UNNS SUBSTRATE RESEARCH PROGRAM · STRUC-PERC-I v2.4.0
CROSS-DOMAIN PERCOLATION CORPUS ANALYSIS
Full pairwise vulnerability graph · IQR-scaled ε · 4-tier verdict system · 81 runs across 14 domains
INSTRUMENT STRUC-PERC-I v2.4.0 DATE 2026-04-05 TOTAL RUNS 81 DOMAINS 14 USL VIOLATIONS 0 (zero hard USL violations) THEOREM 1 TRIGGERED 1 / 81
FULL PERCOLATION: 48 GIANT COMPONENT: 23 TAIL FRAGMENTATION: 9 HARD FRAGMENTATION: 1 ENGINE: IQR-scaled ε · τ-floor · MAX_WINDOW=64
01EXECUTIVE SUMMARY
TOTAL RUNS
81
across 14 domains
FULL PERCOLATION
48
59.3% of corpus
GIANT COMPONENT
23
28.4% · tier 2
TAIL FRAGMENTATION
9
11.1% · inconclusive
HARD FRAGMENTATION
1
TiO₂ density only
THEOREM 1 TRIGGERED
1
USL violation guaranteed
ZERO HARD VIOLATIONS
80
of 81 runs clean
TAIL DOMINANCE >50%
45
outlier-gap-dominated
VERDICT DISTRIBUTION BY DOMAIN all 81 runs · STRUC-PERC-I v2.4.0 · IQR-scaled ε

Domain | N | Tier breakdown | FULL | GIANT | TAIL | HARD | Avg giant ratio | κ_conn range | Theorem 1
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
Molecular (HITRAN) | 6 |  | 0 | 6 | 0 | 0 | 0.9985 | — (GIANT, no conn) | —
Atomic Spectra | 10 |  | 4 | 5 | 1 | 0 | 0.9978 | 2.1×10⁵ – 4.2×10⁵ | —
Biological (QT45) | 7 |  | 7 | 0 | 0 | 0 | 1.0000 | 0.42 – 2.00 | —
Nuclear (ENSDF) | 14 |  | 10 | 0 | 4 | 0 | 0.9945 | 3.8×10⁴ – 4.2×10⁵ | —
Condensed Matter | 12 |  | 11 | 0 | 0 | 1 | 0.9861 | 0.75 – 8.1×10³ | 1 (TiO₂)
CMB | 3 |  | 3 | 0 | 0 | 0 | 1.0000 | 230 – 2389 | —
Cosmic Web | 5 |  | 0 | 2 | 3 | 0 | 0.9954 | — (no direct conn) | —
Atmosphere (ERA5) | 8 |  | 8 | 0 | 0 | 0 | 1.0000 | 0.42 – 2.00 | —
Geodesy (GPS) | 5 |  | 2 | 3 | 0 | 0 | 0.9989 | 253 – 402 | —
Geoid / Gravity | 4 |  | 0 | 4 | 0 | 0 | 0.9983 | — (GIANT, no conn) | —
Solar (GOES-XRS) | 1 |  | 0 | 0 | 1 | 0 | 0.9820 | — (TAIL) | —
Random Matrix (GOE) | 1 |  | 0 | 1 | 0 | 0 | 0.9960 | — (GIANT) | —
Adversarial / Cluster | 3 |  | 1 | 2 | 0 | 0 | 0.9993 | 0.01 | —
Adversarial / Weak | 2 |  | 2 | 0 | 0 | 0 | 1.0000 | 2.7×10⁴ – 2.3×10⁵ | —

02KEY FINDINGS
FINDING 01
Zero Hard USL Violations Across 80 of 81 Runs

Only 1 of 81 runs (TiO₂ density, giant ratio 0.8333) triggers Theorem 1, confirming a USL violation. All remaining 80 domains return verdicts consistent with structural admissibility. This is the central result of the corpus: at the current κ-grid range, real physical ladders are overwhelmingly admissible or inconclusive, with true structural failure appearing only in the one known pathological case.

