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).

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.