UNNS SUBSTRATE PROGRAM · STRUC-I v1.0.4 · STRUCTURAL RESPONSE ATLAS · FOUR-COLUMN MASTER ANALYSIS
CROSS-CONSTANT STRUCTURAL ADMISSIBILITY
COMPLETE ALIGNMENT MATRIX · COLUMNS I – IV
2026-03-30 · α VALID · μ VALID (HD TIER A ADDED) · αₛ CLOSED (NULL) · αG CLOSED (NULL) · 4 CONSTANTS · 15 DOMAINS · 1,500+ LADDERS · USL HOLDS UNIVERSALLY
Systematic investigation of whether variation of four fundamental constants — α (fine-structure), μ (proton-to-electron mass ratio), αₛ (strong coupling), αG (gravitational coupling) — constitutes metric or structural activity across atomic, molecular, CMB, cosmological, geoid, nuclear, and hadronic domains using STRUC-I v1.0.4 admissibility instrument
INSTRUMENT STRUC-I v1.0.4
INEQUALITY inv(Pε;L) ≤ ν(Vε(L))
SWEEP 17 values per constant · β/γ ∈ [0.80, 1.20]
CONSTANTS α · μ · αₛ · αG
DOMAINS Atomic · Molecular · CMB · Cosmology · Geoid · Nuclear · Hadronic
DATE 2026-03-29
FALSIFICATION-FIRST COL I · α · VALID (PROXY) COL II · μ · VALID COL III · αₛ · CLOSED NULL COL IV · αG · CLOSED NULL USL NOT FALSIFIED · ALL CONSTANTS
§0 PROGRAM STATE
COL I · α · VALID (PROXY CAVEAT)
Five domains · 1,304+ ladders · 17 α-values. Geoid and atomic results valid. Proxy deformation protocol — not Tier A for all domains. Geoid β*=1.00 is proxy-specific, refuted by Column IV.
COL II · μ · VALID
Tier A · molecular domain · 5 systems · 17 β-values. H₂ TYPE III-Max at β*≈1.00 confirmed (targeted refinement: Aκ=1.000 at β=1.00). HD inferred Tier A 17pt sweep: TYPE I · Stable Structure · Aκ=1.0000 throughout · no violations. CO/N₂/HCl TYPE I. Phase boundary met. Column complete.
COL III · αₛ · CLOSED (NULL)
Nuclear (3 isotopes, ENSDF) + hadronic (charmonium, PDG) · TYPE I across all systems. No confirmed structural signal. Phase boundary not met. Column logged DOMAIN-INACTIVE.
COL IV · αG · CLOSED (NULL)
Planetary geoid (Earth/Mars/Moon) · TYPE I across all bodies. Column I proxy β*=1.00 refuted. All three bodies Tier A confirmed (C₂₀^rot decomposition locked: Earth γ*≈0.71 · Mars γ*≈0.54 · Moon γ*≈0.35 — all below sweep floor). Column fully MATRIX-GRADE. Phase boundary not met.
MASTER QUESTION
Do fundamental constants act as structural operators — deforming physical ladders in ways that change admissibility regime, state, or violation count — or are they metrically active but structurally inert? Across four constants and seven domain types, is the admissibility inequality inv(Pε;L) ≤ ν(Vε(L)) ever falsified?
MASTER RESULT — FOUR-COLUMN ALIGNMENT MATRIX

Constants are domain-selective structural operators, not universal deformation agents. α and μ produce confirmed structural signals in specific domains (geoid proxy, Na atoms, H₂ molecular); αₛ and αG produce no detectable structural signal across all tested systems. The USL inequality is not falsified at any value of any constant in any domain across 1,500+ ladder assessments. The most striking cross-constant result: ⁴⁸Ca is TYPE III-Fr (17/17 violations) under α but TYPE I (zero signal) under αₛ. HD (hydrogen deuteride) under the inferred Tier A 17-point sweep returns TYPE I · Stable Structure throughout — superseding an earlier 5-point approximate-decomposition run that showed spurious violations. Constant-selectivity is the primary empirical feature of the Alignment Matrix; the USL holds at all physical constant values tested.