FINDING 02
Three Distinct Percolation Regimes Confirmed Across Domains

The corpus cleanly partitions into three structural regimes distinguished by their κ_connect scale:
— Immediate percolation (κ_conn ≤ 2): biological fitness landscapes, atmospheric wind fields, adversarial baselines, some condensed-matter.
— Moderate delayed percolation (κ_conn = 10–8000): condensed matter solids (FeO, KSiO, Cu, VO), CMB spectra (230–2389), GPS crustal displacement.
— Extreme delayed / tail-dominated percolation (κ_conn = 10⁴–4×10⁵): nuclear γ-level schemes, atomic spectra (Na, He-II, Li). These connect only at κ far exceeding the main grid, driven by astronomical gap ratios (max ratio up to 4×10⁵) yet structurally consistent.

FINDING 03
Biological Domain Shows the Lowest κ_connect Across All Physical Domains

All 7 biological runs (QT45 ribozyme fitness landscape) return FULL_PERCOLATION with κ_conn in the range 0.42 – 2.00 — the lowest of any physical domain. This means the fitness landscape gap structure is the most locally homogeneous in Δ-space of all systems tested. The result is consistent with the published biological extension article at unns.tech and confirms the earlier STRUC-BIO-I findings.

FINDING 04
Nuclear Domain Splits: 10 FULL, 4 TAIL — Correlated with Isotope Structure

The four TAIL_FRAGMENTATION nuclei are ²³⁸U, ¹⁵⁰Nd, ¹⁰⁰Mo, ⁴⁸Ca — all isotopes with known structural anomalies (deformed nuclei, shape coexistence, or collectivity transitions). Giant ratios range from 0.976 to 0.988, with tail dominance = 1.000 and astronomically large max ratios (up to 2×10¹⁸). These outliers are single extreme-energy γ-transitions far above the bulk level scheme. Theorem 1 does not apply (TAIL verdict), but the pattern is physically interpretable: structural irregularity in the level scheme produces outlier gaps that resist global connectivity at the tested κ scale.

FINDING 05
Cosmic Web Ladders Are the Only Physical Domain Lacking Direct κ_connect

All 5 cosmic web runs (2MRS, DESI synthetic, DESI sample, SDSS) return GIANT or TAIL verdicts with no κ_connect on the main grid and no adaptive extension result. Giant ratios are tightly clustered at 0.994–0.999. This is structurally different from the nuclear tail cases: the cosmic web ladders have isolated vertices at maximum κ that extension cannot resolve within the tested range, reflecting genuine large-scale void structure in the galaxy distribution. The 3 TAIL cases have tail dominance 0.913, driven by a max ratio of ~10⁶.

FINDING 06
CMB Percolates — But at Extreme Scale (κ_conn up to 2389)

All three CMB spectra (TT, TE, EE) return FULL_PERCOLATION after adaptive extension, but with dramatically different connectivity scales: TE connects at κ ≈ 230, TT at κ ≈ 322, EE at κ ≈ 2389. The large κ required reflects the multi-scale power spectrum structure — acoustic peaks create local gap homogeneity at specific scales, but the overall spectrum is too heterogeneous to bridge within κ ≤ 1. EE polarization requiring 10× larger κ than TE is a new discriminative result consistent with the different noise properties of these spectra.

FINDING 07
TiO₂ Density Is the Only Confirmed Hard Structural Failure

The tio2_density_ladder (cleaned dataset) returns HARD_FRAGMENTATION with giant ratio 0.8333 — the only run triggering Theorem 1. The companion tio2_cleaned_dataset run returns FULL_PERCOLATION at κ_conn = 681, showing that the raw density representation produces genuine structural fragmentation while the cleaned spectral dataset is admissible. This split is scientifically significant: it demonstrates that data pre-processing choice directly affects PRP verdict for the same physical system.

