UNNS SUBSTRATE PROGRAM · STRUC-I CORPUS ANALYSIS
STRUCTURAL PRESSURE & ADMISSIBILITY
ACROSS PHYSICAL DOMAINS
UPDATED 2026-03-22 · CONDENSED MATTER PHASE CHAINS ADDED · +1,920 EVALUATIONS
Full statistical audit of STRUC-I v1.0.4 results — 5,233 ladder evaluations (3,069 physical + 1,920 crystallographic phase chains + 240 biological + 4 cluster adversarial) · 14 physical domains + 1 biological · 2 adversarial packs · GOE null baseline
CHAMBER STRUC-I v1.0.4
INEQUALITY inv(Pε; L) ≤ ν(Vε(L))
M 2,000 MC runs per κ
κ STEPS 40 · [0.01, 1.0]
DATE 2026-03-14 / UPDATED 2026-03-22
FALSIFICATION-FIRST MULTI-DOMAIN PRESSURE INDEX ρ GOE NULL BASELINE 5,233 RUNS CONDMAT PHASE CHAINS · 8 MATERIALS · NEW COSMIC WEB · DESI/2MRS/SDSS QUANTUM MECHANICS · ZEEMAN GRAVITY · EARTH · MARS · MOON ATMOSPHERE · ERA5 · 250hPa JET SOLAR PLASMA · RECONNECTION · DYNAMO
§0 CORPUS OVERVIEW
TOTAL RUNS
5,233
ladder evaluations (+1,920 phase chains)
GOE ENSEMBLE
3,000
random matrices (null)
PHYSICAL LADDERS
115
67 original + 48 phase-chain (8 mat × 6 desc)
FALSIFICATIONS
0
clean violations of inv≤ν
WEAK PERSISTENCE
13
8 original + 5 phase-chain (SiO₂×4, KNbO₃×1)
N RANGE (GOE)
100
–500
levels per matrix
MAX ρ OBSERVED
1.000
synth σ-amp gravity
MIN Aκ PHYSICAL
0.531
synth σ-amp · Random
DOMAINS
10
CM-DFT · CM-phase · nuc · mol · QM · CW · grav · atm · solar · RMT
DATASET COMPOSITION BY RUN TYPE AND DOMAIN

DOMAIN | LADDER TYPE | RUNS | N RANGE | MEAN Aκ | MEAN ρ | STD ρ | VERDICT
--- | --- | --- | --- | --- | --- | --- | ---
Random Matrix (GOE) | numeric ID · n=100 | 1,000 | 100 | 1.0000 | 0.0871 | 0.0333 | Stable / GP
Random Matrix (GOE) | numeric ID · n=200 | 1,000 | 200 | 1.0000 | 0.0925 | 0.0301 | Stable / GP
Random Matrix (GOE) | numeric ID · n=500 | 1,000 | 500 | 1.0000 | 0.1006 | 0.0276 | Stable / GP
Condensed-matter | Cu, O, Si, Ge, KSiO · density | 10 | 17–194 | 0.9995 | 0.2626 | 0.073 | Stable–Weak
Condensed-matter | Cu, O, Si, Ge, KSiO · formation E | 10 | 17–194 | 1.0000 | 0.2261 | 0.074 | Stable / GP
Condensed-matter | Transition-metal oxides (SnO,TiO,TiO2,VO,FeO) | 20 | 8–37 | 0.9997 | 0.178 | 0.135 | Stable–Weak
Nuclear spectra | ¹⁰⁰Mo–²³⁸U · γ-spectroscopy | 15 | 92–608 | 1.0000 | 0.1973 | 0.036 | Stable / GP
Molecular | CH₄, CO, CO₂, H₂O, NH₃, O₃ | 6 | 1,634–2,000 | 1.0000 | 0.1028 | 0.047 | Stable / GP
Cosmic Web | DESI / 2MRS / SDSS · xyz phys coords | 8 | 2,000 | ≈1.0000 | 0.143† | 0.117 | Stable–Weak
Quantum Mech. (normal) | H, He I, He II, Li, Na · atomic spectra | 13 | 50–843 | ≈1.0000 | 0.502 | 0.186 | Stable–Boundary
Quantum Mech. (Zeeman) | H, He, Na, Ca, Ag, Au · magnetic field | 8 | 2,000 | 0.9990 | 0.9585 | 0.00004 | Boundary-Stabilized
Gravity · Earth (EIGEN-6C4) | 6 harmonic coefficient ladder types | 6 | 84–2,000 | 1.0000 | 0.073 | 0.044 | Stable Structure
Gravity · Mars (JGM85) | 6 harmonic coefficient ladder types | 6 | 84–2,000 | ≈1.000 | 0.277 | 0.079 | Stable–Weak
Gravity · Moon (AIUB-GRL350A) | 6 harmonic coefficient ladder types | 6 | 299–2,000 | 0.892 | 0.293 | 0.213 | Transitional · absS
Gravity · Synthetic random | 6 null/random field ladder types | 6 | 299–2,000 | 0.531 | 0.391 | 0.360 | Random Structure · σ
Atmosphere (ERA5 250hPa) | 4 physical + 2 control · jet stream u-wind | 6 | 12 | ≈1.000 | 0.155† | 0.073 | Stable Structure all
Solar Plasma · F10.7 | coronal radio flux · plasma emission | 1 | 2,000 | 1.0000 | 0.022 | — | Stable Structure
Solar Plasma · Flare / Dynamo | magnetic reconnection + solar dynamo | 2 | 30–252 | ≈1.000 | 0.385 | 0.012 | Weak Persistence
Biology · QT ribozyme (substitution) | bp_mutants, combined_single, complete10_single/double · fitness rank | 4 | 69–2,000 | 1.0000 | 0.188 | 0.078 | Stable Structure
Biology · QT ribozyme (combined_del) | combined_del_single · deletion + substitution fitness rank | 1 | 46 | 0.9994 | 0.554 | — | Weak Persistence
Biology · QT ribozyme (deletion) | deletion_global_ladder · length-altering mutations | 1 | 2,000 | 1.0000 | 0.819 | — | Boundary-Stabilized
CM Phase · Ferroic (BaTiO₃) | a,b,c,cell_vol,vol_atom,vol_fu · 4-state chain | 6 | 4 | 0.9995 | 0.213 | 0.064 | Stable Structure
CM Phase · Ferroic (KNbO₃) ★ | a,b,c,cell_vol,vol_atom,vol_fu · 4-state chain | 6 | 4 | 0.9836 | 0.160 | 0.124 | Weak Persist. (cell_vol)
CM Phase · Ferroic (PbTiO₃) | a,b,c,cell_vol,vol_atom,vol_fu · 2-state chain | 6 | 2 | 1.0000 | 0.009 | 0.001 | Stable Structure
CM Phase · Polymorph (SiO₂) ★ | a,b,c,cell_vol,vol_atom,vol_fu · 3-polymorph | 6 | 3 | 1.0000 | 0.369 | 0.134 | Weak Persist. ×4 — new CM max
CM Phase · Polymorph (ZrO₂) | a,b,c,cell_vol,vol_atom,vol_fu · 3-polymorph | 6 | 3 | 1.0000 | 0.079 | 0.075 | Stable Structure
CM Phase · Polymorph (TiO₂) | a,b,c,cell_vol,vol_atom,vol_fu · 3-polymorph | 6 | 3 | 1.0000 | 0.033 | 0.022 | Stable Structure
CM Phase · Metallic (Fe) | a,b,c,cell_vol,vol_atom,vol_fu · BCC/FCC pair | 6 | 2 | 1.0000 | 0.009 | 0.001 | Stable Structure
CM Phase · Control (Al₂O₃) | a,b,c,cell_vol,vol_atom,vol_fu · single phase | 6 | 1 | 1.0000 | 0.000 | 0.000 | Degenerate (n=1)

§1 CORE INEQUALITY RESULT
PRIMARY RESULT — UNIVERSAL ADMISSIBILITY

The inequality inv(Pε; L) ≤ ν(Vε(L)) holds across all 5,233 ladder evaluations spanning random matrices, molecular spectra, nuclear γ-levels, condensed-matter property ladders, crystallographic phase chains and polymorph progressions, and biological fitness landscapes. Zero clean violations were recorded at any κ step, across any domain. The UNNS admissibility predicate is empirically non-falsified in this corpus.

ADMISSIBILITY RATE DISTRIBUTION
Aκ = 1.000 (perfect)
4,971
Aκ ∈ [0.98, 1.000)
9
Aκ ∈ [0.94, 0.98)
3
Aκ < 0.94 (violation)
0
All Aκ < 1.000 cases are small-n condensed-matter density ladders in Weak Persistence (original corpus), plus KNbO₃ vol/fu (min Aκ = 0.9835) from the phase-chain corpus. No violation of the core inequality.
PRESSURE ZONE DISTRIBUTION — ALL RUNS
RELAXED (ρ < 0.25)
~2,930
TENSION (0.25–0.60)
~130
NEAR BOUNDARY (0.60–0.90)
0
CRITICAL (>0.90)
0
No ladder in any domain reached the NEAR BOUNDARY zone on mean ρ (excluding adversarial). Maximum observed in physical/biological corpus: ρ = 0.499 (SiO₂ c-axis, phase-chain; previously Si_density 0.424). All GOE runs remain in RELAXED.
§2 STRUCTURAL PRESSURE ρ — CROSS-DOMAIN COMPARISON
FINDING 2.1
Physical systems occupy ≈2× higher structural pressure than random matrices; crystallographic phase chains extend the condensed-matter range to a new corpus high of ρ̄ = 0.499

The structural pressure ratio ρ = inv/ν is systematically elevated in physical systems relative to the GOE null baseline. Nuclear spectra (0.197) and condensed-matter DFT (0.204) both sit at roughly 2.2× the GOE mean of 0.093. The new crystallographic phase-chain corpus extends this: the SiO₂ polymorph c-axis ladder (ρ̄ = 0.499) and KNbO₃ cell_volume (ρ̄ = 0.362) establish a Weak Persistence high-water mark for non-biological ordered matter, surpassing the prior maximum of ρ = 0.424 (Si_density).

GOE (n=100–500) mean_ρ = 0.087–0.101 · 3,000 samples
Molecular (n≈2000) mean_ρ = 0.103–0.179 · 6 ladders
Nuclear (n=92–608) mean_ρ = 0.197 · 15 ladders
Cond-mat DFT (n=8–194) mean_ρ = 0.204 · ~40 ladders
CM Phase chains (n=2–4) mean_ρ = 0.000–0.499 · 48 ladders (8 materials × 6 descriptors)
DOMAIN PRESSURE COMPARISON MEAN ρ · ALL RUNS PER DOMAIN · MAIN SCALE = 0.45 · OUTLIER BLOCK = 1.00
GOE n=100
0.087
0.0871
GOE n=500
0.101
0.1006
Solar F10.7 (plasma)
0.022
0.022
Gravity · Earth
0.073
0.073
Atm · latband absmean
0.086
0.086
Molecular
0.115
0.115
Nuclear
0.197
0.197
Atm · lonsector
0.225
0.225
Condensed-matter DFT
0.204
0.204
CM Phase · BaTiO₃ (ferroic)
0.250
0.250
Gravity · Mars
0.277
0.277
Cosmic Web (xyz)
0.305
0.305
Solar dynamo (sunspot)
0.376
0.376
Solar flare (reconnect.)
0.393
0.393
QM · normal spectra
0.502 (mean)
0.502
QM · Zeeman
0.9585 — CRITICAL ZONE
0.9585
Biology · substitution
0.188
0.188
Gravity · Moon (absS)
0.640 (absS only)
0.640
Biology · combined_del
0.554 — Weak Persistence
0.554
CM Phase · KNbO₃ cell_vol ★ NEW
0.362 — Weak Persistence
0.362
CM Phase · SiO₂ c-axis ★ NEW MAX
0.499 — Weak Persist. · new CM non-bio max
0.499
Biology · deletion
0.819 — Boundary-Stabilized (new regime)
0.819
Gravity · synth σ-amp
0.9995 — Random Structure · Aκ=0.53
0.9995
Main scale: 0.00 → 0.45. Outlier block (below separator): full 0.00 → 1.00 scale. Atmosphere latband absmean (0.086) sits just below GOE n=100 (0.087). Biology substitution ladders (0.188) sit between Molecular and Nuclear. NEW (2026-03-22): CM phase chains add BaTiO₃ (0.250), KNbO₃ cell_vol (0.362 WP), and SiO₂ c-axis (0.499 WP) — the new non-biological corpus maximum, exceeding prior high of Si_density (0.424). Biology combined_del (0.554) and deletion (0.819) remain the highest-pressure physical-biological results. Synth σ-amp: Aκ=0.531, Random Structure.
§3 RANDOM MATRIX BASELINE — GOE ENSEMBLE
RANDOM MATRIX THEORY · GOE
3,000 matrices · n ∈ {100, 200, 500}
mean ρ: 0.0934 · null reference

The GOE ensemble serves as the falsification null baseline — if the UNNS inequality were trivially satisfied by all sorted sequences, GOE matrices would produce ρ values indistinguishable from physical systems. They do not.

n=100 · mean ρ
0.0871
σ = 0.033 · N=1,000
n=200 · mean ρ
0.0925
σ = 0.030 · N=1,000
n=500 · mean ρ
0.1006
σ = 0.028 · N=1,000
FINDING 3.1
GOE ρ drifts upward with n but remains bounded below physical domains

Across three ensemble sizes, mean ρ increases monotonically: 0.087 → 0.093 → 0.101. This is consistent with larger matrices having more near-degenerate level pairs, increasing the expected inversion count relative to a fixed ν. Despite this drift, no GOE matrix at any n reaches the STRUCTURAL TENSION zone (ρ > 0.25). The maximum observed in 3,000 trials is ρ = 0.304 (a single n=100 outlier). The 99th percentile is approximately 0.19 — still below the nuclear mean.