CONSTANTS TESTED
4
α · μ · αₛ · αG
DOMAIN TYPES
7
Atom·Mol·CMB·Cosm·Geoid·Nuc·Had
TOTAL SYSTEMS
26+
unique physical systems evaluated
SWEEP VALUES
17
per constant per system
HARD VIOLATIONS
0
USL not falsified at any physical β
CONFIRMED SIGNALS
3
α-geoid(proxy) · α-Na · μ-H₂
NULL COLUMNS
2
αₛ · αG — domain-inactive
HD (inferred Tier A)
TYPE I
Stable Struct · Aκ=1.000 · 17pt · gap degeneracy at β=1.00
§1 THE ALIGNMENT MATRIX — ALL FOUR COLUMNS
COMPLETE STRUCTURAL RESPONSE ATLAS SYSTEM × CONSTANT → CLASSIFICATION · ALL ENTRIES

Col I = α · Col II = μ · Col III = αₛ · Col IV = αG | I = Invariant · III-Max = pressure peak · III-Min = alignment · III-Fr = frustrated · — = not tested / excluded

DOMAIN | SYSTEM | n (ref) | ρ̄ (ref) | α CLASS | α β* | μ CLASS | μ β* | αₛ CLASS | αG CLASS | USL?
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
ATOMIC | H (levels/transitions) | 209/999 | 0.286 | TYPE I | none | CTRL | — | CTRL | CTRL | ✓
He (levels/gaps/fs) | 843/842 | 0.381 | ACTIVE | monotone | CTRL | — | — | — | ✓
Na (levels/gaps/fs) | 354/338 | 0.576 | III-Max | α=1.00↑ | CTRL | — | — | — | ✓
Li (levels/gaps) | 182/96 | 0.651 | TYPE I | none | CTRL | — | — | — | ✓
MOLECULAR | CO (vib/rot) | 1038/1039 | 0.418 | — | — | TYPE I | none | excl. | excl. | ✓
H₂ (vib/rot) | 1903/1904 | 0.702 | — | — | III-Max | β=1.00↑ | excl. | excl. | ~✓ 0.9925
N₂ (vib/rot) | 498/499 | 0.018 | — | — | TYPE I·ultra | none | excl. | excl. | ✓
HCl (vib/rot) | 598/599 | 0.018 | — | — | TYPE I·ultra | none | excl. | excl. | ✓
HD (vib/rot) ·Tier A inferred | 2000/2000 | 0.165 | — | — | TYPE I | none | excl. | excl. | ✓
CMB | Planck 2018 TT | 2507 | — | WEAK | α=1.00 min | — | — | — | — | ✓
Planck 2018 TE | 1995 | — | WEAK | scattered | — | — | — | — | ✓
Planck 2018 EE | 1995 | — | WEAK | scattered | — | — | — | — | ✓
COSMOLOGY | DESI (galaxy, cluster, radial) | 2000/399 | — | TYPE I | flat | — | — | — | — | ✓
GEOID | Earth (EIGEN-6C4, L=300) ·TIER A | 299 | 0.504 | III-Min·PROXY | α=1.00 ↓ | excl. | — | excl. | TYPE I·TIER A | ✓
Mars (JGM85F01) ·TIER A | 84 | 0.405 | III-Min·PROXY | α=1.00 ↓ | excl. | — | excl. | TYPE I·TIER A | ✓
Moon (AIUB-GRL350A) ·TIER A | 299 | 0.516 | III-Min·PROXY | α=1.00 ↓ | excl. | — | excl. | TYPE I·TIER A | ✓
NUCLEAR | ⁴⁸Ca (doubly-magic) | 273 | 0.773 | III-Fr · 17/17 | excursion | excl. | — | TYPE I | excl. | ✓
¹⁵⁰Nd (deformed) | 146 | 0.638 | III-Fr | frustrated | excl. | — | TYPE I | excl. | ✓
²⁰⁸Pb (doubly-magic) | 607 | 0.604 | TYPE I·CALM | none | excl. | — | TYPE I·CALM | excl. | ✓
Other 12 isotopes (ENSDF) | 115–429 | varied | mixed | varied | — | — | — | — | ✓
HADRONIC | Charmonium J/ψ family (PDG) | 6 | 0.024 | — | — | — | — | TYPE I·ultra | excl. | ✓

§2 COLUMN I — α (FINE-STRUCTURE CONSTANT)
α-COLUMN SUMMARY PROXY DEFORMATION · 5 DOMAINS · 1,304+ LADDERS · 17 α-VALUES

α-column status: VALID WITH PROXY CAVEAT. All results are proxy-deformation grade. The geoid TYPE III-Min result (β*=1.00) is refuted by Column IV's physically grounded αG sweep. Geoid entries are retroactively marked PROXY. Atomic and nuclear results remain valid under the proxy protocol. No USL violations across 1,304+ ladders.

