local score at each node, but the ordering, connectivity, and ability to remain coherent while temporal scale changes. The decisive conceptual move is that the lower-order node remains true while its exact discovery cell ceases to be the complete identity of the higher-order object.

12.2. The Path Is Not Confined to the Residual-Map Score

The primary raw-strain projection is important because it does not search raw strain for a new path. The path is defined upstream. Its fixed coordinates are then tested in raw H1/L1 strain. The strong empirical separation, survival of every LOEO omission, persistence under no-notch and coordinate-ablation variants, and separate nuisance program jointly support the statement that the path is not merely an internal artifact of one residual-map discovery score.

This remains a cross-representation support claim, not a source-mechanism claim.

12.3. A Smaller p-Value Is Not the Reason the Evidence is Stronger

Paper II contains several finite-tail results at 9.9999 x 10^-6, but the scientific advance is architectural. The evidence is stronger because different potential failure modes are separated: population rarity under a fixed-path null, event influence, conditioning dependence, coordinate dependence, nuisance-family mimicry, conditional phase directionality, detector-coherence level, simple source-parameter attribution, and computational provenance. These are different questions. A single pooled significance would erase that structure. The expanded execution archaeology adds another reason not to compress Paper II into one p-value. The corrected Fixed25 atlas contains both path-bearing and zero-path events; the canonical lock preserves no-replacement and no-best-width rules; strict eventwise coherence remains null after technical repair; the uniform reacquisition reproduces the frozen result exactly while a later stricter XP byte-parity recovery fails. These outcomes live at different scientific and computational layers. Their coexistence is the evidence architecture.

12.4. Negative Coherence and Physical-Attribution Results Strengthen the Boundary

The strict eventwise coherence criterion yields zero FDR05 survivors. The strict eventwise physical-attribution audits also yield zero global FDR05 survivors. These results prevent two overstatements. First, the population-level detector relation cannot be promoted into “every event is individually detector-coherent under the strictest criterion.” Second, the path cannot be reduced to a simple tested source-parameter relation. The correct claim is therefore both stronger and narrower: a frozen path population receives cross-representation support, but its eventwise coherence and physical source identity remain incompletely resolved.

12.5. Single-Detector Transients Remain a Real Nuisance Boundary

The nuisance program is not a ritual appendix in which every alternative conveniently dies.

H1-only and L1-only transient families remain weaker surviving alternatives. That means Paper II does not close the detector-artifact space. The value of the nuisance program lies precisely in this bounded outcome. Four predeclared coherent/independent families fail to reproduce the baseline, while two single-detector families remain scientifically alive.

12.6. Why a Next-Order Dynamics Test is Necessary

Paper II establishes the path as a higher-order observational object and supports it outside the residual-map discovery score. It still does not answer whether path motion itself is recurrent. Repeating the same fixed coordinates in another null would only ask again whether the observed path coordinates are rare. The next-order question is whether independently generated control paths exhibit the same family-level dynamical organization as the observed paths. That question requires control-native freedom. Each control must construct its own map, path, trajectory, and family label. The scientific object therefore changes again, from path existence to pathgenerating dynamics.

13. Limitations

Paper II does not establish:

• a raw-strain free-search discovery;
• a catalog-wide prevalence of the connected path;
• a catalog-wide false-alarm probability;
• a Gaussian null model or a Gaussian-equivalent σ value used as a universal discovery label; the reported σ values are descriptive translations of finite empirical p-values only;
• a unique astrophysical source mechanism;
• independence from every possible waveform or detector-systematic alternative;
• complete exclusion of single-detector transient nuisance classes;
• eventwise coherence under the strict locked-survivor FDR05 criterion;
• a simple eventwise source-parameter law explaining the path;
• an empirical off-source detector-phase null for the final complex-phase statistic;
• a universal rigid path shared identically by all events;
• a universal deterministic trajectory law;
• external independent-team scientific reproduction;
• byte-identical success of the later canonical-XP recovery branch;
• or that every development candidate necessarily contains a path.