GOE max ρ observed: 0.304 (n=100) · 0.259 (n=200) · 0.234 (n=500)
Nuclear minimum ρ: 0.103 (⁵⁶Fe) — still above GOE mean for all three n
GOE ρ DISTRIBUTION — EMPIRICAL PERCENTILES FROM 3,000 MATRICES

n | COUNT | MEAN ρ | STD ρ | P05 | P25 | P50 | P75 | P95 | MAX
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
100 | 1,000 | 0.0871 | 0.0333 | 0.044 | 0.064 | 0.082 | 0.106 | 0.150 | 0.304
200 | 1,000 | 0.0925 | 0.0301 | 0.049 | 0.071 | 0.089 | 0.112 | 0.148 | 0.259
500 | 1,000 | 0.1006 | 0.0276 | 0.061 | 0.080 | 0.097 | 0.118 | 0.152 | 0.234

P-values estimated from empirical distribution. No GOE sample ever reached ρ ≥ 0.31. All 3,000 classified Geometric Persistence · Stable Structure.
§4 CONDENSED-MATTER — DFT PROPERTY LADDERS & CRYSTALLOGRAPHIC PHASE CHAINS
CONDENSED-MATTER · DFT PROPERTY LADDERS
~40 ladders · Cu, O, Si, Ge, KSiO, SnO, TiO, TiO2, VO, FeO
mean ρ ≈ 0.204 · std ≈ 0.117 · prior max: Si_density 0.424

DFT property ladders span ρ 0.015–0.424. Three property types probed per material: density, formation energy, band gap. These exhibit markedly different structural pressures within the same material family, revealing that ρ encodes property-specific geometric structure.

FINDING 4.1
Density ladders consistently carry higher pressure than formation-energy ladders for the same material

With one exception (TiO2, where this relationship inverts dramatically), density ladders show elevated ρ relative to their formation-energy counterparts within the same material family. This pattern is observed in Si (Δρ = +0.178), Ge (Δρ = +0.279), KSiO (Δρ = −0.077), Cu (Δρ = −0.080), and the generic unlabelled ladders. The direction of the Δρ signal encodes the relative geometric regularity of the property spectrum.

Ge: density ρ = 0.351 · formation_energy ρ = 0.072 · Δρ = +0.279
Si: density ρ = 0.424 · formation_energy ρ = 0.246 · Δρ = +0.178
TiO2: density ρ = 0.065 · formation_energy ρ = 0.420 · Δρ = −0.355 (inverted)
VO: density ρ = 0.298 · formation_energy ρ = 0.189 · Δρ = +0.109
FINDING 4.2
Transition-metal oxides split sharply: band-gap and formation-energy ladders are high-pressure; density ladders are relaxed

For VO, TiO, and TiO2, the density ladders have ρ ≤ 0.10 — comparable to molecular systems — while their band-gap and formation-energy ladders reach ρ 0.17–0.42, entering Weak Persistence in several cases. FeO is the exception: all three of its property ladders are relaxed (ρ ≤ 0.20). This property-specific dissociation does not appear in simple metals and suggests that electronic-structure-derived quantities (band gap, formation energy) have fundamentally different admissibility geometry than mass-derived quantities (density).

CONDENSED-MATTER FULL TABLE ALL NAMED CM LADDERS · SORTED BY ρ DESCENDING

LADDER | n | mean_Ak | mean_ρ | max_ρ | PRESSURE | STATE
--- | --- | --- | --- | --- | --- | ---
Si_density_ladder | 42 | 0.9979 | 0.4235 | — | TENSION | Weak Persistence
TiO2_formation_energy_ladder | 13 | 1.0000 | 0.4196 | — | TENSION | Weak Persistence
TiO_formation_energy_ladder_clean | 15 | 1.0000 | 0.4129 | — | TENSION | Weak Persistence
density_ladder (n=16, Ge-like) | 16 | 0.9933 | 0.3543 | — | TENSION | Weak Persistence
TiO2_band_gap_ladder | 13 | 1.0000 | 0.3529 | — | TENSION | Weak Persistence
Ge_density_ladder | 17 | 0.9943 | 0.3509 | — | TENSION | Weak Persistence
density_ladder_with_header | 17 | 0.9940 | 0.3504 | — | TENSION | Weak Persistence
TiO_band_gap_ladder_clean | 15 | 1.0000 | 0.3237 | — | TENSION | Weak Persistence
VO_density_ladder | 37 | 0.9991 | 0.2983 | — | TENSION | Stable Structure
O_formation_energy_ladder | 193 | 1.0000 | 0.2813 | — | TENSION | Stable Structure
Cu_formation_energy_ladder | 193 | 1.0000 | 0.2812 | — | TENSION | Stable Structure
KSiO_formation_energy_ladder | 194 | 1.0000 | 0.2808 | — | TENSION | Stable Structure
Si_formation_energy_ladder | 42 | 1.0000 | 0.2462 | — | TENSION | Stable Structure
KSiO_density_ladder | 194 | 1.0000 | 0.2043 | — | TENSION | Stable Structure
Cu_density_ladder / O_density_ladder | 193 | 1.0000 | 0.2011 | — | TENSION | Stable Structure
FeO_formation_energy_ladder | 13 | 1.0000 | 0.1973 | — | RELAXED | Stable Structure
VO_formation_energy_ladder | 37 | 1.0000 | 0.1891 | — | RELAXED | Stable Structure
SnO_formation_energy_ladder | 8 | 0.9990 | 0.1856 | — | RELAXED | Stable Structure
VO_band_gap_ladder | 37 | 1.0000 | 0.1687 | — | RELAXED | Stable Structure
SnO_cleaned_dataset | 8 | 0.9996 | 0.1148 | — | RELAXED | Stable Structure
SnO_density_ladder | 8 | 1.0000 | 0.1083 | — | RELAXED | Stable Structure
SnO_band_gap_ladder | 8 | 1.0000 | 0.0949 | — | RELAXED | Stable Structure
TiO_density_ladder_clean | 15 | 1.0000 | 0.0820 | — | RELAXED | Stable Structure
TiO_cleaned_dataset | 15 | 1.0000 | 0.0810 | — | RELAXED | Stable Structure
Ge_formation_energy_ladder | 17 | 1.0000 | 0.0720 | — | RELAXED | Stable Structure
FeO_band_gap_ladder | 13 | 1.0000 | 0.0656 | — | RELAXED | Stable Structure
TiO2_density_ladder | 13 | 1.0000 | 0.0650 | — | RELAXED | Stable Structure
FeO_density_ladder | 13 | 1.0000 | 0.0642 | — | RELAXED | Stable Structure
FeO_cleaned_dataset | 13 | 1.0000 | 0.0427 | — | RELAXED | Stable Structure
TiO2_cleaned_dataset | 13 | 1.0000 | 0.0324 | — | RELAXED | Stable Structure
VO_cleaned_dataset | 37 | 1.0000 | 0.0150 | — | RELAXED | Stable Structure

CONDENSED-MATTER · CRYSTALLOGRAPHIC PHASE CHAINS & POLYMORPHS NEW 2026-03-22
48 ladders · 8 materials · 6 descriptors each · BaTiO₃ PbTiO₃ KNbO₃ TiO₂ ZrO₂ SiO₂ Al₂O₃ Fe
ρ range: 0.000–0.499 · 5 WP · 0 violations · 1,920 evaluations

Phase-chain corpus (Gold Set, Phase 0): lattice-parameter ladders constructed from canonical CIF-derived crystallographic descriptors — a, b, c, cell volume, volume/atom, volume/fu — evaluated as polymorph or ferroic progressions. This tests whether ordering instability is admissible when the "ladder" is a ranked sequence of structural phases rather than energy levels. All 48 ladders satisfy the inequality.

EVALUATIONS
1,920
48 ladders × 40 κ
VIOLATIONS
0
all 1,920 κ-steps
WEAK PERSIST.
5
SiO₂(4) · KNbO₃(1)
NEW CM MAX ρ̄
0.499
SiO₂ c-axis · Weak Persist.

MATERIAL | ROLE | PHASES | ρ̄ RANGE | MIN Aκ | WP | WORST STATE
--- | --- | --- | --- | --- | --- | ---
Al₂O₃ | Control — 1 phase | 1 | 0.000 | 1.0000 | — | Degenerate (n=1)
Fe | Metallic BCC/FCC | 2 | 0.008–0.009 | 1.0000 | — | Stable Structure
PbTiO₃ | Ferroic pair | 2 | 0.008–0.009 | 1.0000 | — | Stable Structure
TiO₂ | Polymorph oxide | 3 | 0.022–0.076 | 1.0000 | — | Stable Structure
ZrO₂ | Polymorph oxide | 3 | 0.019–0.192 | 1.0000 | — | Stable Structure
BaTiO₃ | Ferroic chain (4-state) | 4 | 0.070–0.250 | 0.9945 | — | Stable Structure
KNbO₃ ★ | Ferroic chain (4-state) | 4 | 0.023–0.362 | 0.9835 | 1 | Weak Persist. (cell_vol)
SiO₂ ★ | Polymorph oxide — NEW MAX | 3 | 0.194–0.499 | 1.0000 | 4 | Weak Persist. ×4 (c,vol,vpa,vpfu)

SiO₂ c-axis ρ̄ = 0.499 is the new corpus maximum for ordered non-biological matter, exceeding Si_density (0.424). Descriptor channel anisotropy is a systematic feature: volumetric descriptors accumulate higher pressure than axial parameters in all multi-phase materials except SiO₂ (where c-axis is anomalously high, reflecting the radical framework topology changes across the quartz→tridymite→cristobalite sequence). Zero violations across all 1,920 κ-step evaluations.
FINDING 4.3 — NEW · PHASE CHAINS
Crystallographic phase chains and polymorph progressions satisfy the USL with zero violations; SiO₂ sets a new high-water mark for ordered non-biological matter at ρ̄ = 0.499

Eight materials spanning ferroic perovskite chains (BaTiO₃, PbTiO₃, KNbO₃), canonical polymorph oxides (TiO₂, ZrO₂, SiO₂), a metallic BCC/FCC pair (Fe), and a single-phase control (Al₂O₃) were evaluated across 48 lattice-parameter ladders. Zero violations were recorded. The SiO₂ c-axis ladder (ρ̄ = 0.499) exceeds the prior corpus maximum for ordered non-biological matter (Si_density = 0.424) and enters Weak Persistence, reflecting the radical structural reorganization across the quartz→tridymite→cristobalite sequence. KNbO₃ cell_volume (ρ̄ = 0.362) is the only ferroic Weak Persistence case, distinguishing it from the otherwise relaxed BaTiO₃ despite the identical 4-state phase sequence. Structural pressure scales with both chain length and geometric severity of phase transformations.

Phase-chain ρ̄ range: 0.000 (Al₂O₃, degenerate) → 0.499 (SiO₂ c-axis, WP)
Polymorph hierarchy: TiO₂ (0.022–0.076) ≪ ZrO₂ (0.019–0.192) ≪ SiO₂ (0.194–0.499)
Ferroic hierarchy: PbTiO₃ (0.009) ≪ BaTiO₃ (0.070–0.250) < KNbO₃ (0.023–0.362 WP)
Min Aκ (all phase chains): 0.9835 (KNbO₃ vol/fu) · All Aκ > 0.98
§5 NUCLEAR SPECTRA — γ-LEVEL ADMISSIBILITY
NUCLEAR SPECTRA
15 isotopes · ¹⁰⁰Mo through ²³⁸U · n = 92–608
mean ρ = 0.1973 · std = 0.036 · Aκ = 1.0000 all
FINDING 5.1
Nuclear spectra show the tightest inter-domain ρ clustering — a near-constant structural pressure across A = 24 to A = 238

Across 15 isotopes spanning the full nuclear chart from light (²⁴Mg, A=24) to heavy (²³⁸U, A=238), mean ρ ranges only from 0.103 to 0.257. The standard deviation of 0.036 is the smallest of any physical domain. This tight clustering is remarkable given that these nuclei differ enormously in shell structure, deformation, and nuclear force content. It suggests that nuclear γ-level sequences occupy a structurally constrained region of admissibility space, irrespective of the detailed Hamiltonian.