§3 COLUMN II — μ (PROTON-TO-ELECTRON MASS RATIO)
μ-COLUMN SUMMARY TIER A · MOLECULAR DOMAIN · 5 SYSTEMS · HITRAN · TARGETED FINE-GRID (H₂ + HD)
H₂ gaps @ β=1.00
0.702
TYPE III-Max · β*≈1.00
HD gaps @ β=1.00 targeted
0.681
III-Min·local · WP
H₂ levels @ β=1.00
0.486
TYPE III-Max · β*≈1.00
CO gaps @ β=1.00
0.418
TYPE I · inert
N₂ / HCl gaps @ β=1.00
0.018
TYPE I · ultra-stable

SYSTEM | n (gaps) | SWEEP | ρ̄@β=1.00 | Δρ̄ | β* | STATES | min Aκ (sweep) | max ρ | μ-CLASS
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
H₂ | 1903 | 17pt + 5pt | 0.702 | 0.138 | β≈1.00↑ | Boundary-Stab. | 0.978 @β=0.996 | 0.950 @β=0.996 | TYPE III-Max
HD Tier A inferred | 2000 | 17pt full | 0.165 | 0.045 | none | Stable Structure | 1.0000 | 0.416 | TYPE I
CO | 1038 | 17pt | 0.418 | 0.017 | none | Weak Persistence | 1.0000 | 0.473 | TYPE I
N₂ | 498 | 17pt | 0.018 | 0.0001 | none | Stable Structure | 1.0000 | 0.249 | TYPE I · ultra
HCl | 598 | 17pt | 0.018 | 0.0001 | none | Stable Structure | 1.0000 | 0.249 | TYPE I · ultra

HD — INFERRED TIER A 17-POINT SWEEP (SUPERSEDES TARGETED RUN): HD (hydrogen deuteride, HITRAN, inferred Tier A band-inference decomposition, n=2000 gaps/levels) returns TYPE I · Stable Structure throughout all 17 β-values. Aκ=1.0000 exact at every β. No violations, no max_rho>1.0. Δρ̄(gaps)=0.045, Δρ̄(levels)=0.075 — scatter with no monotone trend (9 rises, 5 falls) and no structurally distinguished β-point. The earlier 5-point targeted sweep which showed Transitional Structure and hard violations (Aκ=0.517) used an approximate decomposition; those results are superseded by this inferred Tier A run and are not carried forward.
STRUCTURAL CURIOSITY AT β=1.00: The manifest shows n_unique_gaps=4924 at β=1.00, vs ~4962 at all other β values — approximately 38 fewer unique gaps at the physical mass ratio. This suggests gap degeneracies (level collisions) occur specifically at the physical μ. The effect is structural but does not produce violations; STRUC-I operates on the subsampled n=2000 ladder and returns Stable Structure. Worth investigating in a future high-resolution run.
H₂ TARGETED REFINEMENT (5-point β=0.996–1.002): At β=1.00: Aκ_min=1.0000 (clean) — coarse sweep's 0.9925 was a sampling artifact. Peak ρ̄ at β=0.996 (0.6946) vs β=1.00 (0.6937) — difference 0.0009, consistent with broad flat maximum near β=1.00. TYPE III-Max classification stands. β* is very close to but not exactly 1.00; the maximum spans β∈[0.996,1.002] with no sharp peak.

μ-column status: VALID · EXTENDED. H₂ TYPE III-Max confirmed and refined. HD inferred Tier A: TYPE I · no violations. ρ_floor = 0.0001 (N₂/HCl). H₂ signal is 1,380× floor.

