UNNS SUBSTRATE RESEARCH PROGRAM · UNNS_MULTI_CLOCK_RECURRENCE BRANCH · MC_GRAMMAR_v001 · 2026-08-27

Multi-Clock Recurrence Analytics

Structural-admissibility study of quasi-periodic, two-clock temporal order — a diagnostic representation bake-off across four candidate coordinates, converging on a frozen source-torus-anchored fractional-cover grammar (D_2clk → [P_src; J_frac | M_frac] + C_coll), calibrated on independent golden-ratio two-clock development data, then stress-tested prospectively against a genuinely unseen external-model candidate and retrospectively against the historical TIME-CRYSTAL-I C003 domain-boundary case.

13 CANONICAL RECORDS INGESTED · 4 REPRESENTATIONS BAKED OFF FROZEN GRAMMAR MC_GRAMMAR_v001 · 2026-08-27 DEVELOPMENT REPLAY: DTQC 4/4 SUPPORTED · CONTROLS 0/16 SUPPORTED PROSPECTIVE P001 → CANDIDATE REJECTED (control matched expectation) C003 HISTORICAL RETEST → STILL OUTSIDE GRAMMAR, BUT J_frac=0.911 M_frac=0.596 PRE-FREEZE MALZ–SMITH SPECIFICITY CHALLENGE (v003) → PASS, RECORDS LATER JOINED DEVELOPMENT CORPUS GOLDEN-RATIO DOMAIN ONLY (v001) BLIND → LOCK → REVEAL, SAME FIREWALL AS TIME-CRYSTAL-I C003 EXCLUDED FROM ALL DEVELOPMENT / CALIBRATION / SELECTION SOURCE: RESEARCH_FIREWALL · REP_STUDY_v002 · GRAMMAR_DEV_v001 · GRAMMAR_SELECT_v001 · GRAMMAR_FREEZE_v001 · PROSPECTIVE_P01 · C003_FINAL_v001

Chain: neutral ingestion → representation bake-off → drive-torus diagnostic → locked Zhu v002 transfer test (partial failure) → fractional-cover v003 → locked Malz–Smith specificity challenge (pass) → 4-rep study v002 (Luo + reclassified Zhu + Malz–Smith) → grammar dev (nulls/robustness) → grammar select → grammar freeze → prospective test → C003 historical retest Frozen grammar SHA-256: 8ae9b0d9…39e4bba556d6 Status: grammar FROZEN · prospective CLOSED (failure preserved) · C003 CLOSED (boundary preserved) · Malz–Smith pre-freeze specificity CLOSED (pass, incorporated into development)

§1 BRANCH OVERVIEW — FROM NAIVE RECURRENCE TO A FROZEN MULTI-CLOCK GRAMMAR

NEUTRAL-INGESTION RECORDS

CANDIDATE REPRESENTATIONS TESTED

TEMPORAL-CORE COORDINATES (P_src, J_frac, M_frac)

4/4

DTQC POSITIVES SUPPORTED AT FREEZE

0/16

CONTROLS SUPPORTED AT FREEZE

PROSPECTIVE CANDIDATE — REJECTED

C003 HISTORICAL RETEST — BOUNDARY CONFIRMED

PRE-FREEZE SPECIFICITY CHALLENGE — PASS

0.12

FROZEN M_frac RAW GATE

FINDING §1.1

Three successive representations were tried and abandoned before a specific, robust multi-clock coordinate was found

The branch firewall (00_PROTOCOL/RESEARCH_FIREWALL.md) explicitly quarantines the historical TIME-CRYSTAL-I candidate C003 from every development stage — it may not be used to calibrate, select, or threshold anything, and may only be evaluated once, after the grammar is frozen. Development proceeded through four representation families: source-free vector-depth recurrence (fails — not multi-clock-specific), source-defined frequency-lattice alignment (fails — too permissive), source-defined drive-torus regression at integer depth (fails — DTQC not maximal among regimes), and finally a fractional-cover decomposition beyond the integer parent torus (succeeds — separates DTQC positives from every available control family by AUC 1.0). Each failure was preserved as a negative result rather than quietly discarded, mirroring the TIME-CRYSTAL-I rigidity-sector precedent.

§2 RESEARCH FIREWALL & DEVELOPMENT CHAIN

NON-NEGOTIABLE CONSTRAINTS · 00_PROTOCOL/RESEARCH_FIREWALL.md

neutral ingestion→ representation bake-off→ drive-torus diagnostic→ Zhu transfer test→ fractional-cover v003→ Malz–Smith specificity→ 4-rep study v002→ grammar dev (nulls+robustness)→ grammar select→ grammar freeze→ prospective test→ C003 retest

1. C003 is a historical domain-boundary case and is excluded from development, calibration, representation selection, threshold selection, and validation. 2. ZHU_2026 began as a holdout; a locked v002 transfer test returned PARTIAL_TRANSFER_WITH_COVER_DEPTH_FAILURE, after which ZHU_FIG1_C1/C3 were reclassified as retrospective/development evidence (§5) — not independent validation of the final grammar. 3. MOON_2025 raw data are access-restricted and unused throughout. 4. Development proceeds from independent sources → neutral ingestion → diagnostic representation tests → validation → only then metric/threshold freeze. 5. No target-frequency leakage, synthetic evidence, post-hoc threshold tuning, or retrospective rewriting of the TIME-CRYSTAL-I verdict is permitted. 6. Development-stage methods (v001, v002) are diagnostic-only and emit no admissibility verdicts.