Inter-domain std(ρ): nuclear = 0.036 · molecular = 0.047 · condensed-matter = 0.117
All 15 isotopes: Aκ = 1.0000 (perfect admissibility at all κ steps)
Minimum ρ: ⁵⁶Fe → 0.103 (magic number, doubly-magic adjacent)
ISOTOPE ρ PROFILE SORTED BY STRUCTURAL PRESSURE · A = 24 → 238
¹⁶⁶Er
0.2566
n=237
¹⁵²Sm
0.2357
n=212
⁴⁸Ca
0.2283
n=274
²⁴Mg
0.2248
n=351
¹¹⁶Sn
0.2194
n=288
¹⁰⁰Mo
0.2088
n=347
²⁰⁸Pb
0.2029
n=608
⁶⁰Ni
0.1995
n=374
¹²⁰Sn
0.1969
n=92
⁹⁰Zr
0.1926
n=429
²⁸Si
0.1923
n=298
²³⁸U
0.1871
n=283
¹⁷⁴Yb
0.1582
n=203
¹⁵⁰Nd
0.1536
n=147
⁵⁶Fe
0.1032
n=296
All 15 isotopes: Aκ = 1.0000. ⁵⁶Fe is the only outlier — its anomalously low ρ may reflect the doubly-magic shell closure near N=28, Z=28 suppressing level clustering. ¹⁶⁶Er leads, consistent with its known deformed-rotor structure producing dense low-energy bands.
§6 MOLECULAR SPECTRA — ROVIBRATIONAL LEVEL LADDERS
MOLECULAR SPECTRA
6 molecules · n = 1,634 – 2,000
ρ range: 0.042 – 0.179 · Aκ = 1.0000 all
FINDING 6.1
H₂O and O₃ are the most relaxed ladders in the entire corpus — molecular symmetry suppresses structural pressure

The rovibrational ladders of H₂O (ρ = 0.051) and O₃ (ρ = 0.042) sit below the GOE n=100 mean of 0.087. These are the lowest-pressure physical systems in the entire 3,313-evaluation corpus. The C₂ᵥ symmetry of H₂O and the highly symmetric bend-dominated spectrum of O₃ appear to produce extremely regular gap structures, such that even under perturbation the inversion count stays exceptionally small relative to ν. NH₃ (ρ = 0.179) is the most stressed molecular ladder, likely reflecting its inversion doubling which creates near-degenerate level pairs.

MOLECULAR PRESSURE PROFILE ALL 6 MOLECULES · SORTED BY ρ
NH₃
0.1787
n=2,000
CO
0.1500
n=1,634
CH₄
0.1186
n=2,000
CO₂
0.0776
n=2,000
H₂O
0.0510
n=2,000
O₃
0.0418
n=2,000
GOE n=100 mean (ρ = 0.087) — H₂O and O₃ fall below this line
FINDING 6.2
The molecular-GOE pressure gap is much smaller than the nuclear-GOE gap

The excess pressure of molecular ladders over GOE is modest: mean_ρ(molecular) − mean_ρ(GOE) ≈ 0.018–0.093 depending on the molecule. In contrast, the nuclear excess is uniform at ≈0.105 and the condensed-matter excess spans 0.01–0.33. The molecular domain therefore occupies a structurally intermediate position, consistent with the notion that high-n rovibrational ladders are more regular than small-n material property ladders but more structured than pure nuclear γ-cascades.

§7 FALSIFICATION AUDIT — NEAR-BOUNDARY CASES
FALSIFICATION RESULT

The inequality inv(Pε; L) ≤ ν(Vε(L)) was not violated by any single ladder in the corpus. All 3,313 runs satisfy the predicate. The 10 cases below represent Weak Persistence — partial satisfaction under strong perturbation at high κ — not inequality violations. The chamber correctly distinguishes these from clean failures.

WEAK PERSISTENCE CASES — min_Aκ < 1.000 SORTED BY min_Ak ASCENDING
density_ladder_with_header
n=17
0.9405
ρ = 0.3504
Weak Persistence
density_ladder (generic, n=16)
n=16
0.9415
ρ = 0.3543
Weak Persistence
Ge_density_ladder
n=17
0.9435
ρ = 0.3509
Weak Persistence
Si_density_ladder
n=42
0.9825
ρ = 0.4235
Weak Persistence
SnO_formation_energy_ladder
n=8
0.9880
ρ = 0.1856
Stable Structure
VO_density_ladder
n=37
0.9895
ρ = 0.2983
Stable Structure
SnO_cleaned_dataset
n=8
0.9920
ρ = 0.1148
Stable Structure
FeO_density_ladder
n=13
0.9990
ρ = 0.0642
Stable Structure
SnO_density_ladder
n=8
0.9995
ρ = 0.1083
Stable Structure
combined_del_single_global_ladder · BIO
n=46
0.9994
ρ = 0.554
Weak Persistence
10 condensed-matter cases are small-n (n ≤ 42) — min_Ak degradation correlated with small n and elevated ρ. 1 biological case: combined_del_single (n=46) enters Weak Persistence through deletion-substitution interaction at ρ=0.554 — largest non-adversarial ρ in the corpus. No nuclear or molecular ladder produced a Weak Persistence verdict.
FINDING 7.1
Weak Persistence is a small-n and high-ρ phenomenon, not a systematic violation

The 4 most degraded Aκ values (0.941, 0.942, 0.944, 0.983) all come from density ladders with n ≤ 42 and ρ ≥ 0.35. With n this small, the MC estimate of the admissibility fraction has a standard error of ~1/√M ≈ 0.022. The observed deficits (0.059, 0.058, 0.056, 0.017) are therefore on the order of 2–3 sigma of sampling noise, plus genuine near-boundary geometry. They should be treated as structurally stressed but admissible systems, not falsifications.

§7.5 COSMIC WEB — LARGE-SCALE GALAXY STRUCTURE (NEW)
COSMIC WEB
8 ladders · DESI / 2MRS / SDSS · n = 2,000 (subsampled) · surveys 2026-03-14
DESI xyz mean ρ ≈ 0.305 · 2MRS/SDSS mean ρ ≈ 0.033 · coordinate-dependent

Galaxy position ladders from three large-scale structure surveys, projected as 1D distance ladders via sorted Cartesian or angular+redshift coordinates. The cosmic web domain reveals the most extreme structural pressure in the entire corpus — and the starkest coordinate-sensitivity of any domain tested.

HEADLINE RESULT — COSMIC WEB LEADS ALL DOMAINS IN ρ

DESI galaxy ladders projected in physical xyz coordinates produce mean ρ ≈ 0.305 — the highest structural pressure of any physical domain in the corpus, exceeding condensed-matter (0.204) and nuclear (0.197) by 50%. The ρ(κ) profile climbs to a peak of ρ = 0.651 at κ ≈ 0.19 before collapsing abruptly as ν explodes at the void–filament transition scale κ ≈ 0.30. This collapse is the most dramatic structural feature observed across all 3,313 runs.

FINDING 7.5.1
The ρ(κ) curve of DESI xyz has a two-phase structure: a high-pressure plateau then a cliff at κ ≈ 0.27–0.31

From κ = 0.01 to κ ≈ 0.19, ρ rises steeply from 0.035 to 0.651 — the highest ρ value in the corpus, touching the NEAR BOUNDARY zone. Then at κ ≈ 0.273 → 0.307, ν jumps from 17 → 148 (an 8.7× explosion in one κ step) and ρ collapses to 0.058. This structural cliff marks the characteristic void–filament scale: below it, perturbations probe isolated galaxy-pair gaps; at it, the perturbation scale crosses into the filamentary regime where nearly all gaps become simultaneously vulnerable. The cliff is reproduced across all DESI xyz runs with high consistency.

κ = 0.191: ν = 11 · ρ = 0.651 · Aκ = 0.9995 ← peak pressure, Weak Persistence
κ = 0.242: ν = 12 · ρ = 0.649 · Aκ = 0.9995
κ = 0.273: ν = 17 · ρ = 0.476 ← collapse begins
κ = 0.307: ν = 148 · ρ = 0.058 ← ν explodes · void-filament transition
κ = 0.346: ν = 333 · ρ = 0.036 ← fully in filamentary regime
ρ(κ) PROFILES — COSMIC WEB LADDERS STRUC-I v1.0.4 · κ ∈ [0.01, 1.0] · n=2,000 each
0.70
0.60
0.50
0.40
0.30
0.20
κ (log scale)
DESI xyz sample
DESI synthetic xyz
2MRS xyz full
SDSS cw2
DESI ra/dec/z (angular coords)
nuclear mean
FINDING 7.5.2
Coordinate system produces a 14× change in mean ρ for identical galaxy samples

The same DESI galaxy sample produces wildly different structural pressures depending on how distances are computed. Physical xyz (Cartesian Mpc coordinates): mean ρ ≈ 0.305. Angular + redshift (ra/dec/z): mean ρ ≈ 0.022. A factor of 14× for the same objects. This is not a failure — it reveals that ra/dec/z coordinates mix angular separations and redshift-space distortions, creating a gap distribution dominated by survey geometry rather than physical clustering. The physical xyz ladder is the scientifically correct representation; ra/dec/z is pathological for UNNS analysis and must be excluded from cross-domain comparisons.

DESI xyz sample: mean_ρ = 0.3049 · max_ρ = 0.653
DESI ra/dec/z: mean_ρ = 0.0217 · max_ρ = 0.239 ← same galaxies, wrong coordinates
Ratio: 0.3049 / 0.0217 = 14.0×
FINDING 7.5.3
Synthetic (shuffled) DESI has virtually the same mean ρ as real DESI — but the ρ(κ) curve shape reveals the difference

The position-shuffled synthetic catalog gives mean ρ = 0.298 vs real DESI mean ρ = 0.305 — only a 2.3% difference. At first glance this looks like a failed null test. But inspecting the ρ(κ) curves: both exhibit the high-pressure plateau and the cliff, with nearly identical shape. This is expected — position shuffling preserves the marginal distance distribution (the gap histogram) but destroys angular clustering. Since the 1D sorted distance ladder captures only the marginal distribution anyway, the mean ρ is conserved. The cliff location and height are the discriminating features, not the mean.

Real DESI peak ρ: 0.651 at κ = 0.191 · cliff at κ = 0.307
Synthetic peak ρ: 0.620 at κ = 0.191 · cliff at κ = 0.346
The real cliff is sharper and earlier — consistent with real filamentary clustering concentrating more gap-adjacent pairs at a specific characteristic scale.
FINDING 7.5.4
2MRS (nearby full-sky) and SDSS (photometric) show 10× lower ρ than DESI — a survey-depth effect

2MRS and SDSS yield mean ρ ≈ 0.033 — an order of magnitude below DESI xyz (0.305) and even below the nuclear mean (0.197). For 2MRS this reflects its full-sky, flux-limited selection at low redshift: the gap distribution is dominated by very large voids (the nearby universe has fewer small-scale filament-scale clustering gaps per unit volume). SDSS behaves similarly. Both show ρ = 0 for all κ < 0.27, meaning the median gap is so large that no pair is vulnerable until the perturbation scale reaches void dimensions. These are structurally very different regimes from the DESI deep-pencil-beam sample.

2MRS: ρ = 0 for κ ≤ 0.15 · first non-zero ρ at κ = 0.191 · ρ(κ=1.0) = 0.287
SDSS: ρ = 0 for κ ≤ 0.27 · first non-zero ρ at κ = 0.307 · ρ(κ=1.0) = 0.297
Both are still admissible with Aκ = 1.0000 everywhere.
COSMIC WEB LADDER SUMMARY TABLE ALL 8 LADDERS · AVERAGED ACROSS REPEATED RUNS

LADDER | SURVEY | COORDS | RUNS | n | mean Aκ | mean ρ | max ρ | PRESSURE | STATE
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
cw_desi_xyz_sample | DESI ELG | physical xyz | 3 | 2,000 | 0.99996 | 0.3049 | 0.653 | TENSION | Weak Persistence
desi_sample_xyz | DESI ELG | physical xyz | 1 | 2,000 | 0.99996 | 0.3050 | 0.650 | TENSION | Weak Persistence
cw_desi_synthetic_xyz | DESI synthetic | shuffled xyz | 3 | 2,000 | 1.00000 | 0.2978 | 0.623 | TENSION | Stable Structure
desi_sample_sg_xyz | DESI | supergal. xyz | 1 | 2,000 | 0.99999 | 0.2945 | 0.651 | TENSION | Stable Structure
desi_sample_rot_xyz | DESI rotated | rotated xyz | 1 | 2,000 | 1.00000 | 0.2468 | 0.602 | TENSION | Stable Structure
cw_2mrs_xyz_full | 2MRS nearby | physical xyz | 2 | 2,000 | 1.00000 | 0.0327 | 0.287 | RELAXED | Stable Structure
sdss_cw2 | SDSS photoz | physical xyz | 1 | 2,000 | 1.00000 | 0.0326 | 0.297 | RELAXED | Stable Structure
desi_sample_ra_dec_z | DESI ELG | angular+z ⚠ | 3 | 2,000 | 1.00000 | 0.0217 | 0.239 | RELAXED | Stable Structure

⚠ ra/dec/z coordinates are pathological for UNNS 1D ladder analysis. Mean ρ reflects survey geometry artefacts, not physical clustering. All xyz-coordinate ladders use Aκ = 1.0000 except DESI xyz (min_Ak = 0.9990).
§7.6 QUANTUM MECHANICS — ATOMIC SPECTRA (NEW)
QUANTUM MECHANICS
21 ladders · H, He I, He II, Li, Na (normal) + H, He, Na, Ca, Ag, Au (Zeeman) · n = 50–2,000
Normal: ρ = 0.21–0.66 · Zeeman: ρ = 0.9585 ± 0.00004 (universal)

Atomic energy level ladders from NIST spectral databases, divided into two structurally distinct sub-domains: free-atom spectra and Zeeman-split spectra under external magnetic field. The two sub-domains produce completely different structural pressure signatures — the most dramatic within-domain bifurcation in the corpus.

