Every one of the 34 tested datasets satisfies Theorem 5 of the Phase Mapping Protocol (structural rigidity): for each dataset individually, the verdict V(α,μ) is constant across all tested (α,μ) points, and the giant ratio GR is identical at every grid point within that dataset. No dataset exhibits a phase transition anywhere in the tested parameter space α ∈ [0.80, 1.20] × μ ∈ [0.80, 1.20]. The verdict of a fixed ladder construction is invariant under (α,μ) deformation — though it varies across different representations of the same physical system (Finding 3). This result holds across atomic, molecular, and nuclear domains.
The structural commutator C(α,μ;L) = S(μ(α(L))) − S(α(μ(L))) is zero at every grid point for every dataset: Δ_verdict = 0, Δ_giant = 0.000000, Δ_κ = 0. All tested ladders are structurally commutative (Theorem 2 of Phase Mapping Protocol): the order in which α and μ are applied to a fixed ladder construction does not change the structural outcome. This is a consequence of structural rigidity: a ladder whose GR and verdict do not change under either operator individually cannot produce asymmetry under their composition. Conjecture 3.4 of the Structural Response manuscript (operator non-commutativity) therefore remains untested here — to observe non-commutativity the pipeline must be applied to transitional systems where single-operator response is non-trivial.
The dominant source of structural variation in this corpus is not operator deformation but ladder representation. The same physical system (He, Na, HD) yields categorically different verdicts depending on how the ladder is constructed. This is the dominant signal in this dataset and provides the strongest corpus evidence yet for Theorem 7.5 of the Dual Observability manuscript (representation non-invariance). Intra-grid GR variation (within a single dataset, across all (α,μ) points): zero — GR is constant for every fixed ladder construction. Inter-representation GR variation (across different ladder constructions of the same physical system): up to 0.20 (HD combined vs HD lower: 0.800 vs 1.000). These are different comparison levels: the first is invariance under deformation, the second is sensitivity to encoding.
Even though all systems are verdict-rigid, their κ_conn values differ dramatically within the FULL class: CH4/CO2/H2O/NH3/O3 connect at κ < 0.3 (near-immediate), while He QM-I connects at κ = 10⁶ (maximum adaptive extension). Na QM-I connects at κ = 161,260 and Li at κ = 376,546. These differences are invisible to the verdict but constitute Theorem 3 of the Phase Mapping Protocol (hidden structural variation). The sub-regime metric structure is constant across the grid — it is a property of the ladder, not of the deformation.
Each cell is one (α,μ) grid point. Colour: green=FULL, amber=TAIL, red=HARD. All maps are monochromatic — no system transitions across the grid.
The structural commutator measures whether the order of operator application matters: C(α,μ;L) = S(μ(α(L))) − S(α(μ(L))). A non-zero commutator signals operator-order sensitivity.
Across all 9,826 (forward, reverse) evaluation pairs: Δ_verdict = 0, Δ_giant = 0.000000, Δ_κ = 0 in every single case. The structural commutator is identically zero throughout the corpus. This means: for all tested ladders, the joint (α,μ) deformation is structurally commutative — the order in which α and μ are applied does not change the structural outcome.
This is consistent with (but stronger than) the marginal-sweep operator separation result: if GR does not change under either α alone or μ alone, the composition cannot produce an asymmetry either. The trivial commutator is thus a consequence of structural rigidity, not an independent finding. The combination is degenerate in the sense of Corollary 1 of the Phase Mapping Protocol.
For any fixed ladder construction, the structural class is invariant under (α,μ) deformation (Finding 1). But different ladder constructions of the same physical system can yield categorically different classes. The dominant source of verdict variation across the corpus is representation — not constant deformation — and this corpus substantially extends the helium representation-split first identified in the single-constant sweep.
QM-I ladders uniformly FULL; Zeeman-split ladders uniformly TAIL. The fine structure exposed in Zeeman splitting reveals permanent tail isolation.