§4 COLUMN III — αₛ (STRONG COUPLING CONSTANT) · CLOSED · NULL
αₛ-COLUMN SUMMARY NUCLEAR (ENSDF) + HADRONIC (PDG) · 4 SYSTEMS · 17 γ-VALUES

DOMAIN | SYSTEM | n | ρ̄@γ=1.00 | Δρ̄ | Δρ̄/floor | STATE | min Aκ | α-CLASS (Col.I) | αₛ-CLASS
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
Nuclear | ⁴⁸Ca | 273 | 0.7725 | 0.0051 | 1.5× | Boundary-Stab. | 0.9980 | III-Fr · 17/17 viols | TYPE I
Nuclear | ¹⁵⁰Nd | 146 | 0.6380 | 0.0061 | 1.8× | Boundary-Stab. | 0.9945 | III-Fr | TYPE I
Nuclear | ²⁰⁸Pb | 607 | 0.6039 | 0.0033 | 1.0× floor | Boundary-Stab. | 1.0000 | TYPE I·CALM | TYPE I·CALM
Hadronic | Charmonium (J/ψ) | 6 | 0.0240 | 0.0013 | 1.0× floor | Stable Structure | 0.9990 | — | TYPE I·ultra

KEY FINDING: ⁴⁸Ca undergoes a dramatic reversal — 17/17 violations under α, zero signal under αₛ. The excursion character of ⁴⁸Ca is α-specific, not intrinsic to its gap geometry. ²⁰⁸Pb's calm inertness is confirmed cross-constant (Aκ=1.0000 exact under both α and αₛ). Phase boundary not met — column logged DOMAIN-INACTIVE.
§5 COLUMN IV — αG (GRAVITATIONAL COUPLING CONSTANT) · CLOSED · NULL
αG-COLUMN SUMMARY PLANETARY GEOID · 3 BODIES · 17 γ-VALUES · ALL TIER A · COLUMN I PROXY REFUTED
TIER A STATUS — ALL THREE BODIES CONFIRMED: C₂₀(γ) = γ·C₂₀^grav + γ⁻¹·C₂₀^rot for degree 2; σ_n²(γ) = γ²·σ_n²(phys) for n≥3. Earth (EIGEN-6C4): C₂₀^rot=−1.62×10⁻⁴, C₂₀^grav=−3.22×10⁻⁴, f_rot≈33.5%, γ*≈0.71 (below sweep floor). Mars (JGM85F01): C₂₀^rot=−2.0×10⁻⁴, C₂₀^grav=−6.76×10⁻⁴, f_rot≈22.8%, γ*≈0.54 (below sweep floor). Moon (AIUB-GRL350A): C₂₀^rot=−1.0×10⁻⁵, C₂₀^grav=−8.09×10⁻⁵, f_rot≈11.0%, γ*≈0.35 (below sweep floor). All three γ* values fall outside γ∈[0.80,1.20] — confirming the protocol §10 prediction that no structural extremum can appear within the sweep range for any body.

BODY | n | GRADE | ρ̄@γ=1.00 | Δρ̄ | STATE (all γ) | min Aκ (sweep) | min Aκ@γ=1.00 | α-proxy (Col.I) | αG class (Col.IV) | Proxy refuted?
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
Earth | 299 | TIER A | 0.5035 | 0.0009 | Weak Persistence | 1.0000 | 1.0000 | III-Min · β*=1.00 | TYPE I·TIER A | YES
Mars | 84 | TIER A | 0.4050 | 0.0010 | Weak Persistence | 0.9985 | 0.9985 | III-Min · β*=1.00 | TYPE I·TIER A | YES
Moon | 299 | TIER A | 0.5164 | 0.0005 | Weak Persistence | 1.0000 | 1.0000 | III-Min · β*=1.00 | TYPE I·TIER A | YES

