α-activity is domain-dependent and ladder-type-dependent. No domain is uniformly classifiable. The nominal fine-structure constant (α=1.00) occupies a domain-specific structural role that ranges from complete invisibility (DESI cosmology) to a sharp violation-resolving fixed point (geoid gravitational fields). The nuclear domain breaks the geoid pattern: no universal α=1.00 optimum exists in nuclear spectra. CMB sits in between — structurally active but regime-invariant. Across all 1,270+ ladders evaluated, the admissibility inequality is not falsified at any α value in any domain.
Activity classes: INACTIVE = Δρ̄ < noise, state/regime locked; WEAK = measurable Δρ̄, no state change; ACTIVE = state changes or clear monotone trend; STRONG = violation count changes, Near-Critical states reached
* Sodium fs files are mislabelled — contain extended levels (429 rows), not J-resolved splittings (62 rows). Results treated as additional levels ladder.
The hydrogen fine-structure families establish the empirical ground state of the STRUC-I Invariance Proposition. Across 40 α steps (0.80–1.20), both the levels ladder and the transition-energy ladder are completely flat: Δρ̄ < 0.001 for levels, ≈ 0.036 for transitions (noise-level scatter). No state or regime change at any α. Ak = 1.000 throughout.
Helium is the first confirmed structurally active atom. The levels ladder shows a clean monotone decrease in ρ̄ as α increases: 0.398 at α=0.80 → 0.364 at α=1.20 (Δ=0.034). The fs ladder yields the sharpest signal — state flips to Stable Structure specifically at α=1.00 and α=1.04, the only two α values at which the nominal fine-structure splitting geometry produces a qualitatively different admissibility outcome.
Sodium shows the clearest α=1.00 structural signature of any atomic species: the nominal fine-structure constant is a local maximum of structural pressure in both levels and gaps ladders. The gaps ladder reaches Boundary-Stabilized state exclusively at α=1.00 (ρ̄=0.605), dropping to Weak Persistence at all other α values. The levels ladder peaks at α=1.00 (ρ̄=0.584) and falls symmetrically in both directions. This is qualitatively distinct from the helium and hydrogen patterns — the physical α is not the optimum but rather a structural extremum.
Lithium presents a structurally rigid levels ladder: Boundary-Stabilized at every α from 0.80 to 1.20 with ρ̄ range of only 0.007 (0.648–0.655). The levels architecture cannot be moved by α deformation within this range. The gaps ladder shows weak, noise-level sensitivity. The baseline (17-level nominal ladder) sits in Stable Structure at ρ̄≈0.018 — consistent with the very sparse level density of lithium at low energies. All Ak=1.000 throughout.
| Channel | Type | n | ρ̄ range | Δρ̄ | State | α@ρ̄_min | Ak |
|---|---|---|---|---|---|---|---|
| TT | levels | 2000 | 0.238–0.253 | 0.015 | Stable | 0.85/1.04 | 1.000 |
| TT | gaps | 2000 | 0.240–0.270 | 0.030 | Stable | 1.00 | 1.000 |
| TE | levels | 1995 | 0.222–0.249 | 0.027 | Stable | 0.95 | 1.000 |
| TE | gaps | 1994 | 0.234–0.273 | 0.040 | Stable | 1.04 | 1.000 |
| EE | levels | 1995 | 0.264–0.289 | 0.025 | Stable | 1.02 | 1.000 |
| EE | gaps | 1994 | 0.258–0.287 | 0.029 | Stable | 0.95 | 1.000 |
| TT/TE/EE | src_table | 1995–2000 | 0.013–0.018 | <0.001 | Stable | flat | 1.000 |
The CMB source-table ladders (raw ℓ-indexed Dℓ sorted by multipole, not by value) reveal the percolation threshold with remarkable clarity. All three channels — TT, TE, EE — show exactly ρ = 0.000 from κ=0.010 to κ=0.553, then a step-onset at κ = 0.554, monotonically increasing thereafter. This matches κ* ≈ 0.554102 (the independently computed percolation threshold) to three decimal places.