CORPUS_v001 — 13 INGESTED + 3 RESERVED · 04_CORPUS/CORPUS_v001.json

Source	Domain	Records	Role(s)	Status
HUANG_2025	continuous-time spin dynamics	3 (LC, QP, CHAOS)	periodic / quasiperiodic / chaotic controls	INGESTED
LUO_2026	Rydberg-chain numerical DTQC	10 (M, F, EE×LOW/DTQC/HIGH, N=12/32/40)	DTQC development + secondary + breakdown/decoupled controls	INGESTED
MOON_2025	experimental Fibonacci DTQC / time rondeau	—	experimental development, pending	ACCESS_RESTRICTED
ZHU_2026	experimental DTQC	—	holdout → locked transfer test → reclassified development	VALIDATION_CONSUMED → DEVELOPMENT (see §5)
TC_PROSPECTIVE_01 (HIST_C003)	experimental Z2 DTQC	—	historical boundary case	DO_NOT_CALIBRATE

CORPUS_STATUS_v002.json addendum: ZHU_2026 state moved HOLDOUT → VALIDATION_CONSUMED (locked ZHU_VALIDATION_v001, outcome PARTIAL_TRANSFER_WITH_COVER_DEPTH_FAILURE) → ZHU_FIG1_C1/C3 subsequently reclassified DTQC_POSITIVE development evidence in REP_CORPUS_v002 (C1's M_frac=0.138611 sets the frozen gate's lower bound, §10) · HIST_C003 remains DO_NOT_CALIBRATE, unchanged.

REP_CORPUS_v002 — CORE + AUXILIARY PANELS

Core panel (full source-defined golden-ratio two-clock records, source_ratio = φ = 1.618033988749895 throughout): Luo 2026 (6 entanglement trajectories: LOW/DTQC/HIGH × N=12/32), Zhu 2026 (2 experimental DTQC: Fig.1 C1, C3 — reclassified from holdout after a locked v002 transfer test, §5; not independent validation), Malz–Smith 2021 (12 experimental qubit-tomography trajectories: 6 topological, 6 trivial — reclassified as development controls after a pre-freeze locked specificity challenge, §7).

Auxiliary panel (source-free cross-chart controls, JPR/FL/FC undefined — no explicit two-clock source): Huang 2025 (LC, QP, CHAOS), TIME-CRYSTAL-I integer-recurrence records (C001, C002, C002_CTRL, NRCTRL) — evaluated only via the source-free vector-depth representation, for cross-chart comparison. C003 fully excluded from this corpus.

20 core + 7 auxiliary = 27 records in REP_CORPUS_v002 · all core records share the golden-ratio branch; different-ratio robustness explicitly untested at this stage (REP_STUDY_v002 limitation).

§3 REPRESENTATION BAKE-OFF v001 — FIRST-PASS, DIAGNOSTIC ONLY

RESULT — NONE OF THE THREE FIRST-PASS REPRESENTATIONS IS SUFFICIENT ALONE · 13 records · frequency lattice / observable-derived torus / vector recurrence

id	role	lattice gain (bins)	torus R² (2d)	rec best-pair
LUO_M_N32	DTQC_DEVELOPMENT	23.6635	—	0.8068
LUO_F_N40	DTQC_DEVELOPMENT_SECONDARY	23.7627	—	0.7193
LUO_EE_DTQC_N32	DTQC_DEVELOPMENT	19.0464	0.7171 (not maximal!)	0.7153
LUO_EE_LOW_N32	LOW_FREQ_APERIODIC_CONTROL	2.5947 (weak)	0.9254	0.4433
LUO_EE_HIGH_N32	HIGH_FREQ_DECOUPLED_CONTROL	16.4302 (strong despite decoupling!)	0.9235	0.8849 (exceeds DTQC!)
HUANG_QP	ADJACENT_QUASIPERIODIC	—	—	0.9991 (near-saturated, non-specific)
HUANG_CHAOS	CHAOTIC_CONTROL	—	—	0.9990 (near-saturated, non-specific)

Full corpus in BAKEOFF_RESULTS.csv (13 records × 29 columns) · negative result reproducible across N=12/N=32 · conclusion: need a source-defined drive-phase torus plus a coupling-sensitive coordinate, not a purely spectral or purely recurrent one.

FINDING §3.1

Every single-coordinate first-pass representation is fooled by at least one control

Frequency-lattice alignment ranks the high-frequency decoupled control (16.4302) above the DTQC-positive record on the same statistic family, and vector recurrence ranks the same decoupled control (0.8849) above DTQC (0.7153) — while the observable-derived torus fails to make DTQC's R² maximal among the three regimes at all (0.7171, below the low-frequency control's 0.9254). None of these three representations is discarded as useless — frequency-lattice and vector-depth both survive as supporting, non-gating charts in the eventual compact grammar (§10) — but none is DTQC-specific on its own.

§4 DRIVE-TORUS + COUPLING DIAGNOSTIC v002 — SOURCE-DEFINED PHASE TORUS

SOURCE-DEFINED TORUS (θ_L = 2πx, θ_R = 2πφx) VS OBSERVABLE-DERIVED TORUS · six Luo Fig.3 entanglement trajectories

Regime	N	best depth	full R² (cv)	axis-only R²	mixed interaction gain	mean S
Low-frequency aperiodic	32	2	0.1213	0.0697	0.0515	0.5853
Intermediate DTQC	32	2	0.9409	0.7167	0.2242	0.2425
High-frequency decoupled	32	2	0.6976	0.5972	0.1004	0.0719

N=12 values near-identical to N=32 (not a finite-size artifact) · all six records independently select cover depth d=2, so depth alone is not DTQC-specific · ordering DTQC > high > low survives ×5/×10 downsample, first-75% truncation, and 2% RMS noise

