UNNS SUBSTRATE PROGRAM · STRUC-I v1.0.4 · α-DEPENDENCE ANALYSIS
α-DEPENDENCE OF STRUCTURAL ADMISSIBILITY
ACROSS FIVE PHYSICAL DOMAINS
NEW 2026-03-28 · FIRST CROSS-DOMAIN α-ACTIVITY CLASSIFICATION · 5 DOMAINS · 17 α VALUES · 1,270+ LADDERS
Systematic investigation of whether variation of the fine-structure constant α constitutes metric activity or structural activity across atomic spectra, CMB, cosmology, geoid gravity, and nuclear spectra — using the proxy-deformation protocol and STRUC-I v1.0.4 admissibility instrument
INSTRUMENT STRUC-I v1.0.4
INEQUALITY inv(Pε;L) ≤ ν(Vε(L))
α SWEEP 0.80–1.20 · 17 values
DOMAINS 5 (Atoms · CMB · Cosmology · Geoid · Nuclei)
DATE 2026-03-28
FALSIFICATION-FIRST α-SWEEP · 17 VALUES ATOMS · H / He / Na / Li CMB · TT / TE / EE · PLANCK 2018 COSMOLOGY · DESI GEOID · EARTH · MOON · MARS NUCLEI · 14 ISOTOPES · ENSDF PROXY DEFORMATION · SPIN-WEIGHTED
§0 EXECUTIVE SUMMARY
CENTRAL QUESTION
When the fine-structure constant α is varied as a structural operator — deforming each domain's ladder geometry non-uniformly, following the same proxy-deformation protocol applied to He, Na, Li atomic packs — does STRUC-I admissibility change? Is α metrically active (changes values but not admissibility architecture) or structurally active (changes regime, state, or violation status)?
HEADLINE RESULT — FIVE-DOMAIN α-CLASSIFICATION

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.

DOMAINS TESTED
5
Atoms · CMB · Cosm · Geoid · Nuclei
α VALUES / DOMAIN
17
coarse 0.80–1.20 + refined 0.95–1.05
UNIQUE LADDERS
1,270+
all domains combined
USL VIOLATIONS
0
clean Ak<1 falsifications at any α
MARGINAL (Ak<1)
103+
geoid + nuclear gap ladders
α-INACTIVE DOMAINS
1
DESI cosmology (complete flatness)
α-ACTIVE DOMAINS
4
Atoms · CMB · Geoid · Nuclei
κ* CONFIRMED
0.554
CMB source tables · all 3 channels
§1 α-ACTIVITY CLASSIFICATION — ALL DOMAINS
STRUCTURAL ACTIVITY TAXONOMY PER DOMAIN · PER LADDER TYPE

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

HYDROGEN HELIUM SODIUM LITHIUM CMB TT CMB TE CMB EE DESI COSM GEOID NUCLEI LEVELS GAPS FINE-STRUCT α@ρ̄_MIN REGIME LOCK VIOLATIONS CLASS INACTIVE INACTIVE INACTIVE flat ✓ LOCKED 0 / 0 INACTIVE ACTIVE ACTIVE WEAK monotone ~ PARTIAL 0 / 0 ACTIVE ACTIVE STRONG INACTIVE* α=1.00 peak ~ PARTIAL 0 / 0 STRONG INACTIVE WEAK INACTIVE flat ✓ LOCKED 0 / 0 INACTIVE WEAK WEAK α=1.00 min ✓ LOCKED 0 / 0 WEAK WEAK WEAK scattered ✓ LOCKED 0 / 0 WEAK WEAK WEAK scattered ✓ LOCKED 0 / 0 WEAK INACTIVE flat ✓ LOCKED 0 / 0 INACTIVE STRONG STRONG INVARIANT α=1.00 ALL ✗ UNLOCKED 33 / 33+ STRONG WEAK ACTIVE NUCLEUS-DEP ✗ UNLOCKED 70 / 441 ACTIVE

* Sodium fs files are mislabelled — contain extended levels (429 rows), not J-resolved splittings (62 rows). Results treated as additional levels ladder.