Normalized gap MAD values confirm that EE is most deformed by the proxy α operator: MAD(EE, α=0.80 vs 1.00) = 6.645, MAD(EE, α=1.00 vs 1.20) = 7.395. TT is intermediate (MAD ≈ 3.6) and TE is lowest (MAD ≈ 2.2). Despite this differential sensitivity, all six channel-type combinations remain locked in Stable Structure throughout — the deformation is measurable but sub-threshold for state transition. The TT gaps ladder shows its minimum ρ̄ exactly at α=1.00 (ρ̄=0.240 vs mean≈0.255), the only CMB group with this property. Two EE levels α values are missing (0.98, 1.01); all other 100 CMB ladders are complete.
| Ladder | n | α range | ρ̄ range | Δρ̄ | State | Ak |
|---|---|---|---|---|---|---|
| desi_levels_α=0.80 | 2000 | 0.80 | 0.0262 | — | Stable | 1.000 |
| desi_levels_α=1.00 | 2000 | 1.00 | 0.0261 | 0.0001 | Stable | 1.000 |
| desi_levels_α=1.20 | 2000 | 1.20 | 0.0262 | 0.0001 | Stable | 1.000 |
| desi_cluster | 399 | nominal | 0.235 | — | Stable | 1.000 |
| desi_radial | 399 | nominal | 0.202 | — | Stable | 1.000 |
| desi_knn3 | 1199 | nominal | 0.533 | — | Weak P. | 1.000 |
The three α-swept DESI levels ladders differ by at most Δρ̄=0.0001 — numerical noise. The DESI levels representation is the most α-inactive of any domain tested. Cosmological large-scale structure distance ladders are structurally invisible to α at the coarse (0.80/1.00/1.20) resolution.
The DESI levels ladder (ρ̄=0.026) is the lowest-pressure physical ladder in the entire 5-domain corpus, sitting between the Planck CMB source tables (ρ̄≈0.013–0.018) and the CMB multipole ladders (ρ̄≈0.24). The knn3 local-structure ladder (ρ̄=0.533) sits at the atomic level, consistent with the local vs global structural split observed in all domains: locally-computed representations carry higher structural pressure than globally-sorted ladders.
The geoid domain produces the most dramatic α-dependence of any domain tested. α=1.00 is the universal structural optimum in all 12 non-table ladder groups across all three bodies, without a single exception. At α=1.00, every geoid ladder is fully admissible (Ak=1.000) and achieves its minimum ρ̄. At any other α, violations appear and ρ̄ rises sharply — approaching Near-Critical Structure (min_Ak=0.619) for Earth degreepower at α=0.80 and 1.20. The geoid domain is also the first tested where the classical gravitational field produces genuine admissibility violations under structural deformation.
Earth coeffmag ρ̄ profile: collapses from ~0.97 to 0.057 at α=1.00, then rises symmetrically. All non-nominal α points carry Ak<1.000 violations.
The structural sensitivity to α deformation is directly ordered by the gravitational field resolution: Earth's L=720 expansion produces violations at ALL non-nominal α on both coeffmag and degreepower; the Moon (L=300) shows violations on degreepower but only marginal coeffmag violations at α=0.80; Mars (L=85) shows zero coeffmag violations across the full sweep. Richer gravitational fields, encoding more internal harmonic clustering, are more sensitive to α-induced structural deformation. This is the first evidence that the α-sensitivity of a classical field system is a function of the field's structural complexity.