⚑ MIXED INTERACTION GAIN IS THE FIRST COORDINATE TO RANK DTQC ABOVE BOTH SAME-SYSTEM CONTROLS — AND COUPLING RUNS BACKWARDS

DTQC's mixed interaction gain (0.2242) is ≈2.23× the high-frequency decoupled control's (0.1004) and ≈4.35× the low-frequency control's (0.0515) — the first single number in the entire branch that ranks DTQC strictly above both controls. But the physical coupling-strength proxy (mean entanglement entropy S) runs in the opposite order: low=0.5853 > DTQC=0.2425 > high=0.0719. Multi-clock torus organization and collective coupling strength are not the same axis — echoing the "temporal structure ≠ collective admissibility" lesson later stated explicitly in the project summary. Specificity against generic (non-DTQC) two-clock quasiperiodicity was not yet established at this stage — that came later, from the Malz–Smith specificity challenge (§7).

§5 LOCKED ZHU v002 TRANSFER TEST — HOLDOUT CONSUMED, NOT INDEPENDENT VALIDATION

PRE-REGISTERED TRANSFER TEST OF THE v002 DRIVE-TORUS METHOD ON A HELD-OUT EXTERNAL SOURCE · ZHU_VALIDATION_v001 · 04_CORPUS/CORPUS_STATUS_v002.json

ZHU_2026 entered the branch as a protected holdout under the research firewall (00_PROTOCOL/RESEARCH_FIREWALL.md: "ZHU_2026 is reserved as a holdout and must not be analyzed during development"). Before any Zhu data were analyzed, a locked test — ZHU_VALIDATION_v001 — was pre-registered to ask a narrow external-transfer question: does the source-defined drive-torus + coupling method frozen at v002 (§4) generalize to an independent experimental DTQC source it had never seen? This was designed and executed as a genuine holdout test, not as a source of development evidence.

PARTIAL

PARTIAL TRANSFER — COVER-DEPTH FAILURE

HOLDOUT →
CONSUMED

CORPUS STATE TRANSITION

v002

METHOD UNDER TEST

RECORDS RELEASED (FIG.1 C1, C3)

CORPUS_STATUS_v002.json addendum, ZHU_2026: previous_state = HOLDOUT / DO_NOT_ANALYZE → current_state = VALIDATION_CONSUMED · locked_validation = ZHU_VALIDATION_v001 · validation_outcome = PARTIAL_TRANSFER_WITH_COVER_DEPTH_FAILURE · v003_use = RETROSPECTIVE_ONLY_NOT_VALIDATION.

WHAT "PARTIAL TRANSFER" MEANT · 05_METHODS/FRACTIONAL_COVER/FC_SPEC_v003.md rationale note

The locked test found that Zhu's joint source-clock organization does transfer — the underlying two-clock structure the v002 method is built to detect is genuinely present in the external data. But v002's own cover-depth selector, argmax_d full_r2_cv(d), failed to pick it out correctly on the Zhu records: a method that worked by construction on the Luo development set did not transfer cleanly to an independent source. This mixed outcome — real signal, wrong selector — is exactly what motivated fractional-cover v003 (§6), whose spec note reads verbatim: "Successor to v002 after Zhu external-transfer test showed joint source-clock organization transfers but v002's argmax_d full_r2_cv(d) depth selector does not."

FINDING §5.1

Zhu is retrospective development evidence, not independent validation — the label matters

Because ZHU_VALIDATION_v001 returned a partial, method-revising result rather than a clean pass or fail, the two released records (Fig.1 C1, C3) were reclassified DTQC_POSITIVE and folded into REP_CORPUS_v002 as development evidence (§2) — most consequentially, ZHU_FIG1_C1's fractional mixed gain (M_frac = 0.138611, §6) became the empirical lower bound that sets the frozen grammar's raw M_frac gate (§10). A record that helps define a threshold cannot also be used to independently validate that threshold: Zhu's role in this branch is retrospective and formative, not confirmatory. The only genuinely pre-freeze, held-out specificity test in this branch is the Malz–Smith challenge (§7).

§6 FRACTIONAL-COVER DECOMPOSITION v003 — BEYOND THE INTEGER PARENT TORUS

MAIN DEVELOPMENT TABLE · N=32 · six Luo Fig.3 trajectories

Regime	Parent R²	Frac. axis gain	Frac. mixed gain	Frac. total gain
LOW	0.002072	0.076870	0.051774	0.128645
DTQC	0.012515	0.717709	0.224366	0.942075
HIGH	0.035786	0.598050	0.100526	0.698576

d=1 = integer parent lattice B_Z; d=2,3,4 fractional; robustness at base d_frac: downsample ×5/×10, first 75%, noise σ=0.02·std, seed 20260826.

ZHU 2026 RETROSPECTIVE · motivated v003, not independent validation (Zhu was used to design v003)

Record	Parent R²	Frac. axis gain	Frac. mixed gain	Frac. total gain
C1	0.600458	−0.043640	0.138611	0.094971
C3	0.522323	−0.011398	0.300533	0.289135

Zhu's fractional-mixed-gain values are strongly positive despite near-zero/negative fractional axis gain — the first hint that the mixed term, not the axis or total term, is where the specific signal lives. This directly motivated M_frac = FC_frac_mixed_gain becoming the eventual primary decision coordinate (§10).

§7 PRE-FREEZE SPECIFICITY CHALLENGE — MALZ–SMITH 2021 TOPOLOGICAL FLOQUET QUBIT

FALSIFICATION TEST OF FRACTIONAL-COVER v003, NOT A POSITIVE-DTQC VALIDATION · 12 experimental IBM-qubit tomography runs, two incommensurate drive clocks

Pre-analysis lock written before reading any numeric data. Question: does merely having two incommensurate clocks — with no DTQC order at all — automatically produce v003's positive fractional-cover signature? If yes, v003 is not specific and the whole branch's central claim collapses.