FINDING 08
Geoid / Gravity and Molecular HITRAN Show Persistent GIANT — Never Crossing to FULL

Six molecular species (H₂O, NH₃, CO₂, CH₄, O₃, CO) and all four gravity field models (Earth EIGEN-6C4, Mars JGM85, Moon AIUB-GRL350A, Earth spatial 2°) return GIANT_COMPONENT_PERCOLATION with giant ratios 0.997–0.9997, but no κ_connect on the main grid and no adaptive extension. This is the most structurally coherent GIANT cluster in the corpus: persistent near-complete connectivity without a small number of extreme outliers preventing closure. It suggests these systems are structurally admissible at scale but have isolated tail gaps that resist connection even at κ = 1.

03κ-CONNECT STRUCTURE ACROSS DOMAINS
κ_connect VALUES — ALL 48 RUNS WITH DIRECT CONNECTIVITY log-scale ordering · lower = more homogeneous gap structure

Run ID | Domain | Verdict | κ_connect | Giant ratio | Tail dominance | Max ratio
--- | --- | --- | --- | --- | --- | ---
uniform_baseline_ladder | Adversarial / Cluster | FULL | 0.01000 | 1.0000 | 0.000 | 0.0
era5_latband_absmean_labels | Atmosphere (ERA5) | FULL | 0.4217 | 1.0000 | 0.00 | —
complete10_double_global | Biological (QT45) | FULL | 0.4217 | 1.0000 | — | —
complete10_single_global | Biological (QT45) | FULL | 0.4217 | 1.0000 | — | —
bp_mutants_global | Biological (QT45) | FULL | 0.4217 | 1.0000 | — | —
combined_single_global | Biological (QT45) | FULL | 0.4217 | 1.0000 | — | —
era5_global_lonsector_labels | Atmosphere (ERA5) | FULL | 0.5623 | 1.0000 | — | —
double_mutant_global | Biological (QT45) | FULL | 0.5623 | 1.0000 | — | —
era5_latband_absmax_u | Atmosphere (ERA5) | FULL | 0.7499 | 1.0000 | — | —
deletion_global | Biological (QT45) | FULL | 0.7499 | 1.0000 | — | —
sno_density | Condensed Matter | FULL | 0.7499 | 1.0000 | — | —
era5_latband_signedmean_u | Atmosphere (ERA5) | FULL | 1.000 | 1.0000 | — | —
era5_top12_jet_absmean_u | Atmosphere (ERA5) | FULL | 2.000 | 1.0000 | — | —
era5_top12_jet_labels | Atmosphere (ERA5) | FULL | 2.000 | 1.0000 | — | —
era5_latband_absmean_u | Atmosphere (ERA5) | FULL | 2.000 | 1.0000 | — | —
era5_global_lonsector_absmean_u | Atmosphere (ERA5) | FULL | 2.000 | 1.0000 | — | —
combined_del_single_global | Biological (QT45) | FULL | 2.000 | 1.0000 | — | —
vo_density | Condensed Matter | FULL | 2.000 | 1.0000 | — | —
si_density | Condensed Matter | FULL | 2.000 | 1.0000 | — | —
tio_density_clean | Condensed Matter | FULL | 10.000 | 1.0000 | — | —
ge_density | Condensed Matter | FULL | 10.000 | 1.0000 | — | —
feo_cleaned | Condensed Matter | FULL | 30.08 | 1.0000 | — | —
ksio_density | Condensed Matter | FULL | 94.04 | 1.0000 | — | —
o_density | Condensed Matter | FULL | 94.04 | 1.0000 | — | —
cu_density | Condensed Matter | FULL | 94.04 | 1.0000 | — | —
cmb_te_spectrum | CMB | FULL | 229.68 | 1.0000 | — | —
p811_crustal_displacement | Geodesy (GPS) | FULL | 252.87 | 1.0000 | — | —
cmb_tt_spectrum | CMB | FULL | 321.88 | 1.0000 | — | —
cac2_crustal_displacement | Geodesy (GPS) | FULL | 402.21 | 1.0000 | — | —
tio2_cleaned_dataset | Condensed Matter | FULL | 681.36 | 1.0000 | — | —
cmb_ee_spectrum | CMB | FULL | 2389.12 | 1.0000 | — | —
vo_cleaned_dataset | Condensed Matter | FULL | 8072.55 | 1.0000 | — | —
weak_ladders_summary | Adversarial / Weak | FULL | 26 758 | 1.0000 | 0.999 | 26 759
ensdf_116sn | Nuclear (ENSDF) | FULL | 37 586 | 1.0000 | 0.966 | 37 586
ensdf_90zr | Nuclear (ENSDF) | FULL | 39 623 | 1.0000 | 0.946 | 39 623
ensdf_174yb | Nuclear (ENSDF) | FULL | 59 068 | 1.0000 | 0.982 | 59 068
ensdf_208pb | Nuclear (ENSDF) | FULL | 64 262 | 1.0000 | 0.968 | 64 262
ensdf_166er | Nuclear (ENSDF) | FULL | 106 377 | 1.0000 | 0.983 | 106 377
ensdf_28si | Nuclear (ENSDF) | FULL | 157 499 | 1.0000 | 0.990 | 157 499
ensdf_152sm | Nuclear (ENSDF) | FULL | 176 211 | 1.0000 | 0.987 | 176 211
sodium_spectrum_qm1 (×2) | Atomic Spectra | FULL | 206 721 | 1.0000 | — | 206 721
weak_ladders_profiles | Adversarial / Weak | FULL | 227 428 | 1.0000 | 0.996 | 227 428
heliumii_spectrum_qm1 | Atomic Spectra | FULL | 329 031 | 1.0000 | — | 329 031
ensdf_56fe | Nuclear (ENSDF) | FULL | 329 108 | 1.0000 | 0.996 | 329 108
ensdf_60ni | Nuclear (ENSDF) | FULL | 397 204 | 1.0000 | 0.993 | 397 204
ensdf_24mg | Nuclear (ENSDF) | FULL | 418 677 | 1.0000 | 0.992 | 418 677
lithium_spectrum_qm1 | Atomic Spectra | FULL | 420 033 | 1.0000 | — | 420 033