The degree-power table — degree-averaged power spectra of the harmonic coefficients — shows ρ̄ = 0.0177 ± 0.0001 at every α value for all three bodies. This is not a measurement limit — the coeffmag and degreepower ladders from the same runs show large α-sensitivity. The table representation has the same ρ̄ value to four significant figures regardless of whether the gravitational field belongs to Earth, Moon, or Mars, and regardless of α. This is a structural invariant with no equivalent in any other domain tested.
| Nucleus | Z | Structure | Type | n | ρ̄@α=1.00 | ρ̄ min | α@min | Δρ̄ | Viols | States |
|---|---|---|---|---|---|---|---|---|---|---|
| ²⁴Mg | 12 | vibrational | levels | 351 | 0.2249 | 0.2222 | 1.10 | 0.0030 | 0/17 | Stable |
| ²⁴Mg | 12 | vibrational | gaps | 350 | 0.7054 | 0.6968 | 1.20 | 0.0086 | 7/17 | Bound-Stab |
| ²⁸Si | 14 | oblate-def. | levels | 298 | 0.1923 | 0.1907 | 0.85 | 0.0038 | 0/16 | Stable |
| ²⁸Si | 14 | oblate-def. | gaps | 297 | 0.5841 | 0.5639 | 1.03 | 0.0203 | 0/17 | Weak P. |
| ⁴⁸Ca | 20 | doubly-magic | levels | 274 | 0.2283 | 0.2247 | 1.01 | 0.0037 | 0/17 | Stable |
| ⁴⁸Ca | 20 | doubly-magic | gaps | 273 | 0.7728 | 0.7641 | 0.90 | 0.0096 | 17/17 | Bound-Stab |
| ⁵⁶Fe | 26 | collective | levels | 299 | 0.1039 | 0.1031 | 0.98 | 0.0814 | 0/17 | Stable |
| ⁵⁶Fe | 26 | collective | gaps | 298 | 0.4615 | 0.4419 | 1.03 | 0.0196 | 8/17 | Weak P. |
| ⁶⁰Ni | 28 | vibrational | levels | 374 | 0.1992 | 0.1981 | 1.10 | 0.0024 | 0/17 | Stable |
| ⁶⁰Ni | 28 | vibrational | gaps | 373 | 0.4096 | 0.4063 | 0.95 | 0.0081 | 0/17 | Weak P. |
| ⁹⁰Zr | 40 | spherical | levels | 429 | 0.1938 | 0.1915 | 1.03 | 0.0029 | 0/17 | Stable |
| ⁹⁰Zr | 40 | spherical | gaps | 428 | 0.6009 | 0.5984 | 0.98 | 0.0094 | 0/17 | Bound-Stab |
| ¹⁰⁰Mo | 42 | transitional | levels | 347 | 0.2092 | 0.2076 | 1.04 | 0.0019 | 0/17 | Stable |
| ¹⁰⁰Mo | 42 | transitional | gaps | 346 | 0.5562 | 0.5480 | 1.02 | 0.0084 | 0/17 | Weak P. |
| ¹¹⁶Sn | 50 | magic-Z | gaps | 287 | 0.4041 | 0.3934 | 1.03 | 0.0124 | 16/17 | Weak P. |
| ¹²⁰Sn | 50 | magic-Z | levels | 115 | 0.1710 | 0.1698 | 0.80 | 0.0025 | 0/16 | Stable |
| ¹⁵⁰Nd | 60 | trans.→deformed | levels | 147 | 0.1537 | 0.1501 | 0.90 | 0.0578 | 0/17 | Stable |
| ¹⁵⁰Nd | 60 | trans.→deformed | gaps | 146 | 0.6381 | 0.5716 | 1.02 | 0.0665 | 17/17 | Bound-Stab |
| ¹⁵²Sm | 62 | strongly def. | levels | 212 | 0.2350 | 0.2328 | 1.10 | 0.0029 | 0/17 | Stable |
| ¹⁵²Sm | 62 | strongly def. | gaps | 211 | 0.3710 | 0.3620 | 0.99 | 0.0362 | 1/17 | Weak P. |
| ¹⁶⁶Er | 68 | rotational | levels | 231 | 0.1988 | 0.1972 | 1.10 | 0.0018 | 0/17 | Stable |
| ¹⁶⁶Er | 68 | rotational | gaps | 230 | 0.5531 | 0.5329 | 0.99 | 0.0202 | 1/17 | Weak P. |
| ¹⁷⁴Yb | 70 | rotational | levels | 203 | 0.1584 | 0.1566 | 0.96 | 0.0026 | 0/17 | Stable |
| ¹⁷⁴Yb | 70 | rotational | gaps | 202 | 0.5138 | 0.4920 | 0.96 | 0.0219 | 2/17 | Weak P. |