PASS

PASS_STRONG_SPECIFICITY_CHALLENGE

EXPERIMENTAL SOURCE RECORDS

SCALAR COMPONENT RECORDS (X/Y/Z)

CLASSIFICATION THRESHOLDS / VERDICTS EMITTED

MEDIAN parent_full_r2_cv BY REGIME & COMPONENT

Component	Topological (C=−1)	Trivial (C=0)
X	0.863377	0.404665
Y	0.864897	0.310316
Z	0.899285	−0.136661

Both regimes anchor strongly to the integer parent torus (as expected — genuine two-clock experimental data) — this is precisely the P_src signal, not the M_frac signal.

MEDIAN frac_total_gain BY REGIME & COMPONENT — THE SPECIFICITY RESULT

Component	Topological (C=−1)	Trivial (C=0)
X	−0.004914	−0.026201
Y	−0.007267	−0.029130
Z	−0.008673	−0.161759

Every cell negative — contrasts sharply with Luo DTQC (frac_total ≈ +0.942) and Zhu retrospective (also positive). Two incommensurate clocks alone do NOT produce v003's fractional-cover signature — a direct falsification test the grammar's core intuition survived.

FINDING §7.1

v003 is not fooled by mere two-clock incommensurability, but this is not proof it IS a DTQC classifier

Across 0.6 ≤ M ≤ 3.4 (both the topological C=−1 and trivial C=0 regimes of the Malz–Smith Floquet qubit), median fractional total gain stayed negative in all 6 regime×component cells, and robustness checks (downsample ×5/×10, first-75% truncation, 2% noise) kept the regime medians negative throughout. Sign counts: only 1/18 topological and 2/18 trivial scalar records showed a positive frac_total_gain at all — consistent with noise, not signal. This is a genuine specificity result, but the report is explicit about its own limit: it demonstrates v003 does not fire on generic two-clock quasiperiodicity, not that it is a validated DTQC detector in general.

This was a locked, pre-freeze challenge — executed and interpreted before REP_STUDY_v002 — not an independent validation run against the finished grammar. Its 12 records were subsequently incorporated into REP_CORPUS_v002 as development controls (§2), most visibly anchoring the QP_NON_DTQC_TOPO and QP_NON_DTQC_TRIVIAL role rows in the Four-Representation Study's role-level medians (§8). Nothing in this branch was checked against Malz–Smith data held out after the grammar was frozen.

§8 FOUR-REPRESENTATION STUDY v002 — VECTOR-DEPTH · JPR · FREQUENCY-LATTICE · FRACTIONAL-COVER

ROLE-LEVEL MEDIANS ACROSS ALL FOUR REPRESENTATIONS · REP_STUDY_v002/GROUP_SUMMARY.json

Role	VD family contrast	JPR parent gain (P_src)	JPR cover advantage (J_frac)	FL peak contrast	FC parent R²	FC mixed gain (M_frac)	FC total gain
DTQC_POSITIVE	0.0611	0.3088	0.3616	2.8956	0.2674	0.2244	0.6156
MULTICLOCK_BREAKDOWN_LOW	0.0022	0.1427	0.3981	1.0812	0.0020	0.0528	0.1300
MULTICLOCK_DECOUPLED_HIGH	0.0144	0.0699	0.6025	5.4555	0.0358	0.1005	0.6986
QP_NON_DTQC_TOPO	0.2699	0.6309	−0.2552	1.3103	0.8798	−0.0092	−0.0163
QP_NON_DTQC_TRIVIAL	0.0816	0.1575	−0.0771	0.6620	0.1883	−0.0549	−0.0898

FC mixed gain (M_frac) is the only column where DTQC_POSITIVE sits strictly above every control role · JPR cover advantage (J_frac) separates DTQC from the two Malz-Smith non-DTQC roles but not from the two Luo same-system controls · frequency lattice remains too permissive (HIGH control's 5.4555 exceeds DTQC's 2.8956).

M_frac (FRACTIONAL MIXED-CLOCK GAIN) — ROLE MEDIANS

DTQC_POSITIVE

0.2244

MULTICLOCK_DECOUPLED_HIGH

0.1005

MULTICLOCK_BREAKDOWN_LOW

0.0528

QP_NON_DTQC_TOPO

−0.0092

QP_NON_DTQC_TRIVIAL

−0.0549

Frozen raw gate M_frac ≥ 0.12 (dashed reference) sits between DTQC (0.2244) and every control's median.

REPRESENTATION AGREEMENT · fractional depth chosen by JPR vs FC · 20 core records

All 6 Luo records and both Zhu records: JPR and FC agree on fractional depth, and both prefer the fractional cover over the integer parent. All 12 Malz–Smith records: JPR still selects an integer/fractional depth mechanically, but does not prefer the fractional cover over the parent torus, and FC's fractional gain is negative — 18/20 core records show consistent cross-representation behavior; only 2 (MS_M0p8, MS_M1p4) show a depth disagreement between JPR and FC, both still correctly non-DTQC by FC sign.

Median absolute JPR-cover-advantage change under origin shift = 0.001629, clock exchange = 0.002242, affine transform = 0.000000 — the invariance checkpoint that let J_frac later serve as a stable (if non-gating) chart coordinate.