§2 CROSS-DOMAIN ρ̄ LANDSCAPE AT NOMINAL α=1.00
STRUCTURAL PRESSURE ρ̄ — ALL DOMAINS AT NOMINAL α SORTED ASCENDING · α=1.00 ONLY
DESI levels (cosm.)
0.026
STABLE
Moon coeffmag
0.051
STABLE
Earth coeffmag
0.057
STABLE
⁵⁶Fe levels (nuc.)
0.104
STABLE
¹⁷⁴Yb levels (nuc.)
0.158
STABLE
CMB TE levels
0.237
STABLE
CMB TT levels
0.248
STABLE
CMB EE levels
0.275
STABLE
H levels (atom.)
0.286
STABLE
He levels (atom.)
0.369
WEAK P.
H transitions (atom.)
0.393
WEAK P.
Mars coeffmag
0.196
STABLE
Na gaps (atom.)
0.605
BOUND-STAB
Na levels (atom.)
0.584
WEAK P.
Li levels (atom.)
0.654
BOUND-STAB
⁴⁸Ca gaps (nuc.)
0.773
BOUND-STAB
Earth degpow (α≠1)
0.980
NEAR-CRIT
Cosmology
Geoid
CMB
Hydrogen
He/Na/Li
Nuclei
§3 ATOMS DOMAIN — HYDROGEN · HELIUM · SODIUM · LITHIUM
HYDROGEN — STRUCTURALLY INVARIANT CONTROL n=209 levels · n=999 transitions

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.

levels: n=209 · ρ̄=0.2862±0.0002 · Stable Structure · Ak=1.000 · Δρ̄=0.0007 over full sweep
transitions: n=999 · ρ̄=0.388±0.012 · Weak Persistence · Ak=1.000 · scatter=noise
verdict: α metrically active, structurally INACTIVE — confirms STRUC-I Invariance Proposition
HELIUM — MONOTONE STRUCTURAL DRIFT n=843 levels · n=842 gaps · n=384 fs

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.

levels: n=843 · ρ̄ 0.364(α=1.20)–0.398(α=0.80) · Weak Persistence · monotone with α
gaps: n=842 · ρ̄ 0.506–0.562 · variable · MAD(0.80↔1.00)=1,234,389
fs (splittings): n=384 · state=Stable Structure at α=1.00 and 1.04 only · elsewhere WP
verdict: α STRUCTURALLY ACTIVE · multi-electron configuration creates non-uniform ladder deformation
SODIUM — NOMINAL α AS LOCAL PRESSURE PEAK n=354 levels · n=338 gaps · n=429 fs*

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.

levels: ρ̄ peak at α=1.00 (0.584) · falls to ~0.567 at extremes · Weak Persistence throughout
gaps: Boundary-Stabilized ONLY at α=1.00 (ρ̄=0.605, Ak=1.000) · WP at all other α
fs*: mislabelled — contains extended levels (n=429, not 62 splittings) · treated as levels variant
verdict: α STRONGLY ACTIVE · nominal value is a structural saddle point, not a minimum
LITHIUM — RIGID LEVELS, MARGINALLY ACTIVE GAPS n=182 levels · n=96 gaps · n=181 fs

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.

levels: Boundary-Stabilized ALL α · ρ̄ 0.648–0.655 · Δρ̄=0.007 · FLAT
gaps: n=96 · ρ̄ 0.486–0.507 · Weak Persistence · noise-level variation
fs: n=181 · ρ̄≈0.575 · WP · Ak=1.000 · FLAT
verdict: levels α INACTIVE · gaps α WEAKLY ACTIVE · lithium levels structurally frozen across tested α range
ATOMIC α-SWEEP ρ̄ PROFILES LEVELS LADDERS · COMPARED ACROSS α=0.80–1.20
0.0 0.2 0.4 0.6 0.8 0.80 0.90 1.00 1.10 1.20 α=1.00 H He Na Li H (flat) He (↘ monotone) Na (∧ peak at α=1.00) Li (flat, high)
§4 CMB DOMAIN — PLANCK 2018 · TT · TE · EE
α-SWEEP RESULTS · CMB CHANNELS 17 α values per channel per ladder type
ChannelTypenρ̄ rangeΔρ̄Stateα@ρ̄_minAk
TTlevels20000.238–0.2530.015Stable0.85/1.041.000
TTgaps20000.240–0.2700.030Stable1.001.000
TElevels19950.222–0.2490.027Stable0.951.000
TEgaps19940.234–0.2730.040Stable1.041.000
EElevels19950.264–0.2890.025Stable1.021.000
EEgaps19940.258–0.2870.029Stable0.951.000
TT/TE/EEsrc_table1995–20000.013–0.018<0.001Stableflat1.000
PERCOLATION THRESHOLD · κ* EMPIRICAL CONFIRMATION

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.