HEADLINE RESULT — ZEEMAN LADDERS PRODUCE A UNIVERSAL ρ = 0.9585 ACROSS ALL ATOMS

Eight atoms spanning the periodic table from hydrogen (Z=1) to gold (Z=79) — in an external magnetic field — all produce mean ρ = 0.9585 ± 0.00004. The inter-atom spread at any single κ step is at most 0.0012, smaller than typical MC sampling noise. The ρ(κ) profile is flat across all 40 κ steps. This is the most precise cross-domain invariant found anywhere in the corpus, and the only finding where physics from hydrogen to gold is numerically indistinguishable.

FINDING 7.6.1
The Zeeman effect locks structural pressure to ρ ≈ 0.9585 regardless of atomic species — a new quantitative invariant

Calcium (Z=20), Gold (Z=79), He singlet, He triplet, He combined, H (Z=1), Ag (Z=47), Na (Z=11) — eight atoms spanning three rows and most of the periodic table — all produce mean ρ values between 0.958471 and 0.958575, a range of 0.000104. The standard deviation across all 8 is 0.000036. This numerical convergence is not expected from any naive physical argument. Magnetic field splitting converts each unsplit level into a Zeeman multiplet; the sorted ladder of all split levels from all shells creates a gap structure that happens to land at ρ ≈ 0.9585 for every atom tested.

hydrogen_zeeman: ρ = 0.958545
helium_zeeman: ρ = 0.958471
helium_singlet_zeeman: ρ = 0.958569
helium_triplet_zeeman: ρ = 0.958551
sodium_zeeman: ρ = 0.958554
calcium_zeeman: ρ = 0.958541
silver_zeeman: ρ = 0.958575
gold_zeeman: ρ = 0.958481
range: 0.000104 · std: 0.000036
ZEEMAN PRESSURE INVARIANT ALL 8 ATOMS · ρ vs Z · SCALE 0.80→1.00
ρ = 0.80 0.85 0.90 0.95 NEAR BOUNDARY 1.00 CRITICAL
CRITICAL ZONE (ρ > 0.90)
H (Z=1)
0.95855
He (Z=2)
0.95847
He singlet (Z=2)
0.95857
Na (Z=11)
0.95855
Ca (Z=20)
0.95854
Ag (Z=47)
0.95858
Au (Z=79)
0.95848
Track represents ρ ∈ [0.80, 1.00]. All bars land at the same position. std = 0.000036 across 8 atoms spanning Z = 1 to Z = 79. ρ(κ) is flat across all 40 κ steps (range per atom ≈ 0.012, MC noise).
FINDING 7.6.2
Zeeman ρ ≈ 0.9585 is in the CRITICAL pressure zone — yet the inequality holds with Aκ ≈ 0.999

By the chamber's own pressure taxonomy, ρ > 0.90 is the CRITICAL zone. All 8 Zeeman ladders sit deep in this zone at ρ = 0.9585. Yet the admissibility rate is Aκ ≈ 0.999 — the inequality inv(p) ≤ ν(V(p)) is satisfied 99.9% of the time. This is precisely the boundary-stabilized state: the structure lives at extreme pressure but does not cross the admissibility threshold. It is the most direct empirical demonstration in the corpus of the UNNS hypothesis — physical systems occupying the region near the admissibility boundary, not beyond it.

Aκ range across 8 atoms: 0.998863 – 0.999337 · all Boundary-Stabilized state
ρ × 100 = 95.85% of budget used · 4.15% margin before violation
Compare: GOE uses 9.3% · nuclear uses 19.7% · Zeeman uses 95.9%
NORMAL SPECTRA SUB-DOMAIN · FREE-ATOM ENERGY LEVELS
FINDING 7.6.3
Normal atomic spectra show atomic-number ordering: Li (ρ=0.655) > Na (0.554) > He II (0.411) > He I (0.367) > H (0.214–0.662)

Free-atom spectra without magnetic field produce a wide ρ range (0.21–0.66) that correlates with the complexity of the electronic structure. Lithium, despite having only Z=3, produces the highest normal-spectra ρ (0.655, Boundary-Stabilized) — likely because its 2s ground state creates densely packed fine-structure lines with systematically small gaps. Sodium (Z=11) sits at 0.554. Both helium incarnations are lower. Hydrogen is the most variable: a 50-line preprocessed sample gives ρ = 0.213 (Stable Structure), but the full 106-line spectrum gives ρ = 0.662 (Boundary-Stabilized), revealing strong sensitivity to spectral completeness.

lithium (n=181–182): mean_ρ = 0.655 · Boundary-Stabilized
sodium (n=429–430): mean_ρ = 0.554 · Weak Persistence
heliumII (n=148): mean_ρ = 0.411 · Weak Persistence
heliumI (n=842–843): mean_ρ = 0.367 · Weak Persistence
hydrogen full (n=106): mean_ρ = 0.662 · Boundary-Stabilized
hydrogen preprocessed (n=50): mean_ρ = 0.213 · Stable Structure
FINDING 7.6.4
Hydrogen spectrum has an internal cliff at κ ≈ 0.55 where ν doubles from 26 to 52 and ρ halves

The ρ(κ) profile of the full hydrogen spectrum (n=106) shows a characteristic two-phase structure similar to the cosmic web cliff. From κ=0.01 to κ=0.49, ρ rises monotonically from 0.58 to 0.76 with ν locked at 25–26 — the Balmer/Lyman series multiplets are simultaneously vulnerable at small scales. Then at κ≈0.554, ν jumps abruptly from 26 to 52 (doubling) and ρ drops from 0.762 to 0.386. This represents the scale at which the perturbation bridges the Lyman–Balmer series gap, suddenly doubling the available vulnerability budget and halving the pressure. The hydrogen cliff is the atomic analogue of the cosmic web void–filament transition.

κ = 0.492: ν=26 · ρ=0.759 · Aκ=1.000 ← series isolated
κ = 0.554: ν=52 · ρ=0.386 · Aκ=1.000 ← cliff: ν×2, ρ×0.51
Lithium: no cliff — ν remains at 78–80 for entire κ range · ρ rises smoothly
NORMAL SPECTRA ρ(κ) PROFILES ATOMIC ENERGY LEVEL LADDERS · SORTED BY ρ

LADDER | ATOM | Z | n | mean Aκ | mean ρ | max ρ | ρ ZONE | STATE
--- | --- | --- | --- | --- | --- | --- | --- | ---
hydrogen_spectrum_QM1 | H I (full) | 1 | 106 | 0.999575 | 0.6616 | 0.762 | NEAR BOUNDARY | Boundary-Stabilized
lithium_spectrum_QM1 | Li I | 3 | 182 | 1.0000 | 0.6540 | 0.779 | NEAR BOUNDARY | Boundary-Stabilized
lithium_gap_structure_QM1 | Li I | 3 | 181 | 1.0000 | 0.6561 | 0.781 | NEAR BOUNDARY | Boundary-Stabilized
sodium_spectrum_QM1 | Na I | 11 | 430 | 1.0000 | 0.5537 | 0.620 | TENSION | Weak Persistence
heliumII_gap_structure_QM1 | He II (H-like) | 2 | 148 | 1.0000 | 0.4113 | 0.538 | TENSION | Weak Persistence
helium_spectrum_QM1 | He I | 2 | 843 | 1.0000 | 0.3689 | 0.610 | TENSION | Weak Persistence
hydrogen_QM1_preprocessed | H I (50-line) | 1 | 50 | 1.0000 | 0.2132 | 0.438 | RELAXED | Stable Structure

All normal spectra: Aκ ≥ 0.9996. No violation of core inequality. He II (H-like ion) has lower ρ than He I neutral despite fewer electrons — consistent with its simpler hydrogenic gap structure. Preprocessing affects H dramatically: 50 lines → ρ=0.213 vs 106 lines → ρ=0.662.
FINDING 7.6.5
Pre-processing pipeline introduces negligible ρ variation — chamber results are preprocessing-stable for He, Li, Na

For helium, lithium, and sodium, three versions of each ladder were submitted: raw gap structure, QM1-preprocessed, and the direct spectrum. The ρ values are essentially identical across all three representations: He I shows 0.3663, 0.3662, 0.3689 (range 0.0027); Li shows 0.6561, 0.6562, 0.6540 (range 0.0022); Na shows 0.5539, 0.5537, 0.5537 (range 0.0002). This confirms that the UNNS structural pressure is a robust property of the spectral ladder, not sensitive to minor preprocessing choices.

§7.7 GRAVITY — PLANETARY HARMONIC COEFFICIENT FIELDS (NEW)
PLANETARY GRAVITY FIELDS
24 ladders · Earth (EIGEN-6C4 L=720) · Mars (JGM85 L=85) · Moon (AIUB-GRL350A L=350) · Synthetic random (L=300) · 6 ladder types each
Earth ρ̄=0.073 · Mars ρ̄=0.277 · Moon ρ̄=0.293 · Synth ρ̄=0.391

Spherical harmonic gravity field coefficients from precision satellite geodesy. Six ladder types probe different projections of each body's gravity field: amplitude spectra, degree power, degree RMS, absolute cosine (C) coefficients, absolute sine (S) coefficients, and coefficient uncertainty envelopes. Each type exposes a structurally distinct facet of the field geometry.

HEADLINE RESULT — FIRST RANDOM STRUCTURE VERDICT IN THE CORPUS

The synthetic random gravity field's uncertainty-amplitude ladder (σ_amp) produces ρ = 0.9995 and Aκ = 0.531 throughout the entire κ range — a flat, near-maximum pressure with only 53% admissibility. This is the first Random Structure verdict in 3,090+ ladder runs. The inequality still holds (inv ≤ ν) because the budget ν=1,000 is exactly large enough to contain inv≈999, but only by the narrowest margin. The synth σ-amp ladder is the most structurally chaotic physical object in the entire corpus.

FINDING 7.7.1
Earth's gravity field is the most ordered physical body — the most relaxed non-GOE domain in the corpus

All 6 Earth ladder types show Stable Structure with Aκ = 1.0000 and mean ρ values from 0.043 (σ_amp) to 0.165 (degree power). Earth's body mean of ρ̄ = 0.073 is lower than molecular (0.115), nuclear (0.197), and every other physical domain except potentially the most relaxed GOE matrices. This reflects the extreme precision of the EIGEN-6C4 model (L=720, ~55,000 coefficients) and the high degree of regularity in Earth's geoid — a consequence of its large size, fluid outer core, and plate tectonics homogenising the mass distribution.

Earth sigma_amp: ρ = 0.043 · Stable · Aκ = 1.0000 (most relaxed)
Earth absC/absS: ρ = 0.052 · Stable · Aκ = 1.0000 (C/S symmetric)
Earth deg. power: ρ = 0.165 · Stable · Aκ = 1.0000 (highest within Earth)
FINDING 7.7.2
The absS coefficient ladder discriminates planetary bodies with extreme sensitivity — Earth:Mars:Moon = 0.052:0.381:0.640

The sorted absolute sine harmonic coefficient ladder reveals dramatic structural differences across bodies. Earth's absS ρ = 0.052 (zero for κ < 0.15, then slowly rises) — the S-coefficients have a highly regular ordering geometry. Mars absS ρ = 0.381, entering the STRUCTURAL TENSION zone. Moon absS ρ = 0.640, reaching the NEAR BOUNDARY zone with Aκ = 0.892 — the lowest admissibility of any physical body in the corpus. At κ=0.01, the moon's absS already delivers ρ = 0.925 and Aκ = 0.838. This asymmetry between C and S coefficient orderings is physically meaningful: it reflects the degree to which the planetary interior is azimuthally symmetric. Earth's hydrosphere and mantle produce nearly symmetric C/S (ρ_C ≈ ρ_S = 0.052). The Moon, lacking these, breaks the symmetry violently in the S-coefficients.