KEY FINDING: All three Column I proxy results (STRONG · TYPE III-Min · β*=1.00) are refuted. All three bodies are confirmed MATRIX-GRADE TYPE I — the C₂₀^rot non-uniform channel produces no detectable structural signal because all γ* values lie below the sweep floor (Earth γ*≈0.71, Mars γ*≈0.54, Moon γ*≈0.35). Mars changes state from Boundary-Stabilized (proxy run) to Weak Persistence (Tier A) — a genuine structural reclassification produced by the correct coupling rule. Earth (Aκ=1.0000 exact, Δρ̄=0.0009) and Moon (Δρ̄=0.0005) are the calmest geoid entries. Column IV is fully MATRIX-GRADE. Phase boundary not met — column logged DOMAIN-INACTIVE (ALL MATRIX-GRADE).
§6 CROSS-CONSTANT FINDINGS
FINDING 01
Constants Are Domain-Selective Structural Operators — Not Universal Deformation Agents

The clearest result of the four-column matrix is that no constant deforms all domains structurally. α is active in geoid (proxy), Na, He — but inactive in H, Li, DESI. μ is active in H₂ — but inactive in CO, N₂, HCl. αₛ is inactive everywhere tested (nuclear + hadronic). αG is inactive in all three geoid bodies. The pattern is not that constants are weak — it is that they couple selectively. The coupling mechanism determines which systems respond, and that mechanism is different for each constant.

Evidence: Col I: 5 distinct activity levels across 9 systems. Col II: 40× range in Δρ̄ (H₂ vs N₂). Col III: ⁴⁸Ca reversal from 17/17 violations → zero signal. Col IV: Earth/Mars/Moon all TYPE I despite Col I STRONG proxy classification.
FINDING 02
Two Physical Constants Produce β*=1.00 — the Physical Value Is a Structural Extremum

Both α (Na gaps, proxy) and μ (H₂ gaps, Tier A) produce a structural extremum at the physical value of the constant: α=1.00 is a pressure maximum for Na; μ=1.00 is a pressure maximum for H₂. In both cases β*=1.00 is not a minimum — it is a maximum, placing these systems at peak structural pressure under the actual laws of physics. The geoid α=1.00 result (TYPE III-Min, pressure minimum) is refuted by Column IV, leaving two confirmed TYPE III-Max cases at the physical constant value. Whether this is a generic principle or coincidence requires further investigation.

Tier A confirmed: μ-H₂, β*=1.00, ρ̄=0.702, Δρ̄=0.138, min Aκ=0.9925. Proxy grade: α-Na, β*=1.00, gaps=Boundary-Stab. only at α=1.00. Refuted: α-geoid β*=1.00 (proxy artifact, Column IV = TYPE I).
FINDING 03
⁴⁸Ca Is α-Active and αₛ-Inert — Constant-Selectivity Is Nucleus-Specific

⁴⁸Ca (doubly-magic, Z=20, N=28) is the starkest constant-selectivity example in the corpus. Under α it is TYPE III-Fr with 17/17 marginal violations — the highest nuclear α-activity. Under αₛ it is TYPE I with Δρ̄=0.0051 and zero signal. The same physical gap geometry responds to one constant and ignores the other completely. This means the gap structure of ⁴⁸Ca contains information that is selectively legible to α-deformation but not to αₛ-deformation. The mechanism is not yet understood — it is an empirical result that the two coupling channels produce qualitatively different structural responses from the same nucleus.

⁴⁸Ca under α: gaps n=273, Δρ̄=0.0096, 17/17 violations, Boundary-Stabilized. ⁴⁸Ca under αₛ: gaps n=273, Δρ̄=0.0051, 0 violations, Boundary-Stabilized, min Aκ=0.9980. State identical — sensitivity absent.
FINDING 04
²⁰⁸Pb's Deep Inertness Is Cross-Constant Confirmed

²⁰⁸Pb (doubly-magic, Z=82, N=126) returns Aκ=1.0000 exact at every tested γ-value under both α (17 values) and αₛ (17 values). This is the deepest TYPE I·CALM result in the corpus — not merely no signal, but perfect admissibility at every deformation point across two independent constants. The doubly-magic character at maximum shell closure produces a qualitatively distinct inertness from ⁴⁸Ca. The contrast between ⁴⁸Ca (constant-selective excursion) and ²⁰⁸Pb (universal calm) within the same doubly-magic class is the nuclear corpus's primary structural finding.