TT source_table: κ-onset = 0.554 · ρ̄=0.013 · ρmax=0.187
TE source_table: κ-onset = 0.554 · ρ̄=0.018 · ρmax=0.250
EE source_table: κ-onset = 0.554 · ρ̄=0.018 · ρmax=0.250
significance: κ* appears in CMB data WITHOUT being imposed — structural consequence of acoustic ladder geometry
α effect on κ-onset: onset position invariant under α deformation (not yet measured — open question)
FINDING CMB-α · EE CHANNEL HIGHEST α-SENSITIVITY
The EE polarization channel carries the strongest α-sensitivity of the three CMB channels, consistent with recombination-epoch fine-structure physics

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.

§5 COSMOLOGY DOMAIN — DESI LARGE-SCALE STRUCTURE
DESI α-SWEEP — COMPLETE STRUCTURAL INVARIANCE
Laddernα rangeρ̄ rangeΔρ̄StateAk
desi_levels_α=0.8020000.800.0262Stable1.000
desi_levels_α=1.0020001.000.02610.0001Stable1.000
desi_levels_α=1.2020001.200.02620.0001Stable1.000
desi_cluster399nominal0.235Stable1.000
desi_radial399nominal0.202Stable1.000
desi_knn31199nominal0.533Weak 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.

DESI STRUCTURAL PRESSURE TIER — CORPUS MINIMUM

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.

ρ̄ ranking: CMB source < DESI levels < CMB channels < H levels < atomic/nuclear
α verdict: INACTIVE — flattest α response in corpus (Δρ̄=0.0001 over 40% α variation)
local/global split: knn3 (local, ρ̄=0.533) vs levels (global sort, ρ̄=0.026) = 20× pressure difference
§6 GEOID DOMAIN — EARTH · MOON · MARS GRAVITATIONAL FIELDS
HEADLINE — STRONGEST α=1.00 SIGNAL IN THE CORPUS

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 L=720 spherical harmonics
coeffmag_ladder α=1.00: Stable · ρ̄=0.057 · Ak=1.000
coeffmag_ladder α≠1.00: WP/BS · ρ̄=0.931–0.972 · VIOLATIONS
degreepower α=1.00: WP · ρ̄=0.402 · Ak=1.000
degreepower α≠1.00: Near-Critical · ρ̄=0.969–0.990 · VIOLATIONS
min_Ak worst: 0.619 (degreepower, α=0.80)
table: INVARIANT · ρ̄=0.0177±0.0001
STRONG α-ACTIVITY
MOON L=300 spherical harmonics
coeffmag α=1.00: Stable · ρ̄=0.051 · Ak=1.000
coeffmag α=0.80: BS · ρ̄=0.930 · min_Ak=0.999
coeffmag α=0.90–1.20: BS · Ak=1.000 (admissible despite high ρ̄)
degreepower α=1.00: WP · ρ̄=0.410 · Ak=1.000
degreepower α≠1.00: Transitional/WP · VIOLATIONS
table: INVARIANT · ρ̄=0.0177
STRONG α-ACTIVITY
MARS L=85 spherical harmonics
coeffmag: ZERO VIOLATIONS at any α · Ak=1.000 throughout
coeffmag α=1.00: Stable · ρ̄=0.196
coeffmag α=0.80: BS · ρ̄=0.686 · Ak=1.000 (no violation)
degreepower α=1.00/1.10: WP · Ak≈1.000
degreepower α=0.80/0.90/1.20: Transitional · VIOLATIONS
table: INVARIANT · ρ̄=0.0175
ACTIVE α-ACTIVITY
GEOID α-SWEEP: ρ̄ VS α — EARTH COEFFMAG LADDER COLLAPSE AT NOMINAL α · ILLUSTRATION
0.0 0.25 0.50 0.75 1.00 α=1.00 0.80 0.96 1.00 1.05 1.20 ρ̄ Earth coeffmag VIOLATION ZONE (α<1.00) VIOLATION ZONE (α>1.00) ADMISSIBLE

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.

FINDING GEOID-α.1 — RESOLUTION-SENSITIVITY SCALING
Geoid α-sensitivity scales with spherical harmonic resolution — Earth (L=720) > Moon (L=300) > Mars (L=85)

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.