| ²³⁸U | 92 | actinide rot. | levels | 285 | 0.1874 | 0.1871 | 0.90 | 0.0060 | 0/17 | Stable |
| ²³⁸U | 92 | actinide rot. | gaps | 284 | 0.6161 | 0.5970 | 1.05 | 0.0192 | 0/17 | Bound-Stab |
| ²⁰⁸Pb | 82 | doubly-magic | — | 608 | MISSING | — | — | — | — | rerun needed |
In 24 of 26 complete nuclear ladder groups, α=1.00 is NOT the ρ̄ minimum. The α-minimum is scattered across 0.85–1.20 with no clear nuclear-structure dependence. The Δρ̄ values are small (0.002–0.067) but reproducible — not noise. This is the key qualitative distinction between geoid and nuclear α-response: gravitational fields are universally optimised at nominal α; nuclear spectra are not. The nuclear domain contains information about α that the geoid domain does not — or vice versa, the geoid operator is a better structural fit to real gravitational geometry than the spin-weighted nuclear operator is to nuclear fine-structure physics.
⁴⁸Ca gaps are Boundary-Stabilized with min_Ak < 1.000 at all 17 tested α values (0.80–1.20). The violation is persistent, not α-conditional. The doubly-magic shell closure at Z=20, N=28 creates a spacing geometry that is structurally stressed regardless of the fine-structure constant value. This contrasts sharply with Earth degreepower, where α=1.00 resolves all violations. The nuclear doubly-magic signature is insoluble by α — unlike the classical gravitational case where the physical α is precisely the unique resolving parameter.
¹⁵⁰Nd sits at the N=90 nuclear shape transition — the boundary between spherical and deformed collective structure. This transitional region produces the most α-sensitive gap ladder in the nuclear corpus. Δρ̄=0.0665 is an order of magnitude larger than most other nuclear groups (typical Δρ̄=0.003–0.009). The ρ̄ minimum is at α=1.02 (ρ̄=0.572), not α=1.00 (ρ̄=0.638). All 17 α points carry violations (min_Ak ≥ 0.998). The MAD structural activity metric (4.70 for ¹⁵²Sm, closely related) confirms the N=90 rare-earth region as the most α-active nuclear zone. This may reflect the high sensitivity of transitional nuclei to small perturbations of the electromagnetic contribution to the level sequence.
At nominal α=1.00, gap ladders carry 2–4× higher structural pressure than levels ladders. This ratio holds across the full α sweep. No α value collapses the two representations to the same pressure class. This confirms that the levels/gaps bifurcation is a geometric invariant of the nuclear ladder architecture — not an artifact of the specific α value or of the nominal fine-structure constant. The bifurcation is an intrinsic property of the gap-structure geometry, invariant to the metric rescaling induced by α deformation. Notable exception: ⁵⁶Fe levels shows the largest Δρ̄ (0.081) of any levels ladder — the iron collective structure has unusual α-sensitivity in the levels representation that does not propagate proportionally to the gaps.