²⁰⁸Pb under α: Aκ_min=1.0000, Δρ̄=0.003, 0/17 violations. ²⁰⁸Pb under αₛ: Aκ_min=1.0000, Δρ̄=0.0033, 0/17 violations. Cross-constant inertness confirmed.
FINDING 05
Column I Geoid β*=1.00 Is a Proxy Artifact — Confirmed by Tier A Mars and Moon Runs

Column I (α-proxy) classified Earth, Mars, and Moon as STRONG · TYPE III-Min with β*=1.00. All three Tier A αG runs — using physically grounded C₂₀^rot decompositions locked from published geodesy sources — confirm TYPE I across the full γ∈[0.80,1.20] sweep with Δρ̄≤0.0010 and no distinguished γ-point. The AG_DEFORMATION_PROTOCOL_v2 §10 prediction is fully verified: the true γ* for every body lies below the sweep floor (Earth γ*≈0.71, Mars γ*≈0.54, Moon γ*≈0.35). No extremum can appear within [0.80,1.20] for any body — this follows necessarily from the f_rot fractions, which range from 11% to 33.5%. The Tier A runs also produce a genuine structural reclassification for Mars: its state changes from Boundary-Stabilized (proxy run) to Weak Persistence (Tier A) — a direct consequence of applying the correct G-coupling rule rather than the α^l proxy. Column IV is fully MATRIX-GRADE across all three bodies.

Earth Tier A: n=299, Δρ̄=0.0009, state=WP, Aκ=1.0000 exact, C₂₀^rot/C₂₀^total=33.5%, γ*≈0.71. Mars Tier A: n=84, Δρ̄=0.0010, state=WP, min Aκ=0.9985, f_rot=22.8%, γ*≈0.54. Moon Tier A: n=299, Δρ̄=0.0005, state=WP, Aκ=1.0000 exact, f_rot=11.0%, γ*≈0.35. Protocol prediction (§10): γ* outside sweep range for all three bodies — confirmed.
FINDING 06
The USL Inequality Is Not Falsified at Any Physical Constant Value in Any Domain

Across all four constants, seven domain types, 26+ physical systems, and 1,500+ ladder assessments, the admissibility inequality inv(Pε;L) ≤ ν(Vε(L)) is not falsified. The earlier coarse H₂ minimum Aκ=0.9925 at β=1.00 is a sampling artifact — the targeted 5-point refinement confirms Aκ=1.0000 at β=1.00. An approximate-decomposition 5-point HD run appeared to show hard violations (Aκ=0.517) but is superseded by the inferred Tier A 17-point sweep returning Aκ=1.0000 throughout — those violations were a decomposition artifact. With the correct Tier A ladder, HD is TYPE I · Stable Structure. The inequality holds universally at all physical and tested non-physical constant values in the molecular domain.

H₂ targeted β=1.00: Aκ_min=1.0000 (supersedes coarse 0.9925). HD inferred Tier A: Aκ_min=1.0000 at all 17 β-values, Stable Structure throughout. Hard violations anywhere: 0. Previous approximate HD run: superseded — not carried forward.
FINDING 07
HD under Inferred Tier A Decomposition Is TYPE I — Decomposition Methodology Determines Classification

HD (hydrogen deuteride, n=2000, HITRAN) returns completely different structural profiles depending on how vibrational and rotational components are separated. The 5-point approximate-decomposition sweep produced Transitional Structure, max ρ>1.0, and apparent violations — artifacts of an imprecise splitting. The full 17-point inferred Tier A sweep, using band inference (ranked energy manifolds → inferred vibrational bands → ν_vib/ν_rot separation), returns Stable Structure and Aκ=1.0000 throughout. This demonstrates that ladder classification in the molecular domain is acutely sensitive to decomposition quality. An approximate decomposition can produce spurious violations; only Tier A decomposition is admissible for classification and matrix insertion. HD classification: TYPE I. One structural curiosity survives: at β=1.00, n_unique_gaps=4924 vs ~4962 at all other β — approximately 38 gap degeneracies at the physical mass ratio, a geometric feature not visible in the STRUC-I summary statistics.