04TAIL FRAGMENTATION — 9 INCONCLUSIVE RUNS
TAIL FRAGMENTATION DETAIL giant ratio ≥ 0.95 · no large secondary cluster · Theorem 1 does NOT apply

Run ID | Domain | Giant ratio | Isolated | Tail dominance | Max Δ/med | Interpretation
--- | --- | --- | --- | --- | --- | ---
ensdf_238u | Nuclear (ENSDF) | 0.9829 | — | 1.000 | 2.5 × 10⁹ | Single extreme high-E γ transition
ensdf_150nd | Nuclear (ENSDF) | 0.9758 | — | 1.000 | 2.2 × 10¹⁸ | Shape-coexistence deformed nucleus
ensdf_100mo | Nuclear (ENSDF) | 0.9884 | — | 1.000 | 7.0 × 10¹⁸ | Double β-decay candidate, outlier γ
ensdf_48ca | Nuclear (ENSDF) | 0.9761 | — | 1.000 | 2.5 × 10¹⁸ | Doubly-magic + deformation excitation
cw_desi_xyz_sample | Cosmic Web | 0.9943 | — | 0.913 | 1.01 × 10⁶ | Void radial gap > 1M × median
cw_desi_synthetic | Cosmic Web | 0.9943 | — | 0.913 | 1.01 × 10⁶ | Void radial gap > 1M × median
desi_ra_dec_z_redshift | Cosmic Web | 0.9942 | — | 0.913 | 1.01 × 10⁶ | Redshift ladder void structure
goes-xrs-2014 | Solar (GOES-XRS) | 0.9820 | — | 1.000 | 3.2 × 10¹⁰ | Solar flare spike dominates gap budget
gold_zeeman | Atomic Spectra | 0.9944 | — | 0.998 | 4.8 × 10⁶ | Au Zeeman fine-structure outlier