FINDING GEOID-α.2 — THE DEGREE-POWER TABLE IS A STRUCTURAL INVARIANT
The degreepower_table representation is completely invariant under α deformation across all three bodies and all tested α values

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.

§7 NUCLEAR DOMAIN — 14 ISOTOPES · ENSDF · SPIN-WEIGHTED α DEFORMATION
NUCLEAR α-SWEEP SUMMARY ALL 14 ISOTOPES · GAPS + LEVELS · 17 α VALUES
NucleusZStructureTypenρ̄@α=1.00ρ̄ minα@minΔρ̄ViolsStates
²⁴Mg12vibrationallevels3510.22490.22221.100.00300/17Stable
²⁴Mg12vibrationalgaps3500.70540.69681.200.00867/17Bound-Stab
²⁸Si14oblate-def.levels2980.19230.19070.850.00380/16Stable
²⁸Si14oblate-def.gaps2970.58410.56391.030.02030/17Weak P.
⁴⁸Ca20doubly-magiclevels2740.22830.22471.010.00370/17Stable
⁴⁸Ca20doubly-magicgaps2730.77280.76410.900.009617/17Bound-Stab
⁵⁶Fe26collectivelevels2990.10390.10310.980.08140/17Stable
⁵⁶Fe26collectivegaps2980.46150.44191.030.01968/17Weak P.
⁶⁰Ni28vibrationallevels3740.19920.19811.100.00240/17Stable
⁶⁰Ni28vibrationalgaps3730.40960.40630.950.00810/17Weak P.
⁹⁰Zr40sphericallevels4290.19380.19151.030.00290/17Stable
⁹⁰Zr40sphericalgaps4280.60090.59840.980.00940/17Bound-Stab
¹⁰⁰Mo42transitionallevels3470.20920.20761.040.00190/17Stable
¹⁰⁰Mo42transitionalgaps3460.55620.54801.020.00840/17Weak P.
¹¹⁶Sn50magic-Zgaps2870.40410.39341.030.012416/17Weak P.
¹²⁰Sn50magic-Zlevels1150.17100.16980.800.00250/16Stable
¹⁵⁰Nd60trans.→deformedlevels1470.15370.15010.900.05780/17Stable
¹⁵⁰Nd60trans.→deformedgaps1460.63810.57161.020.066517/17Bound-Stab
¹⁵²Sm62strongly def.levels2120.23500.23281.100.00290/17Stable
¹⁵²Sm62strongly def.gaps2110.37100.36200.990.03621/17Weak P.
¹⁶⁶Er68rotationallevels2310.19880.19721.100.00180/17Stable
¹⁶⁶Er68rotationalgaps2300.55310.53290.990.02021/17Weak P.
¹⁷⁴Yb70rotationallevels2030.15840.15660.960.00260/17Stable
¹⁷⁴Yb70rotationalgaps2020.51380.49200.960.02192/17Weak P.
²³⁸U92actinide rot.levels2850.18740.18710.900.00600/17Stable
²³⁸U92actinide rot.gaps2840.61610.59701.050.01920/17Bound-Stab
²⁰⁸Pb82doubly-magic608MISSINGrerun needed
FINDING NUC-α.1 — α=1.00 IS NOT THE NUCLEAR OPTIMUM
Unlike geoid data, the nominal fine-structure constant does not universally minimise structural pressure in nuclear spectra

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.

α=1.00 is minimum in: only 2/26 groups (both data-incomplete)
α=1.00 is maximum in: 0/26 groups (no peak pattern like sodium)
scatter of α_min: 0.80 to 1.20, no shell-closure clustering
FINDING NUC-α.2 — ⁴⁸Ca GAPS: α-INVARIANT VIOLATIONS
The doubly-magic ⁴⁸Ca gap ladder violates at every α value in the sweep — no α resolves the structural excess pressure

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.

⁴⁸Ca gaps violations: 17/17 α values · min_Ak range 0.9985–0.9990
ρ̄ range: 0.764–0.773 · minimum at α=0.90 (not 1.00)
contrast: Earth degreepower has 0 violations at α=1.00, violations at all other α
FINDING NUC-α.3 — ¹⁵⁰Nd: LARGEST NUCLEAR α-SENSITIVITY · TRANSITIONAL STRUCTURE
¹⁵⁰Nd gaps show the largest α-induced ρ̄ range (Δ=0.067) in the nuclear corpus, and persistent violations at all 17 tested α values

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.