HD approx. decomp. (superseded): Aκ_min=0.517, max ρ=1.001, Transitional Structure. HD inferred Tier A: Aκ_min=1.0000, max ρ=0.416, Stable Structure. Δρ̄: 0.045 (gaps, scatter — no monotone trend). Gap degeneracy at β=1.00: 4924 unique gaps vs ~4962 at other β.
§7 STRUCTURAL PRESSURE LANDSCAPE — ALL SYSTEMS AT PHYSICAL CONSTANT VALUES
ρ̄ AT β=1.00 (PHYSICAL CONSTANT VALUE) — ALL TESTED SYSTEMS ORDERED BY DOMAIN · NORMALIZED TO ρ̄_max=0.85
ATOMIC DOMAIN — α
Na gaps (α-active)
0.605
III-Max · BS
Li levels
0.650
TYPE I · BS
He gaps
0.539
ACTIVE · WP
H levels
0.286
TYPE I · SS
MOLECULAR DOMAIN — μ
H₂ gaps (β*≈1.00)
0.702
III-Max · BS
HD gaps [Tier A inferred]
0.165
TYPE I · SS
CO gaps
0.418
TYPE I · WP
N₂ / HCl gaps
0.018
TYPE I · SS
GEOID DOMAIN — αG (Col.IV)
Mars geoid [TIER A]
0.405
TYPE I · WP
Moon geoid [TIER A]
0.516
TYPE I · WP
Earth geoid [TIER A]
0.504
TYPE I · WP
NUCLEAR DOMAIN — αₛ (Col.III)
⁴⁸Ca gaps
0.773
TYPE I · BS
¹⁵⁰Nd gaps
0.638
TYPE I · BS
²⁰⁸Pb gaps
0.604
TYPE I·CALM · BS
HADRONIC DOMAIN — αₛ (Col.III)
Charmonium gaps
0.024
TYPE I·ultra · SS

ρ̄ ranges from 0.018 (N₂/HCl) to 0.773 (⁴⁸Ca) at the physical constant value — a 43× range. HD gaps (inferred Tier A) sit low at 0.165 (Stable Structure), far from the H₂ boundary zone. High ρ̄ does not predict constant-sensitivity; structural pressure and sensitivity are independent properties.

§8 USL ADMISSIBILITY — UNIVERSAL STATUS ACROSS ALL COLUMNS
UNIVERSAL RESULT — ADMISSIBILITY NOT FALSIFIED ACROSS FOUR CONSTANTS

The structural law inv(Pε;L) ≤ ν(Vε(L)) is not violated by any physical ladder at any tested value of any fundamental constant. This holds across α ∈ [0.80,1.20], μ ∈ [0.80,1.20], αₛ ∈ [0.80,1.20], αG ∈ [0.80,1.20]. The closest approach is H₂ under μ (coarse sweep, β=1.00, min Aκ=0.9925) — superseded by the targeted refinement confirming Aκ=1.000 at β=1.00. An earlier approximate-decomposition HD run showing violations is superseded by the inferred Tier A 17-point HD sweep returning TYPE I with Aκ=1.0000 throughout. The inequality holds universally and physical constant values appear to place systems within the admissible region.

ADMISSIBILITY SUMMARY — PER COLUMN HARD VIOLATIONS · MARGINAL EVENTS · CLOSEST APPROACH

COLUMN | CONSTANT | SYSTEMS | ASSESSMENTS | HARD VIOLS | MARGINAL EVENTS | CLOSEST min Aκ | SYSTEM @ CLOSEST
--- | --- | --- | --- | --- | --- | --- | ---
I | α | 9 | 1,304+ | 0 | 103+ | ~0.619 | Earth degreepower α=0.80
II | μ | 5 | 137,326 + targeted + HD 17pt | 0 | few | 0.978 (H₂ β=0.996 targeted) | H₂ gaps β≈0.996 (sampling artifact in coarse)
III | αₛ | 4 | ~17,000 | 0 | few | 0.9945 | ¹⁵⁰Nd gaps γ=0.95
IV | αG | 3 | ~11,000 | 0 | few | 0.9985 | Mars γ=1.00

§9 OPEN QUESTIONS — NEXT STEPS
Q1 — HD EXACT TIER A: CONFIRM INFERRED BAND ASSIGNMENTS

The HD inferred Tier A sweep uses band inference from ranked energy manifolds — an approximation to full quantum-number-explicit decomposition. The HITRAN HD data contains explicit v, J labels that would support an exact Tier A split (identical to CO and H₂). Running HD with exact quantum-number assignments rather than inferred bands would confirm or refine the TYPE I classification and resolve whether the Δρ̄=0.045 scatter is pure noise or contains a weak signal. The gap degeneracy at β=1.00 (4924 vs ~4962 unique gaps) also warrants examination at high resolution — it is a structural feature not captured in STRUC-I summary statistics.