⚠ INCONCLUSIVE — All 9 cases have tail dominance ≥ 0.913: isolated vertices are produced by single extreme outlier gaps, not structural fragmentation. Theorem 1 (necessary direction) does NOT apply. USL violation is not guaranteed. Diagnostic action: refine domain adapter to exclude pathological outlier transitions, or extend κ to astronomical scales.
05HARD FRAGMENTATION — THEOREM 1 TRIGGER
TiO₂ DENSITY LADDER — THE ONLY USL VIOLATION CONFIRMED tio2_density_ladder · giant ratio = 0.8333 · Theorem 1 triggered
The raw TiO₂ density-of-states ladder returns HARD_FRAGMENTATION with giant ratio 0.8333 — the only run in the corpus triggering PRP Theorem 1. The companion `tio2_cleaned_dataset` run returns FULL_PERCOLATION at κ_conn = 681, demonstrating that the raw density representation contains genuine structural discontinuities that the cleaned spectral dataset does not.
This split confirms: PRP verdict is sensitive to data pre-processing for the same physical system. The raw DOS has large energy-range gaps not present in the cleaned spectral peaks ladder.
GIANT RATIO
0.833
VERDICT
HARD
THEOREM 1
YES
USL violation guaranteed
COMPANION
FULL
at κ_conn = 681
06DOMAIN PROFILES
MOLECULAR — HITRAN
6 runs · H₂O, NH₃, CO₂, CH₄, O₃, CO
All GIANT · avg GR = 0.9985
All six HITRAN molecular energy level ladders return GIANT_COMPONENT_PERCOLATION with giant ratios 0.9975–0.9993. No κ_connect is found on the main grid or via adaptive extension, indicating that isolated outlier energy gaps resist connection even at κ = 1. The giant component is near-complete — the overall HITRAN level structure is highly admissibility-consistent. The absence of FULL percolation (vs. earlier instrument versions) reflects the more discriminative IQR-scaled ε: these ladders have dense bulk level spacings (small IQR) but one or two extreme outlier gaps from high-energy vibrational series.
ATOMIC SPECTRA — QM-I/II
10 runs · H, He, Li, Na, Ca, Ag, Au + Zeeman variants
Mixed: 4 FULL · 5 GIANT · 1 TAIL
The atomic domain splits by measurement type. Zeeman ladder runs (Ca, H, He-singlet, Ag, Na) predominantly return FULL_PERCOLATION at extreme κ_conn (2×10⁵ – 4×10⁵), reflecting the fine-structure outlier transitions. QM-I series spectra (He-II, Li, Na duplicates) also return FULL at large κ. Au Zeeman is the single TAIL case (GR = 0.9944, tail dominance 0.998): the Au fine-structure ladder has a single extremely large outlier gap that dominates the gap budget completely. The universal Zeeman invariant (ρ ≈ 0.96 from prior corpus) is consistent with the κ_connect clustering in the 2×10⁵ – 4×10⁵ range for multiple elements.
BIOLOGICAL — QT45 RIBOZYME
7 runs · fitness landscape mutant classes
All FULL · κ_conn range 0.42 – 2.00
The QT45 ribozyme fitness landscape is the most immediately percolating physical domain in the corpus. All seven mutant class ladders (bp_mutants, single, double, deletion, combined) return FULL_PERCOLATION with κ_connect between 0.42 and 2.00 — the lowest connectivity threshold of any domain. This confirms and extends the published UNNS biological extension (March 2026): living matter fitness landscapes have the most homogeneous gap structure of any tested system, requiring only a 0.42× IQR tolerance to achieve global graph connectivity.
NUCLEAR — ENSDF γ-LEVELS
14 isotopes · ²⁴Mg through ²³⁸U
10 FULL · 4 TAIL · κ_conn: 3.8×10⁴ – 4.2×10⁵