The 13-event canonical path population, the separate 16-event family-dependence role, the separate 16-event physical-attribution role, the 62,400-job nuisance program, the 2,600 selector executions, the 100,000-draw fixed-path null, and the 100,000-draw conditional phase null belong to different denominators and scientific questions. They must not be pooled. The 68-execution archive is likewise a provenance universe, not a 68-test multiplicity denominator. It contains construction, hardening, negative criteria, reruns, alternative representations, and reproducibility work.

14. Conclusions

MIsCORE–RIUD Paper II begins where Paper I ends: with recurrent merger-local morphology already defined as a bounded observational object. Paper II raises that lower-order object into an ordered connected multiscale path, then changes the evidential task from construction to validation. A corrected 25-event atlas shows that the path rule does not force a path in every event. The canonical lock then fixes 13 events, 172 nodes, 26 H1/L1 raw-strain files, the −0.12 to +0.03 s center window, and the complete 10–60 ms width family without post-outcome event replacement, threshold retuning, or BEST_WIDTH selection.

After the path is frozen, raw-strain search freedom is closed. The residual-derived coordinates are projected directly into raw H1/L1 strain. The primary equal-weight statistic is 1.171221 with 0/100,000 empirical-null exceedances, p(+1)=9.9999×10⁻⁶ (≈4.265σ one-sided only as a descriptive translation). Every leave-one-event-out omission remains supported; no-notch conditioning and locked-coordinate ablation also remain supported. A separate 62,400-job nuisance program constrains four predeclared coherent or independent artifact families while preserving weaker H1-only and L1-only transient alternatives. A distinct selector-aware phase analysis yields 0/100,000 conditional-null exceedances under its own non-poolable contract.

the raw-strain search freedom. The residual-derived coordinates were projected directly into raw H1/L1 strain without reselection. The primary equal-weight fixed-path statistic was 1.171221, with 0 exceedances in 100,000 empirical-null draws. Every leave-one-event-out omission remained supported. The result persisted under no-notch conditioning and locked-coordinate ablation. A separate 62,400-job nuisance program constrained several coherent and independent artifact families while retaining weaker H1-only and L1-only transients as surviving alternatives. A distinct selector-aware complex-phase analysis showed directional support under its own 100,000-draw conditional null. Recovered execution history also preserved two important negative results: the strict eventwise coherence criterion did not produce FDR05 survivors, and a large multiplicity-controlled physical-attribution audit did not identify a simple source-parameter explanation. The Paper-II endpoint is therefore:

The resulting claim is deliberately bounded: a frozen residual-derived multiscale path population receives fixed-coordinate support in raw H1/L1 strain, with robustness to single-event deletion, conditioning change, coordinate ablation, several nuisance families, and a separate selector-aware phase test. This does not establish a raw free-search discovery, eventwise coherence under the strictest FDR05 gate, exhaustive detector-artifact closure, or a unique physical source law. Paper II therefore closes the path-survival question while leaving the next-order problem, recurrent path-generating dynamics, to a different scientific object and null design.

contracts. The result does not require a raw free-search discovery or a unique physical identity. The remaining question has changed. It is no longer whether the path is present outside the residual-map score. It is whether the way such paths are generated and organized is itself recurrent. That next-order dynamics question lies beyond the scope of Paper II.

Data, code, and reproducibility

The analysis uses publicly released gravitational-wave strain and catalog products distributed through the Gravitational Wave Open Science Center [4,5,6,7,8,9,10]. The frozen Paper-II evidence package preserves the canonical event panel, residual-path provenance, raw-strain scorer outputs, nuisance-calibration panels, selector-aware control summaries, conditional phase-null assets, source bindings, and SHA-256 manifests. The internal clean-reproduction branch restores the frozen selector and rule sources and forbids prior final-result tables from functioning as upstream numerical inputs. That establishes an internal executable-reproduction tier. It does not substitute for an external team independently implementing the frozen scientific contract from public strain to final figures.