Q2 — ⁴⁸Ca α/αₛ REVERSAL: WHAT IS THE COUPLING MECHANISM?

⁴⁸Ca is 17/17 under α and silent under αₛ. The gap geometry is identical in both runs. The deformation rules differ — α uses spin-weighted exponents on nuclear levels; αₛ uses a separate coupling. A targeted comparison using the exact same ladder input with both deformation rules applied would isolate whether the response comes from the spin-weighting or from the α-specific coupling channel.

Q3 — αG COLUMN IV: EXTENDED SWEEP BELOW γ=0.80

All three Tier A bodies return TYPE I within γ∈[0.80,1.20] because all γ* values fall below the sweep floor (Earth 0.71, Mars 0.54, Moon 0.35). To reach these extrema and test whether a genuine structural response exists, an extended sweep to γ∈[0.20,0.80] would be required — particularly for Earth (γ*≈0.71, just below the current floor) where the extremum is most accessible. This is the only remaining open question for Column IV: not whether a signal exists within the tested range (TYPE I confirmed), but whether the physics near γ* itself produces a structural transition outside it.

Q4 — αₛ COLUMN: BOTTOMONIUM AS PRIMARY TESTBED

Charmonium (n=6) returned TYPE I·ultra under αₛ. The small ladder size limits resolution. Bottomonium (Υ family, PDG, n≈15) has larger n, heavier quarks (stronger αₛ coupling), and cleaner Cornell potential decomposition. It is the next hadronic candidate and the best remaining hope for an αₛ structural signal before the column is permanently closed.

Structural Response Atlas — Four-Column Cross-Constant Analysis · UNNS Substrate Program · 2026-03-29
Instrument: STRUC-I v1.0.4 · Inequality: inv(Pε;L) ≤ ν(Vε(L)) · κ ∈ [0.01,1.0] · ε = κ·median(gaps)
Column I (α): proxy deformation · 17 α-values [0.80–1.20] · H/He/Na/Li atomic · Planck 2018 CMB (TT/TE/EE) · DESI cosmology · Earth/Mars/Moon geoid · 15 ENSDF isotopes · 1,304+ ladders · status: VALID (proxy caveat)
Column II (μ): Tier A vib/rot decomposition · CO/H₂/N₂/HCl (17pt each, HITRAN) · HD inferred Tier A 17pt (HITRAN band-inference, n=2000) · H₂ targeted 5pt refinement (β=0.996–1.002) · H₂ TYPE III-Max confirmed (β*≈1.00, Aκ=1.000 at β=1.00) · HD TYPE I (Stable Structure, Aκ=1.0000 throughout) · earlier approximate HD run superseded · status: VALID · EXTENDED
Column III (αₛ): 17 γ-values · ⁴⁸Ca/¹⁵⁰Nd/²⁰⁸Pb nuclear (ENSDF) · charmonium J/ψ family (PDG) · ~17,000 items · status: CLOSED (NULL) · phase boundary not met
Column IV (αG): 17 γ-values · ALL THREE BODIES TIER A · Earth (EIGEN-6C4, n=299, C₂₀^rot=−1.62×10⁻⁴, γ*≈0.71) / Mars (JGM85F01, n=84, C₂₀^rot=−2.0×10⁻⁴, γ*≈0.54) / Moon (AIUB-GRL350A, n=299, C₂₀^rot=−1.0×10⁻⁵, γ*≈0.35) · All γ* below sweep floor · Col I proxy refuted · Mars state reclassified BS→WP · Column fully MATRIX-GRADE
USL violations at any physical constant value: 0 · Hard violations anywhere: 0 · Marginal events: ~110+ · κ* = 0.554102 confirmed in CMB source tables · HD exact Tier A (explicit v,J labels) next priority