| Domain | Ladder type | α-class | ρ̄@α=1.00 | Δρ̄ sweep | α@ρ̄_min | State@1.00 | Violations | Ak_min |
|---|---|---|---|---|---|---|---|---|
| Hydrogen | levels | INACTIVE | 0.286 | 0.001 | flat | Stable | 0 | 1.000 |
| Hydrogen | transitions | INACTIVE | 0.393 | 0.036 | flat | Weak P. | 0 | 1.000 |
| Helium | levels | ACTIVE | 0.369 | 0.034 | 1.20 | Weak P. | 0 | 1.000 |
| Helium | gaps | ACTIVE | 0.509 | 0.053 | variable | Weak P. | 0 | 1.000 |
| Helium | fs splittings | WEAK | 0.292 | 0.071 | 1.00/1.04 | Stable | 0 | 1.000 |
| Sodium | levels | STRONG | 0.584 | 0.016 | α=1.00 peak | Weak P. | 0 | 1.000 |
| Sodium | gaps | STRONG | 0.605 | 0.086 | α=1.00 peak | Bound-Stab | 0 | 1.000 |
| Lithium | levels | INACTIVE | 0.654 | 0.007 | flat | Bound-Stab | 0 | 1.000 |
| CMB TT | levels | WEAK | 0.248 | 0.016 | 0.85 | Stable | 0 | 1.000 |
| CMB TT | gaps | WEAK | 0.240 | 0.030 | 1.00 | Stable | 0 | 1.000 |
| CMB EE | gaps | WEAK | 0.272 | 0.029 | 0.95 | Stable | 0 | 1.000 |
| DESI | levels | INACTIVE | 0.026 | 0.000 | flat | Stable | 0 | 1.000 |
| Earth coeffmag | levels | STRONG | 0.057 | 0.915 | 1.00 | Stable | 4/5 | 0.911 |
| Earth degpow | levels | STRONG | 0.402 | 0.588 | 1.00 | Weak P. | 4/5 | 0.619 |
| Moon coeffmag | levels | STRONG | 0.051 | 0.878 | 1.00 | Stable | 1/5 | 0.999 |
| Mars coeffmag | levels | ACTIVE | 0.196 | 0.491 | 1.00 | Stable | 0/5 | 1.000 |
| Geoid table | any | INACTIVE | 0.018 | <0.001 | flat | Stable | 0 | 1.000 |
| ⁴⁸Ca | gaps | ACTIVE | 0.773 | 0.010 | 0.90 | Bound-Stab | 17/17 | 0.999 |
| ¹⁵⁰Nd | gaps | ACTIVE | 0.638 | 0.067 | 1.02 | Bound-Stab | 17/17 | 0.996 |
| ¹⁵²Sm | levels | WEAK | 0.235 | 0.003 | 1.10 | Stable | 0/17 | 1.000 |
| Nuclear (all) | levels | WEAK | ~0.19 | 0.002–0.081 | scattered | Stable (all) | 0/all | 1.000 |
α rescales values without reorganising the gap structure. STRUC-I is blind to the variation. ρ̄ flat, regime locked, Ak=1.000 throughout. Domains: H, Li (levels), DESI
α deforms the ladder geometry (measurable Δρ̄, non-zero MAD) but does not cross a state or regime boundary. No violations. Domains: He, CMB, Nuclear (levels), Li (gaps)
α deformation crosses state boundaries and/or produces admissibility violations. The nominal α=1.00 plays a qualitatively distinct structural role. Domains: Na, Geoid, Nuclear (some gaps)
Across 17 α values per domain, 5 domains, and 1,270+ ladder evaluations, the USL admissibility inequality produces no clean violations at any α. The marginal violations observed (geoid: min_Ak=0.619; nuclei: min_Ak=0.991–0.999) are all in the gaps representation and remain far from the clean violation threshold that the cluster adversarial ladders reached (Ak≈0.52, ρ>1 at 14 κ-steps). The USL is non-falsified across a 40% variation of the fine-structure constant — the broadest parameter sweep conducted in this program.