§9 GRAMMAR DEVELOPMENT v001 — NULLS, ROBUSTNESS, CONTROL DISCRIMINATION

PAIRWISE AUC — M_frac (FC MIXED GAIN) IS THE ONLY PERFECT DISCRIMINANT

Metric	vs Luo LOW	vs Luo HIGH	vs Malz-Smith topological	vs Malz-Smith trivial
JPR_cover_advantage (J_frac)	0.500	0.500	1.000	1.000
FC_frac_mixed_gain (M_frac)	1.000	1.000	1.000	1.000
FC_frac_total_gain	0.750	0.500 (fails)	1.000	1.000
JPR_parent_gain (P_src, raw)	0.750	0.750	0.125 (inverted)	0.708

M_frac AUC = 1.000 against every one of the four control families — the only metric with this property, which is exactly why it became the primary decision coordinate rather than a supporting chart · P_src's raw magnitude is not discriminative on its own (Malz-Smith topological QP has AUC 0.125, i.e. inverted) — this is why P_src is retained only as a null-significance gate, never a raw-magnitude threshold (§10).

NULL-MODEL PERCENTILES · median DTQC-positive percentile under each null

JPR_cover_advantage · phase-label-permute null percentile — DTQC 0.803 · Luo-low 0.990 · Luo-high 1.000 · Malz-topo 0.000 · Malz-trivial 0.045
FC_frac_mixed_gain · median DTQC percentile — TIME_PERMUTE 1.000 · BLOCK_SHUFFLE 1.000 · FOURIER_PHASE 0.990

FC_frac_total_gain, despite strong raw values, fails DTQC-vs-Luo-HIGH discrimination (AUC 0.500) — dropped from the compact grammar for this reason (§10).

ROLE COORDINATES · 08_OUTPUTS/GRAMMAR_DEV_v001/ROLE_COORDINATES.json

role	JPR parent gain	JPR cover adv.	FC parent R²	FC mixed gain	FC total gain
DTQC_POSITIVE	0.3088	0.3616	0.2674	0.2244	0.6156
MULTICLOCK_BREAKDOWN_LOW	0.1427	0.3981	0.0020	0.0528	0.1300
MULTICLOCK_DECOUPLED_HIGH	0.0699	0.6025	0.0358	0.1005	0.6986
QP_NON_DTQC_TOPO	0.6309	−0.2552	0.8798	−0.0092	−0.0163
QP_NON_DTQC_TRIVIAL	0.1575	−0.0771	0.1883	−0.0549	−0.0898

§10 COMPACT GRAMMAR SELECTION — SELECT / DROP DECISIONS

COORDINATE_SELECTION.csv — ALL 8 CANDIDATE COORDINATES, FULL DERIVATION

Coordinate	Symbol	Same-system min AUC	Generic-QP min AUC	Decision
JPR_parent_gain	P_src	0.750	0.125	SELECT — NULL ANCHOR GATE
JPR_cover_advantage	J_frac	0.500	1.000	SELECT — CHART COORDINATE, NOT A GATE
FC_parent_full_r2	P_FC	0.500	0.0833	DROP — REDUNDANT (ρ≈0.887 with P_src)
FC_frac_mixed_gain	M_frac	1.000	1.000	SELECT — PRIMARY TEMPORAL GATE
FC_frac_total_gain	T_frac	0.500	1.000	DROP — REDUNDANT (ρ≈0.949 with M_frac)
vector-depth family	VD	—	—	SUPPORTING CHART ONLY
frequency lattice	FL	—	—	SUPPORTING CHART ONLY
source-native collective evidence	C_coll	—	—	PHYSICAL EXTENSION, NOT A TEMPORAL GATE

"Same-system min AUC" = worst-case discrimination against the physically closest control (Luo LOW/HIGH); "generic-QP min AUC" = worst-case against the physically unrelated Malz-Smith quasiperiodic controls · full reasoning strings preserved in COORDINATE_SELECTION.csv.

M_frac RAW-GATE DERIVATION · GRAMMAR_CANDIDATE_v001.json

min(DTQC-positive development) = 0.138611 (Zhu C1, reclassified from the locked v002 transfer-test holdout, §5)
max(control development) = 0.100526
midpoint = 0.119568
rounded candidate gate = 0.12

development_replay at selection: DTQC-positive core satisfied 4/4; control core satisfied 0/16.

M_GATE ROBUSTNESS · 30-row perturbation sweep; one raw-gate crossing found

Role	Variant	Records	Gate count	Median M
DTQC_POSITIVE	all 6 variants	4	4/4	0.2243 – 0.2291
MULTICLOCK_BREAKDOWN_LOW	all 6 variants	2	0/2	0.0527 – 0.0544
MULTICLOCK_DECOUPLED_HIGH	5 of 6 variants	2	0/2	0.1005 – 0.1005
MULTICLOCK_DECOUPLED_HIGH	PREFIX_0.75	2	2/2 (crosses raw gate)	0.1480
QP_NON_DTQC_TOPO / TRIVIAL	all 6 variants	12	0/12	negative throughout

The only robustness transformation in which a control crosses the raw M_frac threshold: truncating the high-frequency decoupled control to its first 75% pushes both records' raw M_frac over 0.12 (median 0.1480 vs. 0.12). This is not a frozen-grammar false positive: both HIGH records already fail the base M_frac≥0.12 condition before this transform is applied. The robustness transforms are an additional stability requirement on the base classification, not alternative classification inputs in their own right — the crossing is preserved as a documented boundary condition of the raw gate considered in isolation.

§11 THE FROZEN GRAMMAR — MC_GRAMMAR_v001

GRAMMAR STRUCTURE · FROZEN 2026-08-27 · pre-prospective

D_2clk → [ P_src ; J_frac | M_frac ] + C_coll

D_2clk — exactly two externally specified incommensurate source clocks; genuine time-domain observable; ratio from source metadata, not fitted; no inverse-FFT/synthetic reconstruction. Supported ratio domain v001: golden-ratio branch only.