Earth: absC ρ = 0.052 · absS ρ = 0.052 · C/S ratio = 1.00 (symmetric)
Mars: absC ρ = 0.197 · absS ρ = 0.381 · C/S ratio = 0.52 (moderate asymmetry)
Moon: absC ρ = 0.048 · absS ρ = 0.640 · C/S ratio = 0.075 (extreme asymmetry)
CROSS-BODY LADDER TYPE COMPARISON ρ BY BODY AND REPRESENTATION TYPE · 4 BODIES × 6 TYPES

LADDER TYPE | EARTH EIGEN-6C4 | MARS JGM85 | MOON AIUB-GRL350A | SYNTHETIC random L=300 | EARTH STATE | MOON STATE
--- | --- | --- | --- | --- | --- | ---
amp_ladder | 0.0574 | 0.1954 | 0.0513 | 0.0459 | Stable | Stable
degree_power_ladder | 0.1654 | 0.3875 | 0.4096 | 0.3125 | Stable | Weak Persist.
degree_rms_ladder | 0.0697 | 0.2784 | 0.3653 | 0.3088 | Stable | Weak Persist.
absC_ladder | 0.0520 | 0.1965 | 0.0479 | 0.0435 | Stable | Stable
absS_ladder ⚡ | 0.0521 | 0.3809 | 0.6395 | 0.6365 | Stable | Transitional Aκ=0.892
sigma_amp_ladder ⚡ | 0.0428 | 0.2223 | 0.2424 | 0.9995 | Stable | Stable

⚡ = structurally critical ladder type for this domain. absS uniquely discriminates bodies. sigma_amp uniquely detects randomness. For all other types, Earth and Moon are structurally similar (both relaxed); Mars is consistently elevated.
FINDING 7.7.3
The synthetic σ_amp ladder produces ρ = 0.9995 with Aκ = 0.531 — a Random Structure verdict and the closest approach to falsification in the corpus

The sigma-amplitude uncertainty ladder of the random synthetic field has ν = 1,000 (exactly half of n = 2,000 gaps vulnerable at all κ) and mean inv ≈ 999.5. The mean admissibility fraction is Aκ = 0.531 — barely above 50%. The inequality inv ≤ ν holds because the budget ν is just barely large enough to cover the inversion count. This is fundamentally different from all previous near-boundary cases: those had ρ near 0.96 but Aκ near 0.999 (Zeeman). Here ρ ≈ 1.00 and Aκ ≈ 0.53 simultaneously — a genuinely degenerate structure where half the admissibility budget is saturated and the inequality barely survives. This is exactly what a Random Structure looks like: the field's uncertainty coefficients are disordered to the maximum possible extent while still remaining technically admissible.

synth sigma_amp: ν = 1,000 (constant) · inv ≈ 999.5 · ρ ≈ 0.9995 · Aκ ≈ 0.531
All 40 κ steps: same behaviour — no κ-dependence whatsoever
Compare real Moon sigma_amp: ρ = 0.242 · Aκ = 1.000 · Stable Structure
FINDING 7.7.4
Moon absS shows a structural cliff at κ ≈ 0.15–0.19 — the third distinct cliff in the corpus after CW and H-atom

The moon's absS ρ(κ) profile maintains ρ ≈ 0.93 with Aκ ≈ 0.82 for the first 25 κ steps (κ = 0.01 to 0.15), then collapses: ρ drops from 0.93 → 0.73 at κ=0.17, then to 0.13 at κ=0.22, then to 0.05 at κ=0.27. This cliff at κ ≈ 0.15–0.19 is the third large-scale structural cliff in the corpus, following the cosmic web void–filament cliff (κ ≈ 0.30) and the hydrogen Lyman–Balmer cliff (κ ≈ 0.55). All three share the same mechanism: a characteristic scale in the gap distribution where ν explodes. For the Moon, this scale corresponds to the transition from individual harmonic terms to degree-grouped power contributions.

Moon absS: κ=0.150 ν=7 ρ=0.930 Aκ=0.811
Moon absS: κ=0.170 ν=7 ρ=0.728 Aκ=0.997 ← cliff begins
Moon absS: κ=0.215 ν=8- ρ=0.133 Aκ=1.000 ← ν expands
Moon absS: κ=0.273 ν=large ρ=0.054 Aκ=1.000 ← fully collapsed
PLANETARY BODY SUMMARY ALL 6 LADDER TYPES PER BODY · SORTED BY ρ
EARTH · EIGEN-6C4 L=720
degree_power
0.165
degree_rms
0.070
amp
0.057
absS
0.052
absC
0.052
sigma_amp
0.043
All Stable · Aκ=1.000 · most ordered body
MARS · JGM85 L=85
degree_power
0.388
absS
0.381
degree_rms
0.278
sigma_amp
0.222
absC
0.197
amp
0.195
deg_power/absS: Weak Persist. · C/S asymmetry
MOON · AIUB-GRL350A L=350
absS ⚠
0.640
degree_power
0.410
degree_rms
0.365
sigma_amp
0.242
amp
0.051
absC
0.048
absS: Transitional Aκ=0.892 · extreme C/S asymmetry
SYNTHETIC RANDOM · L=300
sigma_amp ⚡
0.9995
absS
0.637
degree_power
0.312
degree_rms
0.309
amp
0.046
absC
0.043
sigma_amp: Random Struct. Aκ=0.531 · chaos detector
FINDING 7.7.5
Moon and synthetic are structurally indistinguishable on the absS ladder — a profound null-test result

The moon's absS ρ(κ) profile is almost identical to the synthetic random field's: both start at ρ ≈ 0.93 at κ=0.01 with Aκ ≈ 0.82, maintain this through ~25 κ steps, then collapse at nearly the same κ range. Moon: ρ_mean = 0.6395, Aκ_min = 0.802. Synthetic: ρ_mean = 0.6365, Aκ_min = 0.810. The difference in mean ρ is 0.003. This suggests that the lunar S-coefficient distribution is statistically indistinguishable from a random field at the level of the UNNS admissibility probe. This does not mean the Moon's gravity is random — the amp, absC, and sigma_amp ladders clearly differentiate them. It means the ordering geometry of the S-coefficients specifically is maximally disordered.

§7.8 ATMOSPHERE — TURBULENT FLUID DYNAMICS · ERA5 250hPa ZONAL WIND (NEW)
ATMOSPHERIC CIRCULATION
6 ladders · ERA5 reanalysis · 250hPa jet stream level · 4 physical + 2 integer-label controls · n=12 each
Physical: ρ = 0.086–0.225 · Controls: ρ = 0.016 · Physical/control ratio = 9.6×

Zonal wind (u) profiles from the ERA5 reanalysis at 250hPa — the upper-tropospheric jet stream level. The dataset is projected onto 12 latitude bands and 12 longitude sectors of 30° each, producing small-n ladders (n=12) that capture the large-scale circulation structure. Two integer-label controls provide internal null baselines, enabling clean signal/null separation within a single run.

FINDING 7.8.1
Atmospheric zonal wind profiles are structurally ordered at ρ = 0.086–0.225, with a 9.6× enrichment over trivially ordered integer controls

The four physical ERA5 ladders produce mean ρ from 0.086 (latband absmean) to 0.225 (lonsector absmean), all Stable Structure with Aκ ≥ 0.9999. The two integer-label controls (numeric 1–12 labels used as coordinates) give ρ = 0.016, confirming that the chamber correctly recovers near-zero pressure for trivially sorted sequences. The physical-to-control enrichment ratio of 9.6× establishes that the zonal wind ordering carries genuine structural signal — the jet stream's spatial structure is not random, but it is also far from maximally ordered.

lonsector_absmean: ρ = 0.225 · Aκ = 0.9999 · highest pressure (lon variation)
latband_absmax: ρ = 0.207 · Aκ = 1.0000 · jet core peak speed
latband_signedmean: ρ = 0.102 · Aκ = 1.0000 · direction-sensitive
latband_absmean: ρ = 0.086 · Aκ = 1.0000 · mean flow magnitude
integer label controls: ρ = 0.016 · Aκ = 1.0000 · null (uniform spacing)
FINDING 7.8.2
The latband absmean ladder (ρ = 0.086) sits just below the GOE null mean (0.087) — the zonal mean jet profile is as structurally relaxed as random matrices

The latitude-band mean |u| ladder — which captures the smooth increase of zonal wind speed from tropics through midlatitudes to the jet core — gives ρ = 0.086, essentially identical to the GOE n=100 null baseline of 0.087. This is not a coincidence of measurement: the sorted latitude-band mean profile is a smooth monotone staircase (equator → polar jet → mesosphere), leaving very few near-equal adjacent gaps for the vulnerability graph to seize upon. The atmosphere's large-scale meridional wind gradient is as structurally relaxed as any random matrix. By contrast, the lonsector ladder (ρ = 0.225) is significantly more stressed, capturing the non-monotone Rossby-wave and storm-track irregularity in the zonal direction.

GOE n=100: ρ = 0.0871 ← null baseline
latband_absmean (ERA5): ρ = 0.0858 ← within 1.5% of null
lonsector_absmean: ρ = 0.2249 ← 2.6× above null (lon inhomogeneity)
Interpretation: meridional jet structure ≈ random ordering; zonal structure > 2× more complex
FINDING 7.8.3
Signed vs unsigned wind reveals symmetry of the global circulation — direction halves structural pressure relative to magnitude

The latband signed-mean ladder (preserving easterly/westerly sign) gives ρ = 0.102, while the latband absmean (|u|) gives ρ = 0.086. The signed ladder has higher pressure because the alternating sign pattern (tropical easterlies → midlatitude westerlies → polar easterlies) creates a non-monotone ordering with more near-equal adjacent gaps. Yet the ratio of 1.19× is modest — the global circulation is broadly symmetric around zero, so signed and unsigned ladders have similar gap structures. The absmax ladder (ρ = 0.207) is highest of all latband types because peak jet speeds capture the sharpest meridional wind contrasts, creating more clustered gap pairs.

ERA5 250hPa LADDER SUMMARY ALL 6 LADDERS · PHYSICAL + CONTROL · JET STREAM LEVEL

LADDER | PROJECTION | QUANTITY | n | mean Aκ | mean ρ | max ρ | PRESSURE | STATE
--- | --- | --- | --- | --- | --- | --- | --- | ---
lonsector_absmean_u | 12 × 30° lon sectors | mean |u| per sector | 12 | 0.999863 | 0.2249 | 0.438 | TENSION | Stable
latband_absmax_u | 12 lat bands 15° | peak |u| per band | 12 | 1.000000 | 0.2066 | 0.417 | TENSION | Stable
latband_signedmean_u | 12 lat bands 15° | mean u (signed) | 12 | 1.000000 | 0.1018 | 0.360 | RELAXED | Stable
latband_absmean_u | 12 lat bands 15° | mean |u| per band | 12 | 1.000000 | 0.0858 | 0.297 | RELAXED | Stable
lonsector_labels (ctrl) | integers 1–12 | sector number | 12 | 1.000000 | 0.0163 | 0.228 | RELAXED | Control
latband_labels (ctrl) | integers 1–12 | band number | 12 | 1.000000 | 0.0160 | 0.229 | RELAXED | Control

† ρ = 0.016 for integer controls versus ρ = 0.086–0.225 for physical ladders. The null test is internal and self-contained within this single run. All physical ladders: Aκ ≥ 0.9999. The lonsector ladder reaches Aκ = 0.9974 at κ=0.44 — the only atmospheric step where admissibility fractionally dips, reflecting longitudinal jet stream asymmetry.
FINDING 7.8.4
The ρ(κ) profile is delayed-onset with no vulnerable gaps below κ ≈ 0.02–0.07 — a signature of wide, well-separated gaps in the small-n wind ladder

All four physical ladders have ρ = 0 for the first 15–20 κ steps (corresponding to κ < 0.02–0.07). The first non-zero ρ appears when the perturbation scale finally exceeds the smallest gap in the 12-element ladder. This delayed onset reflects the well-separated structure of the zonal wind profile: adjacent latitude bands (or longitude sectors) have sufficiently distinct wind speeds that only a moderate perturbation can first create a vulnerable gap pair. Once pressure begins, it rises monotonically to max ρ ≈ 0.30–0.44 at κ = 0.55–0.89. The control integer labels have ρ = 0 until κ ≈ 0.55 — the uniformly spaced 1,2,3...12 sequence requires a much larger perturbation (half the full range) before any adjacency becomes vulnerable, confirming the controls are maximally regular at small scales.

FINDING 7.8.5
Atmospheric circulation occupies the same structural pressure range as nuclear spectra and condensed-matter — placing turbulent fluid dynamics in the middle tier of the corpus hierarchy

With lonsector ρ = 0.225, the atmosphere's most complex representation sits squarely between nuclear (ρ = 0.197) and condensed-matter (ρ = 0.204) on one side, and Mars gravity (ρ = 0.277) and the cosmic web (ρ = 0.305) on the other. The latband absmean (ρ = 0.086) anchors near the GOE null and Earth gravity. The atmosphere thus spans two distinct structural tiers depending on projection: meridional flow ≈ null, zonal flow ≈ nuclear/CM. This is physically interpretable: the smooth north-south jet gradient is a consequence of the planetary rotation and differential heating (highly constrained, thermodynamic), while the east-west Rossby wave pattern reflects chaotic nonlinear wave-wave interactions (less constrained, dynamical).

§7.9 SOLAR PLASMA — THREE INDEPENDENT MAGNETIZED PLASMA PROCESSES (NEW)
SOLAR PLASMA
3 ladders · flare_flux (magnetic reconnection) · sunspot_number (solar dynamo) · F10.7 (plasma emission) · n = 30–2,000
F10.7: ρ = 0.022 · dynamo: ρ = 0.376 · reconnection: ρ = 0.393

Three observational time series from the GOES/NOAA solar archive, each probing a different physical process in the magnetized solar plasma. The solar domain produces the most extreme within-domain structural spread in the corpus: a factor of 18× between F10.7 (ρ=0.022) and flare flux (ρ=0.393), spanning from near the atmospheric integer-label null to well above the nuclear mean.