The nuclear corpus is the most internally differentiated domain. The FULL percolating isotopes all connect at large κ (37k – 418k), driven by tail-dominated gap structures (tailDominance 0.946–0.996): a small number of high-energy γ-transitions create extreme outlier gaps but the bulk of the level scheme is structurally coherent. The 4 TAIL cases (²³⁸U, ¹⁵⁰Nd, ¹⁰⁰Mo, ⁴⁸Ca) have tail dominance = 1.000 with astronomical max ratios (10⁹–10¹⁸), indicating single pathological transitions that cannot be bridged within the extension range. Notably, ⁴⁸Ca — previously identified as a cross-constant anomaly in the α-sweep corpus — again appears as a structural outlier here.
CONDENSED MATTER
12 runs · Si, Ge, Cu, FeO, TiO, TiO₂, VO, KSiO, O, SnO
11 FULL · 1 HARD (TiO₂) · κ_conn: 0.75 – 8072
The widest κ_connect range of any domain (0.75 for SnO to 8072 for VO cleaned). Simple elemental metals and oxides (SnO, Si, Ge) connect within κ ≤ 10; complex transition-metal compounds (FeO, Cu, O) require κ ≈ 94; vanadium oxide and TiO₂ cleaned require κ in the hundreds to thousands. The TiO₂ raw density ladder is the corpus's only HARD_FRAGMENTATION case. VO appears twice (density and cleaned): both return FULL, but at very different scales (κ = 2.00 vs. 8072), demonstrating strong representation sensitivity.
CMB — PLANCK 2018 POWER SPECTRA
3 spectra · TT · TE · EE
All FULL · κ_conn: TT=322, TE=230, EE=2389
All three CMB power spectra percolate via adaptive extension. The ordering TE < TT < EE by κ_connect is physically interpretable: TE (cross-correlation) has the most regular spacing between acoustic features; TT (temperature) is slightly more heterogeneous; EE (polarization E-mode) requires 10× larger κ, reflecting lower signal-to-noise and finer multi-pole structure near recombination. The factor of ~10 spread (230 vs. 2389) is a new discriminative result from the IQR-scaled engine that was not visible in earlier median-scaled runs.
COSMIC WEB
5 runs · 2MRS · DESI sample/synthetic · SDSS · redshift
2 GIANT · 3 TAIL · no κ_connect
The cosmic web domain is structurally distinct: no run achieves κ_connect on the main grid or via extension. Giant ratios cluster tightly at 0.994–0.999. The 2MRS and SDSS ladders return GIANT; the three DESI-based ladders (sample, synthetic, redshift) return TAIL with tail dominance 0.913 and max ratio ~10⁶. The DESI ladder void gaps are physically real — large-scale voids create radial separations orders of magnitude larger than typical filament spacings. The cosmic web requires either an adaptive extension far exceeding the current range, or a void-aware adapter that treats void structure separately from filament structure.
07STRUCTURAL PERCOLATION REGIMES
REGIME I — IMMEDIATE
κ_conn < 2
Biological (QT45) · Atmosphere (ERA5) · SnO · Si · Ge · Adversarial baseline
Smooth, locally homogeneous gap structure. Small IQR relative to gap range. No significant outliers. Physical interpretation: systems with near-uniform spacing or very regular excitation ladders.
REGIME II — MODERATE DELAY
κ_conn = 2 – 10³
Condensed matter (most) · CMB (TT, TE) · Geodesy (GPS)
Structured gap distributions with band-gap or multi-scale features. One or more secondary scale regimes. Physical interpretation: systems with competing length or energy scales producing bimodal or multi-modal gap distributions.
REGIME III — EXTREME DELAY
κ_conn = 10³ – 4×10⁵
Nuclear (ENSDF) · Atomic spectra · CMB EE · Weak ladders
Astronomically large tail gaps. Tail dominance > 0.95. Bulk structure is coherent; connectivity blocked only by a handful of extreme outlier transitions. Physical interpretation: discrete energy systems with rare high-energy excitations far above the bulk spectrum.
REGIME CLASSIFICATION SUMMARY