P_src (source-torus anchor, gate) — metric JPR_parent_gain; null PHASE_LABEL_PERMUTE, 99 surrogates; gate p_upper ≤ 0.10; fail → SOURCE_UNANCHORED.

J_frac (chart coordinate, non-gating) — JPR_cover_advantage = max_d=2,3,4 JPR(d) − JPR(1).

M_frac (primary decision coordinate, gate) — FC_frac_mixed_gain at d* = argmax_d FC_frac_total_gain(d); raw gate ≥ 0.12; null FOURIER_PHASE, 49 surrogates, ridge α=1e-9, 5-fold CV; gate p_upper ≤ 0.10.

C_coll — separate physical extension; COLLECTIVE_NOT_ASSESSED / COLLECTIVE_EVIDENCE_UNAVAILABLE / COLLECTIVE_EVIDENCE_AVAILABLE_FOR_DOMAIN_SPECIFIC_REVIEW; no universal numeric gate.

STATE PRECEDENCE LADDER · first match wins

OUTSIDE_CHART_DOMAINnot a genuine two-clock time-domain record

OUTSIDE_EMPIRICALLY_QUALIFIED_RATIO_DOMAINsource ratio outside the golden-ratio branch (v001)

SOURCE_UNANCHOREDfails P_src phase-label-permute null (p_upper > 0.10)

MIXED_ORGANIZATION_WEAKM_frac below raw gate 0.12

MIXED_ORGANIZATION_SPECTRAL_NULLfails M_frac Fourier-phase null (p_upper > 0.10)

MIXED_ORGANIZATION_UNSTABLEraw gate fails to hold under one or more of the 6 stability transforms

TEMPORAL_CORE_SUPPORTEDP_src anchored, M_frac ≥ 0.12, Fourier-null passed, stable under all 6 transforms

Stability transforms: origin shift (0.371 lower-clock cycles), clock exchange, affine state transform (scale 1.7, offset 0.31), ×5 downsample, first 0.75 of trajectory, additive noise σ=0.02·std.

DEVELOPMENT REPLAY AT FREEZE · GRAMMAR_FREEZE_v001/README.md · all pre-freeze data, no prospective/C003 data

4/4

DTQC POSITIVES → TEMPORAL_CORE_SUPPORTED

0/2

LUO LOW CONTROLS SUPPORTED

0/2

LUO HIGH CONTROLS SUPPORTED

0/6

MALZ-SMITH TOPOLOGICAL SUPPORTED

0/6

MALZ-SMITH TRIVIAL SUPPORTED

All 16 controls fail through SOURCE_UNANCHORED or MIXED_ORGANIZATION_WEAK — no post-hoc rule was added to force this separation; it falls out of the frozen P_src / M_frac gates directly. Excluded non-decision quantities (explicitly, by name): vector-depth recurrence, raw frequency-lattice alignment, FC parent full R², FC fractional total gain, any composite scalar, any hard J_frac threshold, fractional-depth agreement.

§12 PROSPECTIVE TEST — P01 · AN EXTERNAL-MODEL CANDIDATE THE GRAMMAR HAD NEVER SEEN

BLIND → LOCK → REVEAL · same firewall discipline as TC_PROSPECTIVE_01

pre-registered transfer parameters→ generate A/B trajectories→ cryptographic mapping commitment→ blind analysis (frozen grammar)→ quantitative lock→ reveal→ commitment verification

Candidate source: an external theoretical model transfer (Marripour–Abouie 2026 Hamiltonian family), not any record used in development. Pre-registered: L=4 periodic chain, J=5.5, h=0.3, 100 paired disorder realizations, initial |+x⟩^⊗L, ensemble-averaged m_x(t), 64 drive periods, 16 samples/period, Ω/ω_d = 1/φ. Two frequency-scale conditions tested blind: ω_d=12 (candidate hypothesis: long-lived prethermal quasiperiodic order) and ω_d=1 (control hypothesis: rapid heating / loss of order). Deliberately adapted to the golden-ratio domain rather than the paper's published √2/4 ratio.

P001_A — REVEALED: LOW_FREQ_BREAKDOWN_CONTROL_HYPOTHESIS

SOURCE_UNANCHORED

P_src p=0.14 · J_frac=−0.0383 · M_frac=−0.0140 · Fourier p=0.60 · M_robust_min=−0.2024

EXPECTATION MET predeclared NOT_TEMPORAL_CORE_SUPPORTED

Worst robustness variant: DOWNSAMPLE_5 (M_frac → −0.2024).

P001_B — REVEALED: HIGH_FREQ_CANDIDATE_HYPOTHESIS

MIXED_ORGANIZATION_WEAK

P_src p=0.01 · J_frac=−0.1664 · M_frac=−0.3300 · Fourier p=1.00 · M_robust_min=−1.0245

EXPECTATION NOT MET predeclared TEMPORAL_CORE_SUPPORTED

Worst robustness variant: PREFIX_0.75 (M_frac → −1.0245, its steepest drop).

⚑ PROSPECTIVE_FAILURE_CANDIDATE_REJECTED — the frozen grammar failed its first genuinely blind test

The pre-registered high-frequency candidate (P001_B, ω_d=12) was expected to show TEMPORAL_CORE_SUPPORTED and instead returned MIXED_ORGANIZATION_WEAK with strongly negative M_frac (−0.330) — the opposite sign from every development-stage DTQC-positive record. The control (P001_A) behaved exactly as predeclared. Per the pre-registered protocol, this failure cannot be repaired by changing thresholds, nulls, robustness transforms, the ratio domain, the representation, or the mapping after the fact — doing so would require declaring a new grammar version, not patching v001. MC_GRAMMAR_v001 therefore cannot be presented as a universal multi-clock temporal grammar. This does not itself disprove the external model's predicted quasiperiodic/prethermal order — it shows only that the frozen UNNS grammar did not admit this pre-registered trajectory.