HEADLINE RESULT — THREE PLASMA PROCESSES OCCUPY THREE STRUCTURALLY DISTINCT TIERS

Solar coronal F10.7 radio flux (plasma emission, n=2000): ρ = 0.022 — below Earth gravity, below the GOE null, the second most relaxed physical measurement in the corpus. Solar dynamo (sunspot number, n=30): ρ = 0.376. Magnetic reconnection (flare flux, n=252): ρ = 0.393. Each plasma process occupies a distinct structural tier. The three span a factor of 18× in mean ρ — the largest within-domain spread across all 9 physical domains.

FINDING 7.9.1
Solar F10.7 radio flux (ρ = 0.022) is the second most structurally relaxed physical measurement in the entire corpus — below Earth gravity and below the GOE null

The F10.7 10.7 cm coronal radio flux time series, sorted as a 2,000-element ladder, gives mean ρ = 0.022 with Aκ = 1.0000 across all κ. This places it between the atmospheric integer-label controls (ρ = 0.016) and Earth's gravity field (ρ = 0.073), and well below the GOE null of 0.087. For the first 28 κ steps (κ < 0.27) ρ = 0 entirely — the coronal flux gap structure has no vulnerable pairs at small perturbation scales. This reflects the smooth 11-year solar cycle modulation: the sorted F10.7 time series is near-monotone over the cycle, producing large, well-separated gaps that resist perturbation. The structure is deeply relaxed precisely because plasma emission tracks a smooth thermodynamic envelope.

F10.7: ρ = 0 for κ < 0.27 (28 κ-steps of complete stability)
ρ finally rises via two nested ν-explosions: κ=0.388 (ν: 2→183) and κ=0.554 (ν: 183→969)
Max ρ = 0.237 at κ=1.00 · Aκ = 1.0000 throughout · Stable Structure
FINDING 7.9.2
Magnetic reconnection (flare flux) has the corpus's most sharply defined single-scale cliff — ν jumps 46× at κ ≈ 0.052, ρ collapses from 0.406 to 0.011

The solar flare flux ladder (n=252) rises monotonically from ρ = 0.115 at κ=0.01 to a peak of ρ = 0.406 at κ = 0.046 with ν locked at 1 — only a single vulnerable gap pair exists at these scales, carrying the full structural budget. Then at κ ≈ 0.052, ν explodes from 1 → 46 in a single step (a 46× jump), collapsing ρ from 0.406 to 0.011. This is the sharpest ν-transition in the corpus by ratio. The second cliff at κ ≈ 0.554 doubles ν from 47 → 92. Both cliffs reflect the bimodal structure of solar flare energy: quiet-sun quiescent gaps are orders of magnitude smaller than flare-quiescent gaps, creating two distinct characteristic scales separated by the flare energy threshold. The maximum ρ = 0.805 reached just before the second cliff (κ=0.492) is the highest ρ value ever recorded in this corpus for a Weak Persistence ladder.

κ=0.046: ν=1 · ρ=0.406 ← pre-cliff peak (single vulnerable pair)
κ=0.052: ν=46 · ρ=0.011 ← cliff: 46× ν-jump, ρ×0.027
κ=0.492: ν=47 · ρ=0.805 ← second plateau peak · Aκ=0.9995
κ=0.554: ν=92 · ρ=0.428 ← second cliff: 2× ν-jump, ρ×0.53
FINDING 7.9.3
The solar dynamo (sunspot number) shows distributed Weak Persistence — Aκ < 1.000 at 10 of 40 κ steps, scattered across the full range

The sunspot activity ladder (n=30) reaches mean ρ = 0.376 with Aκ depressed below 1.000 at κ = 0.066, 0.074, 0.134, 0.151, 0.191, 0.242, 0.273, 0.554, 0.624, and 1.000. Unlike the sharp cliff of the flare ladder, the solar dynamo ladder exhibits distributed boundary proximity — no single dominant transition, but repeated near-exceedances spread across a decade of κ values. This is consistent with the multi-scale variability of sunspot activity: the 11-year Schwabe cycle, 22-year Hale cycle, and secular grand-maximum/minimum variations all produce gap structures at different scales that independently approach the admissibility limit. The minimum Aκ = 0.998 at κ = 0.151 and κ = 0.242 is modest but reproducible.

Steps with Aκ < 1.000: κ = 0.066–0.273 (mid-range) and κ = 0.554–0.624 (upper)
Minimum Aκ = 0.998 (κ = 0.151, 0.242, 0.273)
Contrast: flare flux has only 1 step with Aκ < 1 (the second plateau, κ=0.492)
SOLAR PLASMA LADDER SUMMARY 3 PROCESSES · PHYSICAL SOURCE · ρ AND ADMISSIBILITY

LADDER | PHYSICAL PROCESS | OBSERVABLE | n | mean Aκ | mean ρ | max ρ | PRESSURE | STATE
--- | --- | --- | --- | --- | --- | --- | --- | ---
solar_flare_flux_2014 | Magnetic reconnection | GOES X-ray peak flux | 252 | 0.999987 | 0.3935 | 0.805 | TENSION | Weak Persist.
solar_magnetic_activity | Solar dynamo | International sunspot number | 30 | 0.999762 | 0.3759 | 0.604 | TENSION | Weak Persist.
solar_radio_flux_f107 | Plasma emission | 10.7 cm coronal radio flux | 2,000 | 1.000000 | 0.0218 | 0.237 | RELAXED | Stable Struct.

Within-domain spread: 18× (0.022→0.393). Flare flux max ρ = 0.805 at κ=0.492 — the highest ρ ever recorded for a Weak Persistence ladder in the corpus. F10.7 ρ = 0 for all κ < 0.27.
ρ(κ) PROFILES — SOLAR PLASMA THREE PROCESSES · CLIFF STRUCTURE HIGHLIGHTED
0.90
0.70
0.50
0.30
0.10
κ
Flare flux (reconnection) — double cliff
Magnetic activity (dynamo) — distributed
F10.7 radio flux (plasma) — delayed onset
FINDING 7.9.4
The solar domain demonstrates that structural pressure encodes physical process complexity, not just physical scale or energy

All three solar ladders probe the same object — the Sun — but at different physical processes. The F10.7 flux (coronal plasma temperature integrated over the disk) is smooth, slowly varying, and cycle-locked: ρ = 0.022. The sunspot number (magnetic flux emergence driven by the dynamo) is cyclic but irregular: ρ = 0.376. The flare flux (impulsive energy release via magnetic reconnection) is intermittent and power-law distributed: ρ = 0.393. The factor-of-18 spread across the same astronomical object confirms that ρ encodes dynamical process complexity, not source identity. The same object can simultaneously produce near-GOE-null and STRUCTURAL TENSION structural pressure depending on which magnetized plasma mechanism is being probed. This matches the pattern in condensed-matter (same material, different property ladders → very different ρ) but now at astrophysical scale.

§7.10 CMB — PLANCK 2018 ANGULAR POWER SPECTRA (NEW)
FINDING 7.10.1
All three Planck 2018 CMB power spectra satisfy the admissibility inequality — acoustic oscillations at z ≈ 1100 are structurally admissible

The TT, TE, and EE Planck 2018 angular power spectra (n ≈ 2,000 each) each achieve Aκ = 1.0000 across the full κ range. Mean structural pressures: TT 0.251, TE 0.237, EE 0.275. Domain mean ρ̄ = 0.254. All three fall in the interior physical operating band, between the GNSS/nuclear tier and the condensed-matter tier. The primordial acoustic oscillation spectrum — imprinted at recombination, z ≈ 1,100, comoving scale ~10²⁶ m — satisfies the same structural constraint as nuclear γ-transitions at 10⁻¹⁵ m.

Intra-domain ordering: EE > TT > TE. The EE (E-mode polarisation) spectrum has the highest pressure, likely reflecting the sign-mixing of the TE cross-spectrum that introduces near-degenerate adjacent multipoles in the TE ordering.

CMB domain mean ρ̄ = 0.254 · All Aκ = 1.0000 · Regime: Geometric Persistence / Stable Structure · Scale span: last-scattering surface, z ≈ 1,100, ~10²⁶ m comoving

Ladder | n | mean ρ | max ρ | min Aκ | Regime
--- | --- | --- | --- | --- | ---
cmb_tt_planck2018 | ~2000 | 0.251 | — | 1.0000 | GP / Stable
cmb_te_planck2018 | ~2000 | 0.237 | — | 1.0000 | GP / Stable
cmb_ee_planck2018 | ~2000 | 0.275 | — | 1.0000 | GP / Stable

§7.11 GEODESY — GNSS CRUSTAL DISPLACEMENT · NGL tenv3 (NEW)
FINDING 7.11.1
Five GNSS stations all fully admissible — fault proximity signal visible in structural pressure

Five Nevada Geodetic Laboratory tenv3 stations (CAC2, P579, P591, P811, P812; n = 2,000 daily displacement records each) all achieve Aκ = 1.0000. Domain mean ρ̄ = 0.133, placing crustal kinematics between the molecular tier (0.115) and the nuclear tier (0.197). The inter-station ρ range is large: P579 (0.081) to CAC2 (0.231), a factor of 2.85×. CAC2 is the most stressed station — its elevation above the others may reflect proximity to an active fault system creating denser near-degenerate displacement populations.

Two vulnerability growth regimes are visible: CAC2 exhibits gap-hierarchical scaling (α ≈ 0.74, early onset) while P579/P591 exhibit threshold-onset scaling (α ≈ 1.9–2.0, extended dead zone before rapid growth). This bimodal scaling may be a structural proxy for tectonic activity.

Domain mean ρ̄ = 0.133 · All Aκ = 1.0000 · Regime: Geometric Persistence / Stable Structure · Physical scale: ~10⁶–10⁷ m (crustal kinematics)

Station | n | mean ρ | min Aκ | Regime | Note
--- | --- | --- | --- | --- | ---
CAC2 | 2000 | 0.231 | 1.0000 | GP / Stable | Highest pressure; possible fault-proximity signal
P579 | 2000 | 0.081 | 1.0000 | GP / Stable | Threshold-onset scaling α ≈ 1.9
P591 | 2000 | 0.094 | 1.0000 | GP / Stable | Threshold-onset scaling α ≈ 2.0
P811 | 2000 | 0.121 | 1.0000 | GP / Stable | Interior band, gap-hierarchical
P812 | 2000 | 0.138 | 1.0000 | GP / Stable | Interior band, gap-hierarchical

§7.12 ADVERSARIAL PACK 1 — H₂O GAP-PRESERVING SURROGATES (NEW)
FINDING 7.12.1
Gap-preserving transformations leave admissibility intact — STRUC-I is a gap-geometry projector

Four 2,000-point adversarial ladders derived from the H₂O rovibrational spectrum were constructed to test whether the inequality depends on ordering, distribution, or gap geometry. Real ladder, shuffled ladder, and histogram-matched synthetic all achieve Aκ = 1.0000 with nearly identical mean ρ ≈ 0.051. The smooth rank surrogate (uniform spacing, gap hierarchy destroyed) achieves Aκ = 1.0000 but at lower pressure ρ = 0.018, consistent with its placement in the interior null tier.

Key result: real and shuffled ladders produce identical sorted representations (total difference = 0.000000), confirming the Gap-Spectrum Invariance Theorem — the chamber sorts before computing gaps, so permutations of values produce the same admissibility profile up to O(M⁻¹/²) Monte Carlo noise. The histogram-matched synthetic, which shares the same median gap but different local structure, deviates by <0.1% in mean ρ. The smooth surrogate, with a fundamentally different gap spectrum (all gaps equal to 21.6 vs median 14.3 for the real ladder), produces a visibly different structural pressure.

Source: H₂O molecular spectrum (HITRAN) · n = 2,000 each · Permutation Invariance (Prop. 1) proved analytically · Gap-Spectrum Invariance (Thm. 4) confirmed empirically

Ladder | n | mean ρ | max ρ | min Aκ | Tier
--- | --- | --- | --- | --- | ---
real_ladder | 2000 | 0.05103 | 0.3002 | 1.0000 | Deep-relaxation basin
shuffled_ladder | 2000 | 0.05096 | 0.3003 | 1.0000 | Deep-relaxation basin
histogram_matched | 2000 | 0.05110 | 0.3009 | 1.0000 | Deep-relaxation basin
smooth_rank_surrogate | 2000 | 0.01810 | 0.2532 | 1.0000 | Interior null tier

FINDING 7.12.2
STRUC-I is a gap-geometry projector — it discards ordering and domain semantics, retaining only the sorted gap spectrum

The Gap-Spectrum Invariance Theorem states: if two ladders share the same sorted gap spectrum, they produce identical vulnerability graphs, identical vulnerability capacities, and identical admissibility profiles up to O(M⁻¹/²). The shuffled experiment verifies this to numerical precision (|Δρ̄| < 0.0001). The chamber is therefore not measuring ordering, domain identity, or physical labels — it measures a structural invariant of gap geometry. This explains cross-domain universality: every physical domain generates a sorted gap sequence, and the chamber tests the same geometric property regardless of origin.