FINDING NUC-α.4 — UNIVERSAL LEVELS/GAPS BIFURCATION PERSISTS ACROSS ALL α
The gap/levels ρ̄ ratio (1.38–4.43×) is preserved at every α value — α deformation does not collapse the levels/gaps structural distinction

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.

§8 CROSS-DOMAIN SYNTHESIS — α-DEPENDENCE OF THE UNIVERSAL STRUCTURAL LAW
CROSS-DOMAIN COMPARISON TABLE KEY α-SWEEP METRICS · SUMMARISED
DomainLadder typeα-classρ̄@α=1.00Δρ̄ sweepα@ρ̄_minState@1.00ViolationsAk_min
HydrogenlevelsINACTIVE0.2860.001flatStable01.000
HydrogentransitionsINACTIVE0.3930.036flatWeak P.01.000
HeliumlevelsACTIVE0.3690.0341.20Weak P.01.000
HeliumgapsACTIVE0.5090.053variableWeak P.01.000
Heliumfs splittingsWEAK0.2920.0711.00/1.04Stable01.000
SodiumlevelsSTRONG0.5840.016α=1.00 peakWeak P.01.000
SodiumgapsSTRONG0.6050.086α=1.00 peakBound-Stab01.000
LithiumlevelsINACTIVE0.6540.007flatBound-Stab01.000
CMB TTlevelsWEAK0.2480.0160.85Stable01.000
CMB TTgapsWEAK0.2400.0301.00Stable01.000
CMB EEgapsWEAK0.2720.0290.95Stable01.000
DESIlevelsINACTIVE0.0260.000flatStable01.000
Earth coeffmaglevelsSTRONG0.0570.9151.00Stable4/50.911
Earth degpowlevelsSTRONG0.4020.5881.00Weak P.4/50.619
Moon coeffmaglevelsSTRONG0.0510.8781.00Stable1/50.999
Mars coeffmaglevelsACTIVE0.1960.4911.00Stable0/51.000
Geoid tableanyINACTIVE0.018<0.001flatStable01.000
⁴⁸CagapsACTIVE0.7730.0100.90Bound-Stab17/170.999
¹⁵⁰NdgapsACTIVE0.6380.0671.02Bound-Stab17/170.996
¹⁵²SmlevelsWEAK0.2350.0031.10Stable0/171.000
Nuclear (all)levelsWEAK~0.190.002–0.081scatteredStable (all)0/all1.000
THREE DISTINCT α-RESPONSE ARCHETYPES IDENTIFIED ACROSS THE 5-DOMAIN CORPUS
TYPE I — METRIC ACTIVITY ONLY

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

Consistent with STRUC-I Invariance Proposition — α metrically active, structurally INACTIVE
TYPE II — SUB-THRESHOLD STRUCTURAL ACTIVITY

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 is structurally active at a sub-threshold level — deformation is real but admissibility is not approached
TYPE III — THRESHOLD-CROSSING STRUCTURAL ACTIVITY

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)

The most structurally consequential α-response — the physical α value is structurally distinguished
§9 KEY FINDINGS — α-DEPENDENCE ACROSS DOMAINS
FINDING α.1 — UNIVERSAL NON-FALSIFICATION UNDER α VARIATION
The admissibility inequality inv(Pε;L) ≤ ν(Vε(L)) is not falsified at any tested α value in any domain

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.

FINDING α.2 — GEOID α=1.00 UNIVERSALITY VS NUCLEAR α=1.00 ABSENCE
The geoid domain shows universal α=1.00 optimality across all three bodies and all ladder types; the nuclear domain shows no such pattern — the α minimum is nucleus-dependent and unaligned with the physical value

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.

FINDING α.3 — SODIUM GAPS: NOMINAL α IS A STRUCTURAL PRESSURE MAXIMUM
In sodium gap ladders, α=1.00 is the unique Boundary-Stabilized state across the sweep — a structural maximum rather than a minimum

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.

FINDING α.4 — COSMOLOGY IS α-INERT: SCALE-INVARIANCE CONFIRMED
DESI large-scale structure distance ladders show complete α-invariance to four decimal places — the flattest α response in the corpus

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.