§13 C003 HISTORICAL RETEST — THE DOMAIN BOUNDARY, REVISITED WITH A NEW CHART

UNDER THE OLD TIME-CRYSTAL-I INTEGER GRAMMAR · frozen v1.1.0 record, unchanged

Run	q0	C	F	p	Verdict
C003 (candidate)	6	0.198765	0.093433	0.960199	NO_TEMPORAL_ORDER
C003_CTRL	6	0.094640	0.030874	0.995025	NO_TEMPORAL_ORDER

UNDER THE NEW FROZEN MULTI-CLOCK GRAMMAR · MC_GRAMMAR_v001, evaluated once, after prospective reveal

Run	P_src p	J_frac	M_frac	Fourier p	State
C003 (Fig.1d, candidate)	1.000	0.910688	0.596092	0.020	SOURCE_UNANCHORED
C003_CTRL (Fig.1b, breakdown)	0.010	−0.593846	−0.003958	0.780	MIXED_ORGANIZATION_WEAK

C003 M_robust_min = 0.595093 (stable across all 6 transforms) · C003_CTRL M_robust_min = −0.558942.

⚑ THE BOUNDARY HOLDS — BUT FOR A COMPLETELY DIFFERENT, AND MORE INFORMATIVE, REASON

C003 is still not admitted — C003_REMAINS_OUTSIDE_FROZEN_MULTI_CLOCK_GRAMMAR — but it fails at the source-torus-anchor gate (P_src p=1.000, the worst possible value), not at the mixed-organization gate. Its M_frac (0.596) is nearly 5× the frozen raw gate of 0.12, and its J_frac (0.911) is the largest fractional-cover advantage seen anywhere in this report — the new coordinates cleanly separate C003 from its own matched breakdown control (M_frac 0.596 vs. −0.004; J_frac 0.911 vs. −0.594), which is exactly the opposite pattern from the control, whose parent-torus anchor holds (p=0.010) while its fractional organization is null. State precedence still routes C003 to a rejection (SOURCE_UNANCHORED is checked before M_frac), so this is not a positive admission — it is a sharper diagnosis of why the historical boundary exists: weak parent-torus anchoring + very strong fractional organization, the near-inverse of what MC_GRAMMAR_v001 was built to expect (parent organization → fractional organization). The grammar was not changed for C003, and this result is read alongside the P001 prospective failure as further evidence the grammar is not universally permissive.

§14 STRUCTURAL INTERPRETATION — CLAIMS SUPPORTED vs CLAIMS DELIBERATELY NOT MADE

CLAIM LEDGER

ClaimStatus

M_frac is the strongest development-stage discriminator among the tested coordinates AUC = 1.000 against all four development control families. The fractional-cover representation passed the pre-freeze Malz–Smith specificity challenge (§7); M_frac itself was established afterward, in the four-representation study (§8), where it is negative in the same Malz–Smith controls ✓ DEMONSTRATED

Closure magnitude / raw parent-torus anchoring alone identifies multi-clock admissibility Malz-Smith topological QP anchors as strongly as any DTQC record (parent R² ≈ 0.87–0.90) yet has negative fractional gain throughout ✗ EXPLICITLY NOT CLAIMED

Frozen grammar generalizes to an unseen external theoretical-model candidate P001 — candidate expectation NOT met; PROSPECTIVE_FAILURE_CANDIDATE_REJECTED ✗ NOT DEMONSTRATED (failure preserved)

MC_GRAMMAR_v001 is a universal multi-clock temporal grammar Directly contradicted by the P001 prospective failure ✗ EXPLICITLY NOT CLAIMED

Historical C003 is admitted by the new grammar (retrofitting the old boundary away) C003 remains outside — SOURCE_UNANCHORED — despite very strong M_frac/J_frac; the grammar was not adjusted for it ✗ EXPLICITLY NOT CLAIMED / NOT ATTEMPTED

C003's fractional organization is real and distinguishes it sharply from its own control M_frac 0.596 (candidate) vs −0.004 (control), remains robust across all six frozen transforms; base J_frac 0.911 vs −0.594 strongly separates candidate and control ✓ DEMONSTRATED

Parent-anchored multi-clock order and fractional-primary multi-clock order are the same phenomenon C003 shows weak parent anchoring + very strong fractional organization; its control shows the near-inverse — the two axes can decouple ✗ NOT DEMONSTRATED (evidence for a second structural mode instead)

Multi-clock temporal organization is the same axis as collective physical coupling Drive-torus organization order (DTQC>HIGH>LOW) is reversed relative to mean entanglement entropy order (LOW>DTQC>HIGH) ✗ EXPLICITLY REFUTED

Two incommensurate clocks alone (no DTQC order) produce a positive fractional-cover signature Malz-Smith specificity test: all 6 regime×component medians negative (pre-freeze test; its records were later reclassified as development controls, §7) ✗ EXPLICITLY REFUTED (specificity holds)

§15 REPRODUCIBILITY & PROVENANCE ARCHITECTURE

FROZEN GRAMMAR / STAGE HASHES

Frozen grammar (MC_GRAMMAR_v001 / FROZEN_GRAMMAR.json)
8ae9b0d91dba41332dac58392b84a222f6c63b6323b8ddb616ce39e4bba556d6

Prospective P01 grammar archive (as loaded for blind analysis)
f49d5553c5bd7fa878dbaca5fb04ae5ee5a20fb14564a0fafb4da1d1ebb7c925