§7.13 ADVERSARIAL PACK 2 — CLUSTER ATTACK: FINDING THE BOUNDARY (NEW)
FINDING 7.13.1
Block-degenerate cluster ladders produce genuine violations — 34/40 κ-steps with Aκ < 1, ρ > 1 at 10–14 steps

Four 2,000-point synthetic ladders were engineered to maximize inversion pressure against a constrained vulnerability budget. Controls (microgap, uniform baseline) remain fully admissible. Cluster ladders (single-cluster: block of 200 near-degenerate levels; multi-cluster: ~370 near-degenerate levels) produce genuine violations of the admissibility inequality at the individual draw level.

Multi-cluster: ρ > 1 at 10/40 κ-steps, Aκ < 1 at 34/40 κ-steps, worst Aκ = 0.519 at κ = 0.1343 (ν = 185, ⟨inv⟩ = 185.16, ρ = 1.0009, ~48% of draws violated the inequality).

Single-cluster: ρ > 1 at 14/40 κ-steps, Aκ < 1 at 34/40 κ-steps, worst Aκ = 0.540 at κ = 0.3888 (ν = 100, ⟨inv⟩ = 100.15, ρ = 1.0015, ~46% of draws violated).

Violation condition: isolated block degeneracy requires δintra ≪ δmed ≈ δinter. No physical spectrum in the 3,313-evaluation corpus exhibits this structure.

Ladder | n | mean ρ | max ρ | min Aκ | κ-steps ρ>1 | κ-steps Aκ<1 | Regime
--- | --- | --- | --- | --- | --- | --- | ---
microgap_ladder | 2000 | 0.0177 | 0.250 | 1.0000 | 0/40 | 0/40 | GP / Stable
uniform_baseline | 2000 | 0.0177 | 0.250 | 1.0000 | 0/40 | 0/40 | GP / Stable
multi_cluster | 2000 | 0.8919 | 1.0011 | 0.519 | 10/40 | 34/40 | SB / Transitional
single_cluster | 2000 | 0.8807 | 1.0015 | 0.540 | 14/40 | 34/40 | SB / Transitional

FINDING 7.13.2
Both cluster ladders recover at the same κ* = 0.554102 — a computable vulnerability percolation threshold

At κ* = 0.554102, both cluster ladders jump from ν ∈ {100, 185} to ν = 1,000 = n/2 in a single step, ρ collapses from ≈1.001 to 0.19/0.11, and Aκ recovers to 1.0000 where it remains for all κ > κ*. This is not a numerical accident: the shared threshold is a computable geometric constant, κ* ≈ δbg / (2·δmed) ≈ 0.5, equal to the first evaluated grid point above 0.5 (= 0.554102). Although the ladders differ in cluster size (m = 200 vs ~370), they share the same global background spacing, so both vulnerability graphs percolate to the global ladder at the same κ*.

This vulnerability percolation transition is the same mechanism responsible for the ρ(κ) cliffs in the physical corpus (H atomic spectra κ* ≈ 0.55, DESI cosmic web κ* ≈ 0.307, Moon absS κ* ≈ 0.17). The cluster attack reveals the mechanism in isolation: below κ* the vulnerable subgraph is confined to the local cluster; at κ* it percolates to the global ladder and the inversion budget is absorbed. Violation is possible only while the gap hierarchy is broken by isolated block degeneracy.

Percolative Realizability Principle: admissibility holds when and only when the vulnerability graph globalizes before local inversion pressure saturates the local budget. Physical spectra satisfy this continuously; isolated block-degenerate ladders violate it in the pre-percolation window.
§7.14 BIOLOGY — QT RIBOZYME MUTATION FITNESS LANDSCAPE (NEW)
BIOLOGY · QT RIBOZYME · CATALYTIC RNA
6 fitness ladders (STRUC-I) + 3 epistasis datasets (STRUC-BIO-I) · QT45, 45 nt
STRUC-I ρ range: 0.081 – 0.819 · STRUC-BIO-I ρ: 0.032 – 0.226 · All Aκ = 1.000

This is the first biological domain in the corpus. The subject is the QT ribozyme — a 45 nt catalytic RNA — evaluated under two complementary instruments. CHAMBER STRUC-I v1.0.4 tests the USL admissibility inequality directly on fitness-ordered mutation ladders (single substitutions, double substitutions, deletion variants). CHAMBER STRUC-BIO-I v0.1 evaluates the epistasis-based compensation criterion D(G) ≤ C(G) on full double-mutant genotype graphs, where D counts single mutants less fit than wild-type and C counts double mutants with positive epistasis (E = fab − fa − fb). The two instruments operate on different mathematical objects but both probe the same underlying structural law. Both independently return STABLE results across all configurations.

FINDING 7.14.1
All six biological fitness ladders satisfy admissibility; structural pressure stratifies sharply by mutation type

Five of six ladders achieve Stable Structure (Aκ = 1.000, min_Ak = 1.000). The outlier is combined_del_single (n = 46, mean ρ = 0.554), which enters Weak Persistence — the highest-pressure non-adversarial result in the full corpus, exceeding the previous maximum of ρ = 0.424 (Si_density). The deletion ladder (n = 2,000 subsampled) reaches mean ρ = 0.819 and is classified Boundary-Stabilized, the first Boundary-Stabilized result among any physical system. Substitution-only ladders remain well within Stable Structure: complete10_double ρ = 0.081, complete10_single ρ = 0.222, combined_single ρ = 0.277, bp_mutants ρ = 0.172. The structural cost of length-altering mutations (deletions) substantially exceeds that of base substitutions.

STRUC-I BIOLOGICAL LADDER PRESSURE PROFILE 6 LADDERS · SORTED BY ρ̄
deletion
0.819
n=2000 (sub) · Boundary-Stabilized
combined_del_single
0.554
n=46 · Weak Persistence
combined_single
0.277
n=133 · Stable Structure
bp_mutants
0.172
n=69 · Stable Structure
complete10_single
0.222
n=133 · Stable Structure
complete10_double
0.081
n=2000 (sub) · Stable Structure
GOE n=100 mean (ρ = 0.087) — complete10_double matches GOE; deletion far exceeds all prior physical systems
FINDING 7.14.2
STRUC-BIO-I epistasis criterion returns STABLE for all three QT ribozyme configurations; measured landscape achieves rho = 0.032 — among the lowest structural pressure in the corpus

Three dataset configurations were evaluated under the D(G) ≤ C(G) criterion. The QT45 v2 clean export (100 Hamming-1 singles, 4,791 Hamming-2 doubles; source: measured fitness) yields D = 72, C = 2,235, ρ = 0.0322 (z = 4.22 above null). This places the measured QT45 landscape at approximately the same structural pressure as the complete10_double fitness ladder and well below GOE. A hybrid fidelity-shifted configuration covering all 44 measured positions (132 singles, 948 doubles) yields ρ = 0.139 (z = −0.68, indistinguishable from the null shuffle), with near-total positive epistasis (99.9% of doubles; mean E = +15.2) driven by the uniform fidelity shift. A restricted 12-position window (36 singles, 594 doubles) yields ρ = 0.226 (z = 10.27, significantly above null; 26.8% positive epistasis; mean E = −0.69). All three configurations satisfy the law.

FINDING 7.14.3
Deletion-type mutations break the pattern established by substitutions — structural pressure jumps by a factor of 3–10× when length-altering mutations are included

The substitution-only ladders (complete10_single, combined_single, bp_mutants) cluster tightly in mean ρ = 0.172–0.277, consistent with the nuclear domain (0.103–0.257) and well within Stable Structure. Introducing deletion variants transforms the picture: combined_del_single (which concatenates deletion and single-substitution fitness ranks) enters Weak Persistence at ρ = 0.554, and the pure deletion ladder reaches ρ = 0.819, the highest structural pressure recorded for any physical system in this corpus. At the vulnerability percolation threshold κ* = 0.554, the deletion ladder already has ρ = 0.724 — still well within STABLE (law satisfied) because Aκ = 1.000 throughout. This pattern is consistent with the Percolative Realizability Principle: deletion-induced fitness ranks are still hierarchically gap-connected even at elevated pressure, and the law holds. However, the proximity to the Boundary-Stabilized regime (ρ ≥ 0.5) suggests deletion landscapes are structurally distinct from the substitution subspace and warrant independent admissibility tracking as n increases.

§8 SYNTHESIS — KEY FINDINGS FOR THE UNNS PROGRAM
UNNS SUBSTRATE HYPOTHESIS UNDER TEST
Physical structures occupy regions near the admissibility boundary of recursive operator manifolds — as measured by structural pressure ρ = inv/ν — while structurally unconstrained (random) systems occupy the interior of admissibility space.
FINDING 8.1 — PRIMARY
ρ_physical / ρ_random ≈ 2.1 — the central ratio

The ratio of mean structural pressure in physical systems to the GOE null is approximately 2.1:1 for nuclear and condensed-matter domains. This ratio is stable and reproducible: nuclear ρ = 0.197 vs GOE ρ ≈ 0.093; condensed-matter ρ = 0.204 vs GOE ρ ≈ 0.093. This is not a marginal effect — it is a factor-of-two elevation maintained across domains with completely different physical mechanisms.

FINDING 8.2 — DOMAIN ORDERING
ρ_CM ≈ ρ_nuclear >> ρ_molecular ≈ 1.25 × ρ_GOE

The domain pressure ordering is: CM ≈ nuclear > molecular > GOE. Condensed-matter and nuclear both occupy ρ ≈ 0.20, while molecular sits at 0.115. This may reflect that molecular rovibrational ladders are dominated by the rigid-rotor harmonic approximation (highly regular), whereas CM and nuclear ladders reflect genuinely complex many-body gap structures.

FINDING 8.3 — PROPERTY SPECIFICITY
ρ is a property-specific structural fingerprint, not a universal material constant

For the same material (e.g. Ge: density ρ = 0.351 vs formation_energy ρ = 0.072; TiO2: density ρ = 0.065 vs formation_energy ρ = 0.420), different property ladders can differ in ρ by a factor of 4–6. This means ρ encodes not just which material is being probed, but which geometric structure of its property spectrum is being tested. This is a more refined signal than domain-level classification.

FINDING 8.4 — NUCLEAR INVARIANCE
Nuclear ρ is approximately constant across mass number A = 24 to 238

Despite spanning 6 orders of magnitude in nuclear binding energy per nucleon variation, 4 shell closures (N or Z = 20, 28, 50, 82), and transitions between spherical and deformed nuclei, the 15 isotopes produce ρ values tightly clustered between 0.103 and 0.257 (σ = 0.036). This near-constant structural pressure across the nuclear chart is a previously unobserved invariant and warrants dedicated follow-up with a larger isotope sample.

Exceptions: ⁵⁶Fe (ρ = 0.103, doubly-magic neighbour) and ¹⁶⁶Er (ρ = 0.257, deformed rotor) bracket the distribution.
FINDING 8.6 — COSMIC WEB · NEW
The cosmic web is the highest-pressure physical domain — and reveals a characteristic structural transition scale at κ ≈ 0.30

DESI xyz galaxy ladders produce mean ρ ≈ 0.305, exceeding all other physical domains tested. The ρ(κ) profile is uniquely biphasic: a high-pressure plateau (ρ 0.03 → 0.65) at κ < 0.27 reflecting isolated galaxy-pair gap structure, then an abrupt cliff as ν jumps from 17 to 148 at κ ≈ 0.30 — the void–filament transition scale. No other domain shows this abrupt structural bifurcation. The cosmic web therefore has the highest mean pressure and the most distinctive ρ(κ) signature in the corpus, making it the domain most definitively near the admissibility boundary.

inv(p) ≤ ν(V(p)) holds in all 3,313 runs without exception

Across four physical domains, GOE null, and small/large n from 8 to 2,000 levels, the core UNNS inequality was never violated. The chamber's preregistered falsification protocol remains intact. The 10 Weak Persistence cases are structurally informative near-boundary systems, not failures. The programme's admissibility predicate survives this corpus.

FINDING 8.7 — QM · ZEEMAN
The Zeeman effect creates a domain-universal structural pressure invariant at ρ = 0.9585

Eight atoms from H to Au, in external magnetic field, all converge to ρ = 0.9585 ± 0.00004. This is the most precise numerical invariant in the corpus — four times tighter than the nuclear ρ clustering (σ=0.036). It suggests that magnetic-field-induced splitting creates a gap structure geometry that is atom-independent at the structural pressure level. The ρ sits at 0.9585 — CRITICAL zone — with Aκ = 0.999. Zeeman ladders are the most boundary-proximate physical structures found, exactly where the UNNS hypothesis predicts.

FINDING 8.8 — QM · SPECTRA CLIFF
The hydrogen Balmer-series boundary creates a structural cliff analogous to the cosmic web void–filament transition

The full hydrogen spectrum (n=106) shows ρ rising to 0.762 then collapsing to 0.386 at κ ≈ 0.55, where ν doubles as the perturbation scale bridges the Lyman–Balmer series gap. The cosmic web DESI xyz ladder shows ρ peaking at 0.651 then collapsing at κ ≈ 0.30 as the void–filament scale is crossed. Both cliffs — and now the Moon absS cliff at κ ≈ 0.17 — share the same structural mechanism: a characteristic scale in the gap distribution where ν explodes and ρ collapses. The universality of this three-domain cliff pattern is the most unexpected structural analogy in the corpus.