FINDING α.5 — CMB κ* PERCOLATION THRESHOLD: CROSS-CHANNEL CONSISTENCY
All three CMB polarization channels independently confirm κ* = 0.554 — the USL percolation threshold appears in observational cosmological data without being imposed

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.

FINDING α.6 — GEOID TABLE REPRESENTATION IS A DOMAIN-CROSSING STRUCTURAL INVARIANT
The degreepower_table ladder returns ρ̄ = 0.0177 ± 0.0001 for Earth, Moon, and Mars at every tested α — a four-significant-figure structural constant

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.

FINDING α.7 — HELIUM LEVELS SHOW MONOTONE α-SENSITIVITY: MULTI-ELECTRON STRUCTURAL RESPONSE
Helium levels ρ̄ decreases monotonically as α increases from 0.80 to 1.20 — the first clear directional trend in any atomic levels ladder

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.

FINDING α.8 — LEVELS/GAPS BIFURCATION IS α-INVARIANT IN NUCLEAR SPECTRA
The gap/levels ρ̄ ratio (1.4–4.4×) is preserved at every α in the nuclear domain — structural representation-dependence is independent of the fine-structure constant

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.

FINDING α.9 — DATA ANOMALIES: Mg/Pb FOLDER MISLABELLING, ¹²⁰Sn GAPS INCOMPLETE, ²⁰⁸Pb MISSING
Three data integrity issues identified in the nuclear α-sweep output; ²⁰⁸Pb requires a complete rerun

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.

§10 OPEN QUESTIONS — NEXT EXPERIMENTS
SIX PRIMARY QUESTIONS ARISING FROM α-DOMAIN ANALYSIS
Q1 — GEOID/NUCLEAR ARCHETYPE DISCREPANCY

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.

Q2 — CMB κ* ONSET UNDER α DEFORMATION

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.

Q3 — ²⁰⁸Pb DOUBLY-MAGIC α-SWEEP (MISSING)

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.

Q4 — SODIUM STRUCTURAL MAXIMUM — PHYSICAL INTERPRETATION

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.

Q5 — EXTENDED α RANGE AND FALSIFICATION FRONTIER

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.

Q6 — REFINED NUCLEAR SWEEP ON N=90 TRANSITIONAL NUCLEI

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.

α-Dependence of Structural Admissibility — Cross-Domain Analysis · UNNS Substrate Program · 2026-03-28
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)
α operator: proxy-deformation protocol · envelope + fine-structure residual decomposition · differential exponents per domain
Atomic packs: H (Bohr/fs/transitions) · He (structactive, n=843/842/384) · Na (structactive, n=354/338/429) · Li (structactive, n=182/96/181) · sweep: 17 α values 0.80–1.20
CMB packs: Planck 2018 R3.01 · TT (n=2507) · TE (n=1995) · EE (n=1995) · levels + gaps · 17 α values · proxy: Dl = envelope·α² + residual·α⁴
Cosmology: DESI · desi_levels (n=2000) at α=0.80/1.00/1.20 · desi_cluster (n=399) · desi_radial (n=399) · desi_knn3 (n=1199)
Geoid: Earth (L=720, EIGEN-6C4) · Moon (L=300, JGM85F01) · Mars (L=85, AIUB-GRL350A) · α^l harmonic weighting · 5 α values: 0.80/0.90/1.00/1.10/1.20 · 5 ladder types each
Nuclear: 14 isotopes (²⁴Mg ²⁸Si ⁴⁸Ca ⁵⁶Fe ⁶⁰Ni ⁹⁰Zr ¹⁰⁰Mo ¹¹⁶Sn ¹²⁰Sn ¹⁵⁰Nd ¹⁵²Sm ¹⁶⁶Er ¹⁷⁴Yb ²³⁸U) · ENSDF raw data · spin-weighted exponent q(J)=2+(J/J_max)·2 · 17 α values · ²⁰⁸Pb MISSING (rerun required)
Data anomalies: Na fs mislabelled (extended levels, n=429, not splittings n=62) · Mg/Pb folder mislabelling (content correct) · ¹²⁰Sn gaps 1/17 α only · ²⁰⁸Pb absent
Total unique ladders evaluated: ~1,270 · USL violations: 0 clean (Ak<0.52 threshold) · Marginal violations: 103+ (geoid+nuclear gaps, all Ak≥0.619) · κ* confirmed at 0.554 in CMB TT/TE/EE source tables