P001 mapping commitment (expected == recomputed at reveal)
02b3c7b7dd25f4769d1094bde8db99997e0ad0de1532c6148c38427c8043caf1

Malz–Smith raw archive (qubit-topological-floquet-v2.0.zip, 41,780,798 bytes)
1f1d9b4acbc64ec0facc563bef5884ac50c865bbddee2d464da25146cff6d9b0

FIREWALL DISCIPLINE, BY STAGE

Stage	C003 loaded?	Thresholds changed?	Nulls changed?
Grammar development / selection / freeze	false	—	—
Prospective P01 (blind analysis)	false	false	false
Prospective P01 (reveal)	false	false	false
C003 final evaluation	true (only after P001 reveal)	false	false

C003_FINAL_v001/C003_FINAL_AUDIT.json: adapter_changes_after_metrics=false · C003_used_in_grammar_development=false throughout every prior stage.

§16 SYNTHESIS & NEXT RESEARCH DIRECTION

CURRENT STRONGEST BOUNDED CONCLUSION

A fractional-cover-based multi-clock grammar (MC_GRAMMAR_v001) was developed independently of the historical C003 boundary case. The final development corpus contains four DTQC-positive records from Luo and Zhu and sixteen controls from Luo and Malz–Smith; the selected M_frac coordinate separates all four positives from all sixteen controls in the development replay. The underlying fractional-cover representation also survived a pre-freeze independent-source specificity challenge (Malz–Smith, §7) before M_frac itself was established in the later four-representation study; the Malz–Smith records were subsequently incorporated into grammar development as controls. Its first genuinely prospective test — a pre-registered external theoretical-model candidate — failed: the frozen grammar did not admit the trajectory predicted to show prethermal quasiperiodic order. Re-examined under this same frozen grammar, C003 remains unadmitted, but now for a diagnosed reason — a failed source-torus anchor rather than weak fractional organization — with fractional-cover values (J_frac≈0.91, M_frac≈0.60) among the strongest seen anywhere in this report and cleanly separated from its own matched control.

Neither failure is treated as fatal to the branch: both were pre-registered as possible outcomes, both were preserved rather than repaired after the fact, and both narrow rather than close the research question. The emerging picture is not "one multi-clock grammar" but a distinction between parent-anchored and fractional-primary multi-clock order that the current grammar's state-precedence ladder was not designed to separate — C003 may be the first empirical instance of the second mode.

READY FOR

· independent methodological review
· a second, larger prospective campaign (single blind pairs are underpowered)
· design of a source-anchoring-relaxed grammar variant for the fractional-primary regime
· non-golden-ratio domain extension, tested on new unseen data
· manuscript preparation alongside the TIME-CRYSTAL-I integer-depth result

NEXT SCIENTIFIC QUESTION

Not "does MC_GRAMMAR_v001 detect more DTQCs?" but: is parent-anchored multi-clock order structurally distinct from fractional-primary multi-clock order, and if so, does a grammar variant that gates on M_frac without requiring P_src anchoring admit C003 while still rejecting P001_B and all sixteen development controls? Any such variant is a new grammar version, tested on new unseen candidates — never retrofitted onto C003 or P001 directly.

MULTI-CLOCK RECURRENCE ANALYTICS — UNNS Substrate Research Program · 2026-08-27
Branch: UNNS_MULTI_CLOCK_RECURRENCE · Grammar: MC_GRAMMAR_v001 (frozen, SHA-256 8ae9b0d9…39e4bba556d6) · D_2clk → [P_src; J_frac | M_frac] + C_coll
Development corpus: 13 neutral-ingestion records (Huang 2025 LC/QP/CHAOS · Luo 2026 10 Rydberg-chain DTQC/breakdown/decoupled records) feeding a 27-record representation-study corpus — 4 DTQC positives (Luo×2, Zhu×2, the latter reclassified from a holdout) + 16 controls (Luo×4, Malz–Smith×12, reclassified after a pre-freeze specificity challenge) + 7 auxiliary cross-chart records · development replay at freeze: 4/4 DTQC positives supported, 0/16 controls supported
Prospective campaign: P01 · external Marripour–Abouie 2026 model transfer · outcome PROSPECTIVE_FAILURE_CANDIDATE_REJECTED (control matched expectation, candidate did not) · failure preserved, not repaired
Historical retest: C003 (TIME-CRYSTAL-I domain boundary) re-evaluated once under MC_GRAMMAR_v001 after P001 reveal — outcome C003_REMAINS_OUTSIDE_FROZEN_MULTI_CLOCK_GRAMMAR, state SOURCE_UNANCHORED, with J_frac=0.9107 / M_frac=0.5961 far above every development threshold
Pre-freeze specificity challenge: Malz–Smith 2021 topological Floquet qubit — PASS_STRONG_SPECIFICITY_CHALLENGE, locked before REP_STUDY_v002; its 12 records were then incorporated into the development corpus as controls (not independent validation of the final grammar) — see §7
Zhu 2026: began as a holdout; locked v002 transfer test → PARTIAL_TRANSFER_WITH_COVER_DEPTH_FAILURE; C1/C3 then reclassified as DTQC-positive development evidence (C1 sets the frozen M_frac gate's lower bound, 0.138611) — not independent validation — see §5
Caution: all core development/validation records share the golden-ratio branch — non-golden-ratio robustness is untested · single blind candidate/control pair in P01 is not a statistically powered prospective campaign · C_coll (collective sector) has no universal numeric gate and was not assessed for P001 · MOON_2025 and ZHU_2026 raw provenance remain access-restricted / partially consumed respectively — see §2