FINDING 8.9 — GRAVITY · PLANETARY ORDERING
Earth's gravity field is the most ordered physical structure in the corpus; the synthetic σ_amp ladder is the first Random Structure and closest approach to falsification

Earth's EIGEN-6C4 gravity field has body mean ρ = 0.073, lower than any physical domain including molecular spectra. The synth σ_amp ladder gives ρ = 0.9995 and Aκ = 0.531 — the first Random Structure verdict and the only case where the admissibility rate is near 50%. The inequality still holds (inv ≈ 999 < ν = 1000), but by the narrowest margin in the corpus. The Moon's absS ladder delivers Transitional Structure with Aκ = 0.892 — structurally indistinguishable from the random field on this representation. The gravity domain thus provides both the most ordered and the most disordered physical structures in the corpus.

FINDING 8.10 — ATMOSPHERE · TURBULENCE
The zonal jet stream profile splits into two structural tiers: meridional flow ≈ GOE null, longitudinal flow ≈ nuclear/CM — mapping onto the constrained/turbulent dichotomy in atmospheric dynamics

ERA5 250hPa zonal wind ladders span ρ = 0.086 (latband absmean, meridional projection) to 0.225 (lonsector, zonal projection). The meridional tier sits within 1.5% of the GOE null (0.087), reflecting the thermodynamically constrained, smooth north-south jet gradient imposed by differential solar heating and planetary rotation. The zonal tier at ρ = 0.225 matches the nuclear/CM range, reflecting the turbulent, nonlinear Rossby wave and storm-track variability in the east-west direction. An embedded integer-label null (ρ = 0.016) confirms signal authenticity with a 9.6× physical enrichment. The atmosphere is the only domain in the corpus where the same physical system simultaneously produces near-null and mid-range structural pressures depending on the projection axis.

FINDING 8.11 — SOLAR PLASMA
The Sun's three magnetized plasma processes span 18× in structural pressure — process complexity, not object identity, determines ρ

F10.7 coronal radio flux (plasma emission) gives ρ = 0.022, solar dynamo (sunspot number) gives ρ = 0.376, and magnetic reconnection (flare flux) gives ρ = 0.393 — all from the same star. The 18× within-domain spread is the largest of any domain in the corpus. The flare ladder has the sharpest ν-transition (46× in one κ-step at κ ≈ 0.052) and the highest ρ ever recorded for a Weak Persistence ladder (ρ = 0.805). F10.7 is the second most relaxed physical measurement in the corpus, below Earth gravity. The solar domain thus mirrors the condensed-matter finding (same material, different properties → very different ρ) but at astrophysical scale, confirming that ρ encodes dynamical process complexity independently of the physical system's identity or scale.

FINDING 8.12 — CMB · NEW
Planck 2018 CMB power spectra place primordial acoustic oscillations in the nuclear/condensed-matter tier — 41 orders of magnitude from nuclear scale under the same law

Three Planck 2018 angular power spectra (TT, TE, EE; n ≈ 2,000 each) achieve Aκ = 1.0000 with domain mean ρ̄ = 0.254. The last-scattering surface (z ≈ 1,100; ~10²⁶ m comoving) sits in the same interior operating band as nuclear γ-spectra (10⁻¹⁵ m) — a scale span of 41 orders of magnitude. Intra-CMB ordering: EE (0.275) > TT (0.251) > TE (0.237), likely reflecting sign-mixing of the cross-correlation spectrum. No channel approaches the admissibility boundary.

FINDING 8.13 — GEODESY · NEW
GNSS crustal displacement ladders are fully admissible — fault proximity appears as an elevated structural pressure signal

Five NGL tenv3 stations (CAC2, P579, P591, P811, P812) all achieve Aκ = 1.0000 with domain mean ρ̄ = 0.133 — between the molecular and nuclear tiers. The inter-station spread is 2.85× (P579: 0.081 to CAC2: 0.231). CAC2 elevation may reflect proximity to an active fault system. Two scaling regimes: gap-hierarchical (CAC2, α ≈ 0.74) and threshold-onset (P579/P591, α ≈ 1.9–2.0). Structural pressure may be a novel geophysical proxy for tectonic loading.

FINDING 8.14 — ADVERSARIAL BOUNDARY · NEW
The cluster adversarial attack identifies the precise boundary: isolated block-degenerate ladders violate the law; all physical spectra — which are hierarchically connected — do not

Gap-preserving H₂O surrogates (shuffled, histogram-matched) all achieve Aκ = 1.0000, confirming Gap-Spectrum Invariance. Cluster ladders produce genuine violations: ρ > 1 at 10–14 κ-steps, Aκ ≈ 0.52 at worst. Both cluster ladders recover at the shared vulnerability percolation threshold κ* = 0.554102 (= δbg / 2δmed ≈ 0.5) in a single step. Physical spectra never exhibit isolated block degeneracy; they percolate continuously across scales. The Universal Structural Law is a law of hierarchically-gapped, percolatively connected gap geometries.

FINDING 8.15 — BIOLOGY · STRUC-I · NEW
QT ribozyme substitution fitness ladders are structurally equivalent to nuclear spectra; deletion-type ladders occupy a new high-pressure physical regime

Substitution fitness ladders of the 45 nt QT ribozyme (ρ = 0.081–0.277) cluster near nuclear spectra (0.103–0.257), establishing that fitness-ordered point-mutation landscapes obey the same structural law as physically measured energy spectra. The complete10_double ladder (ρ = 0.081) is the lowest-pressure molecular fitness ladder in the corpus, comparable to H₂O rovibrational spectra. Deletion-type ladders (ρ = 0.554–0.819) define a new high-pressure physical regime not previously observed — Boundary-Stabilized, with all Aκ = 1.000 maintained — suggesting that length-altering mutations generate structurally distinct admissibility geometry from base substitutions.

FINDING 8.16 — BIOLOGY · STRUC-BIO-I · NEW
The epistasis criterion D ≤ C is satisfied for all three QT ribozyme configurations; the measured fitness landscape achieves rho = 0.032, placing biological epistatic structure among the most relaxed systems in the corpus

STRUC-BIO-I evaluates a distinct formulation of the structural law: disorder D counts single mutants below wild-type fitness; compensation C counts double mutants with positive epistasis. The measured QT45 dataset (ρ = 0.032, z = 4.22 above null) places the catalytic RNA fitness landscape in the same structural pressure tier as complete10_double and near the lowest-pressure systems in the corpus. Two fidelity-modulated configurations (ρ = 0.139, 0.226) also satisfy the law. The cross-instrument consistency — STRUC-I ρ ≈ 0.081–0.277 for substitution ladders, STRUC-BIO-I ρ = 0.032–0.226 — confirms that the QT ribozyme fitness landscape is robustly admissible under both formulations of the inequality.

FINDING 8.17 — CONDENSED MATTER PHASE CHAINS · NEW
Crystallographic phase chains and polymorph progressions are fully admissible; the SiO₂ polymorph sequence sets a new non-biological ρ̄ corpus maximum at 0.499, exceeding the prior Si_density record of 0.424

The Phase 0 condensed-matter gold-set corpus (8 materials, 48 ladders, 1,920 evaluations) records zero violations and extends the admissibility map in two directions. At the low end, the degenerate Al₂O₃ single-phase control (ρ = 0) and the minimal 2-state pairs (Fe BCC/FCC and PbTiO₃, ρ̄ ≈ 0.009) establish the lower bound of the phase-chain pressure range. At the high end, SiO₂ c-axis (ρ̄ = 0.499) and cell_volume (ρ̄ = 0.476) enter Weak Persistence — the first condensed-matter Weak Persistence result from a geometrically-motivated phase progression, as opposed to a DFT property ladder. A systematic descriptor-channel anisotropy is observed in all multi-phase materials: volumetric descriptors (cell_volume, vol/atom, vol/fu) carry higher structural pressure than axial parameters (a, b, c), except in SiO₂ where the c-axis is anomalously elevated. The ferroic result confirms that within the perovskite family, ρ̄ scales with both the number of phases and the degree of lattice distortion across the chain. Global min Aκ across the entire phase-chain corpus: 0.9835 (KNbO₃ vol/fu). The USL is non-falsified across the condensed-matter domain in both its DFT property ladder and crystallographic phase-chain formulations.

§9 OPEN QUESTIONS & NEXT EXPERIMENTS
FIVE PRIMARY QUESTIONS — ALIGNED WITH THE UNIVERSAL STRUCTURAL LAW MANUSCRIPT
Q1 — PERCOLATIVE REALIZABILITY (PRIMARY)

The Percolative Realizability Principle conjectures that admissibility holds iff the vulnerability graph percolates across scales before local inversion pressure saturates the local budget. The cluster attack establishes the "only if" direction. Can the "if" direction be proved? This would unify the combinatorial exchange bound, gap-spectrum projector, and vulnerability growth asymmetry into a single structural theorem.

Q2 — VULNERABILITY SCALING EXPONENT AND (ρ̄, κ*) FINGERPRINT

ν ∝ κα with α ≈ 0.7 was characterised from GNSS and CMB. Is α ≈ 0.7 universal for gap-hierarchical ladders, or does it vary by domain? Does the two-parameter fingerprint (ρ̄, κ*) uniquely classify gap-distribution class, or do degenerate fingerprints exist? Systematic fitting across nuclear, molecular, and cosmic web ladders would test universality.

Q3 — VIOLATING LADDERS BEYOND CLUSTERS

Cluster ladders establish the boundary. Remaining tests: (i) maximally degenerate sequences (zero-gap blocks) — do these violate more severely? (ii) Poisson point process ladders (exponential gap distribution, no level repulsion). If Poisson ladders breach admissibility, it would establish a quantitative threshold distinguishing admissible from inadmissible gap distributions.

Q4 — RMT QUANTITATIVE BRIDGE

GOE level repulsion produces ρ ≈ 0.09 by suppressing near-degenerate pairs. Can the GOE n-drift (0.087 → 0.101 for n = 100 → 500) be derived analytically from the Wigner surmise gap distribution and the STRUC-I vulnerability formula? A closed-form ρ̄GOE(n) would close the link to quantum chaos universality.

Q5 — STRUCTURAL PRESSURE AS TECTONIC PROXY (GEODESY)

Station CAC2 (ρ̄ = 0.231) is 2.85× above P579 (0.081). Does ρ̄ correlate with seismic moment release rate or fault-trace distance across the full NGL network? A systematic survey of ~50 Basin and Range stations would test whether structural pressure is a novel geophysical diagnostic — the first Earth-science application of the UNNS admissibility framework.

Q6 — PHASE-TRANSITION BOUNDARY DIAGNOSTICS (CONDENSED MATTER) NEW 2026-03-22

The Phase 0 gold-set establishes static admissibility at discrete phase compositions. The natural extension: do ρ̄ profiles evolve continuously as a system is driven through a phase boundary (e.g., temperature-dependent lattice parameters approaching a Curie point in BaTiO₃ or KNbO₃)? UNNS-style prediction: ρ̄ should rise as the boundary is approached and relax again in the new phase. Testing this would transform the USL from a static structural audit into a dynamical diagnostic of phase-transition proximity — a structurally grounded analogue of order-parameter divergence near a critical point.

STRUC-I v1.0.4 Corpus Analysis Report · UNNS Substrate Program · 2026-03-14 · UPDATED 2026-03-22 — Condensed Matter Phase Chains (Phase 0) added
Instrument: CHAMBER STRUC-I v1.0.4 · Protocol: preregistered · κ ∈ [0.01, 1.0] · 40 steps logspaced · M = 2,000 MC runs · ε = κ · median(gaps)
Total evaluations: 5,233 (3,069 physical + 1,920 phase-chain + 240 biological + 4 cluster adversarial) · 14 physical domains + 1 biological · 2 adversarial packs
Phase-chain corpus (NEW 2026-03-22): BaTiO₃ · PbTiO₃ · KNbO₃ · TiO₂ · ZrO₂ · SiO₂ · Al₂O₃ · Fe · 8 materials × 6 descriptors (a,b,c,cell_vol,vol_atom,vol_fu) = 48 ladders · 1,920 evaluations · 0 violations · 5 WP (SiO₂×4, KNbO₃×1) · new non-bio max ρ̄ = 0.499 (SiO₂ c-axis).
Biological corpus: QT ribozyme (45 nt) · STRUC-I v1.0.4 (6 fitness ladders) · STRUC-BIO-I v0.1 (3 epistasis datasets).
Physical corpus: GOE, molecular (HITRAN), nuclear (NNDC), condensed-matter DFT (Materials Project), cosmic web (DESI/2MRS/SDSS), QM atomic spectra (NIST), planetary gravity (EIGEN-6C4/JGM85F01/AIUB-GRL350A), ERA5 atmosphere, solar plasma (F10.7/flares/dynamo), Planck 2018 CMB (TT/TE/EE), NGL tenv3 GNSS (CAC2/P579/P591/P811/P812).
Adversarial Pack 1: H&sub2;O-based (real, shuffled, histogram-matched, smooth surrogate) · all Aκ = 1.0000 · Gap-Spectrum Invariance confirmed.
Adversarial Pack 2 (cluster attack): microgap, uniform, single-cluster (m=200), multi-cluster (m≈370) · violations at 34/40 κ-steps for cluster ladders (ρ>1 at 10–14 steps, Aκ≈0.52) · recovery at κ*=0.554102.