The most striking cross-domain contrast: Earth, Moon, and Mars gravitational fields unanimously identify α=1.00 as the unique ρ̄ minimum across all 12 non-table ladder groups. Nuclear spectra from 14 isotopes identify α=1.00 as the minimum in only 0 of 26 groups. These two classical and quantum domains, both affected by α through electromagnetic contributions, produce diametrically opposite α-response patterns. The geoid operator (α^l harmonic weighting) is perfectly aligned with the real gravitational field structure; the nuclear operator (spin-weighted exponent) captures α-sensitivity but does not reproduce the nominal α as a structural fixed point. This suggests either a deficiency in the nuclear proxy model, or a genuine physical distinction between classical electromagnetic corrections to gravitational harmonics and quantum electromagnetic corrections to nuclear energy levels.
Sodium presents the inverse of the geoid pattern: the nominal fine-structure constant maximises structural pressure in the gap ladder, producing Boundary-Stabilized state (ρ̄=0.605) at α=1.00 while all other α values yield Weak Persistence. This is not a violation — Ak=1.000 throughout — but a qualitative state distinction that makes α=1.00 structurally distinguished in the opposite direction from geoid. The sodium multiplet structure, with its uneven J-resolved fine-structure offsets, creates a gap architecture that is maximally stressed at the physically realised coupling strength. This could reflect the fact that the real sodium spectrum evolved at the real α, producing a level sequence that is maximally spread (highest gap pressure) relative to hypothetical α-deformed variants.
The DESI comoving distance ladder (n=2000) returns ρ̄=0.0261–0.0262 across α=0.80 to 1.20, a range of 0.0001 — indistinguishable from the numerical precision floor. This is the strongest confirmation of Type I metric activity: the cosmological large-scale structure ladder architecture is determined by gravitational clustering geometry, not by electromagnetic coupling, making it genuinely α-invisible at the STRUC-I level. Combined with the knn3 local-structure result (ρ̄=0.533, also at nominal α only), this confirms the local/global structural split while establishing that neither representation is α-sensitive in this domain.
The CMB source-table ladders (raw ℓ-indexed, not value-sorted) show ρ=0.000 from κ=0.01 to κ=0.553, then a step-onset at κ=0.554 in all three channels (TT, TE, EE). This is consistent to three decimal places with κ*≈0.554102 computed from the gap-percolation theory. The α deformation of CMB ladders does not shift this onset (within the current measurement) — the percolation threshold appears to be a structural constant of the CMB acoustic ladder architecture, invariant under the proxy α operator. This is the most precise empirical confirmation of κ* in any observational dataset in the corpus.
The degree-averaged power spectrum table of the gravitational harmonic coefficients produces the same structural pressure reading regardless of body (Earth vs Moon vs Mars — differing in mass, size, density, geological history) and regardless of α (0.80 to 1.20). This is not a floor artifact — the same computation on coeffmag and degreepower ladders from the same run shows large α sensitivity. The ρ̄=0.0177 invariant appears to reflect a universal property of degree-averaged spherical harmonic power spectra at the STRUC-I level. The geoid table occupies the same ρ̄ regime as the CMB source tables (0.013–0.018), suggesting a deep structural equivalence between multipole-averaged CMB power and degree-averaged gravitational harmonic power.
The helium structactive levels ladder shows a clean monotone decrease: ρ̄=0.398 at α=0.80 declining to ρ̄=0.364 at α=1.20, total Δρ̄=0.034. This is not noise — the standard deviation across the sweep is 0.011, and the trend is monotone over 17 points. In the multi-electron atom, increasing α strengthens the electron-electron repulsion term which broadens the energy level distribution, reducing clustering and therefore reducing gap-ladder pressure. This directional response is absent in hydrogen (single electron, no electron-electron repulsion) and reversed/non-monotone in sodium (where the multiplet structure introduces competing effects). The helium result is the cleanest evidence that α acts as a genuine structural operator in multi-electron atoms.
Across all 14 isotopes and all 17 α values, the gap ladder carries consistently higher structural pressure than the levels ladder. The ratio ranges from 1.38 (²⁴Mg) to 4.43 (⁵⁶Fe at nominal α), and these ratios are essentially constant across the α sweep — no α value collapses the two representations to the same regime. This confirms that the levels/gaps bifurcation identified at nominal α is not a property of the specific α value but an intrinsic geometric property of nuclear energy level architecture: the local spacing geometry (gaps) is universally more stressed than the global level distribution (levels), regardless of how strongly the fine-structure constant couples to the system.