Regime | κ_conn range | Domains | Runs | Tail dominance | Physical signature
--- | --- | --- | --- | --- | ---
I — Immediate | < 2 | Biological, Atmosphere, Simple condensed | 18 | Low (<0.1) | Uniform spacing, no major outliers
II — Moderate | 2 – 1000 | Most condensed matter, CMB TT/TE, GPS | 13 | Moderate | Multi-scale, band gaps, acoustic peaks
III — Extreme | 10³ – 4×10⁵ | Nuclear, Atomic, CMB EE, Weak | 17 | High (>0.9) | Rare high-E transitions dominate gap budget
GIANT (no conn) | — | Molecular, Geoid/Gravity, Cosmic Web (2MRS/SDSS) | 16 | Low–moderate | Near-complete but one or few persistent isolated gaps
TAIL (no conn) | — | Cosmic Web (DESI), Nuclear (4), Solar, Gold Zeeman | 9 | Very high (>0.91) | Single extreme outlier prevents closure
HARD | — | TiO₂ raw density only | 1 | — | True structural fragmentation · Theorem 1 triggered

08THEORETICAL STATUS
NECESSARY DIRECTION ESTABLISHED
PRP Theorem 1 (established): Non-percolation in the full pairwise vulnerability graph ⟹ USL violation guaranteed.
Across 81 runs, Theorem 1 fires exactly once (TiO₂ density, giant ratio 0.8333). All 9 TAIL_FRAGMENTATION cases correctly suppress Theorem 1 — fragmentation is outlier-driven, not structural.
The zero-violation rate across 80 of 81 runs is the central empirical result: physical systems at the constant values of α, μ, αₛ, αG appear to universally satisfy the structural admissibility condition.
SUFFICIENT DIRECTION OPEN CONJECTURE
PRP Conjecture 1 (open): Percolation ⟹ admissibility. Not proven.
All 71 runs returning FULL or GIANT verdicts are consistent with Conjecture 1 but do not prove it. The conjecture remains open. Each percolating run should be verified independently via STRUC-I v1.0.4 to confirm inv(Pε; L) ≤ ν(Vε(L)) directly.
The sufficient direction proof would require showing that the PRP connectivity condition is not only necessary but sufficient for the inversion bound — a result not established by this corpus.
09INSTRUMENT & METHOD NOTES
STRUC-PERC-I v2.4.0 — ENGINE PARAMETERS FOR THIS CORPUS
ε SCALE — κ · IQR(Δ), fallback to κ · median when IQR = 0. Replaces median-only scaling used in v2.3.1 and earlier, which produced false universal percolation.
MICRO-GAP FLOOR τ — 0.1 × median(Δ). Gaps below τ are not connected. Prevents degenerate near-zero gaps from flooding the graph at κ → 0.
MAX_WINDOW = 64 — Limits transitive union reach per step in the sliding window. Prevents false full-percolation via transitive chaining of dense clusters.
OUTLIER THRESHOLD — max(10×med, 5×IQR). Hybrid threshold catches extreme tails even when median is small relative to distribution width.
ADAPTIVE EXTENSION — data-aware targets [2, 10, √scale, scale] where scale = min(maxRatio, 10⁶). Plateau: growth < 10⁻⁴ OR relGrowth < 10⁻³.
HARD_CAP — 200,000 gaps. Larger ladders are downsampled via uniform stride. LARGE_RENDER_THRESHOLD = 20,000 for UI guards.
STRUC-PERC-I v2.4.0 · UNNS Substrate Research Program · 2026-04-05 81 RUNS · 14 DOMAINS · FULL PAIRWISE VULNERABILITY GRAPH · IQR-SCALED ε · MAX_WINDOW=64 PRP Theorem 1: 1/81 triggered · Conjecture 1: OPEN