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.
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. | ✓ |
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.
| 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 |
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.
| 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 |
| 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 |
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.
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.
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.
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.
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.
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.
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.
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.
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.
| 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 |
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.
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.
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.
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.