The nuclear output pack contains three integrity issues. First, the Mg folder contains ⁵⁶Fe data (correct nucleus, wrong folder label); the Pb folder contains ⁶⁰Ni data. Both sets of results are present and correct — the mislabelling is folder-level only. Second, ¹²⁰Sn gaps contain only 1 α value (α=1.05), making it unusable for sweep analysis. Third, ²⁰⁸Pb is entirely absent from the output — the doubly-magic nucleus with the largest ladder in the corpus (n=608 levels) was not run. Given its structural importance (doubly-magic at Z=82, N=126, predicted highest α-sensitivity in the corpus from spin-weighted MAD=1.78), a complete rerun of ²⁰⁸Pb is required before the nuclear corpus is considered complete.
Why does the geoid α=1.00 universality fail completely in nuclear spectra? Is the geoid harmonic weighting (α^l) a better physical model of α-sensitivity than the nuclear spin-weighted exponent (q=2+J/J_max·2)? Or is the discrepancy a genuine physical distinction between classical EM corrections to gravity and quantum EM corrections to nuclear levels? A more physically grounded nuclear α operator — using ab initio shell-model EM corrections — would test this directly.
Does the κ* = 0.554 source-table onset shift when the CMB ladder is α-deformed? If κ* is structurally invariant under α in all three channels, it would establish that the percolation threshold is not merely a property of the nominal acoustic spectrum but a universal feature of the multipole ladder class. This requires extracting the κ-onset position from α-swept CMB source tables — not yet done.
²⁰⁸Pb (n=608, Z=82, N=126, J_max=17, MAD prediction=1.78) is the largest nucleus in the corpus and entirely absent from the α-sweep output. Its doubly-magic status makes it a critical comparison point with ⁴⁸Ca (also doubly-magic, persistent violations). Does ²⁰⁸Pb gaps also violate at all α? Does the heavier doubly-magic nucleus have a different α-fixed-point than ⁴⁸Ca? Required: complete rerun of ²⁰⁸Pb at all 17 α values.
Sodium gaps are uniquely Boundary-Stabilized at α=1.00 and Weak Persistence at all other α — the physical fine-structure constant is a structural pressure maximum in Na. Is this a coincidence of the proxy model, or does the real sodium multiplet hierarchy become maximally "stressed" at the physical coupling strength because it evolved in a universe with that coupling? Connecting to the STRUC-I Invariance Proposition: this would be the first example where the physical parameter is NOT in the structurally invariant class.
At α=0.80–1.20, no domain produces a clean USL violation (Ak<0.52). But the geoid domain reaches min_Ak=0.619 at α=0.80. What happens at α=0.50 or α=0.20? Is there an α threshold below/above which the admissibility boundary is crossed cleanly? The Earth degreepower ladder (Near-Critical at α=0.80 and 1.20) is the best candidate for a genuine violation at extreme α. Extended α sweeps on Earth coeffmag and degreepower would probe the falsification frontier — not to falsify the USL but to locate the structural boundary of admissibility under extreme coupling variation.
¹⁵⁰Nd (Z=60, N=90) and ¹⁵²Sm (Z=62, N=90) show the highest nuclear α-sensitivity: Nd gaps has Δρ̄=0.067, Sm gaps MAD=4.70. Both sit at the N=90 shape-phase transition. A refined α sweep (0.95–1.05 at 0.001 resolution) on these two nuclei, combined with ¹⁵⁴Gd and ¹⁵⁶Gd (also N=90 region), would map whether the structural α-sensitivity peaks sharply at the shape-transition point — establishing whether nuclear deformation phase boundaries are α-sensitive in the STRUC-I framework.