Representation-invariant HARD: all four constructions return HARD at every (α,μ) point. H is robustly HARD regardless of encoding. Principal quantum number shell structure dominates every representation.
Most extreme split: QM-I → FULL at large κ, Zeeman → HARD. Zeeman sub-structure introduces magnetic-field-split outlier gaps that fragment the vulnerability graph severely.
Merging sub-ladders creates permanent fragmentation. The combined ladder incorporates transitions spanning very different energy scales; the resulting extreme gap ratios prevent backbone formation.
Cross-corpus discrepancy: CO returns HARD here but FULL in the previous μ-deformation corpus. Different ladder constructions (rovibrational level selection, gap preprocessing) produce opposite verdicts. Requires canonical ladder investigation.
| Dataset | Domain | Verdict (all γ) | GR at (1,1) | κ_conn | n (gaps) | Phase stable | C(α,μ) |
|---|---|---|---|---|---|---|---|
| calcium_zeeman_ladder | Atomic (Zeeman) | FULL | 1.0000 | 296,875 | 1999 | ✓ | 0 |
| iron_zeeman_ladder | Atomic (Zeeman) | FULL | 1.0000 | 5,960 | 1999 | ✓ | 0 |
| helium_QM1_preprocessed | Atomic (QM-I) | FULL | 1.0000 | 1,000,000 | 1999 | ✓ | 0 |
| helium_gap_structure_QM1 | Atomic (QM-I) | FULL | 1.0000 | 1,000,000 | 1999 | ✓ | 0 |
| helium_spectrum_QM1 | Atomic (QM-I) | FULL | 1.0000 | 271,999 | 1684 | ✓ | 0 |
| heliumII_QM1_preprocessed | Atomic (QM-I) | FULL | 1.0000 | 1,000,000 | 589 | ✓ | 0 |
| heliumII_gap_structure_QM1 | Atomic (QM-I) | FULL | 1.0000 | 1,000,000 | 441 | ✓ | 0 |
| heliumII_spectrum_QM1 | Atomic (QM-I) | FULL | 1.0000 | 329,031 | 296 | ✓ | 0 |
| lithium_QM1_preprocessed | Atomic (QM-I) | FULL | 1.0000 | 376,546 | 460 | ✓ | 0 |
| lithium_gap_structure_QM1 | Atomic (QM-I) | FULL | 1.0000 | 376,546 | 366 | ✓ | 0 |
| lithium_spectrum_QM1 | Atomic (QM-I) | FULL | 1.0000 | 420,033 | 277 | ✓ | 0 |
| sodium_QM1_preprocessed | Atomic (QM-I) | FULL | 1.0000 | 161,260 | 1385 | ✓ | 0 |
| sodium_gap_structure_QM1 | Atomic (QM-I) | FULL | 1.0000 | 161,260 | 1038 | ✓ | 0 |
| sodium_spectrum_QM1 | Atomic (QM-I) | FULL | 1.0000 | 206,721 | 836 | ✓ | 0 |
| gold_zeeman_ladder | Atomic (Zeeman) | TAIL | 0.9705 | none | 1999 | ✓ | 0 |
| helium_zeeman_ladder | Atomic (Zeeman) | TAIL | 0.9575 | none | 1999 | ✓ | 0 |
| helium_singlet_zeeman_ladder | Atomic (Zeeman) | TAIL | 0.9840 | none | 1999 | ✓ | 0 |
| helium_triplet_zeeman_ladder | Atomic (Zeeman) | TAIL | 0.9655 | none | 1999 | ✓ | 0 |
| silver_zeeman_ladder | Atomic (Zeeman) | TAIL | 0.9555 | none | 1999 | ✓ | 0 |
| hydrogen_zeeman_ladder | Atomic (Zeeman) | HARD | 0.9750 | none | 1999 | ✓ | 0 |
| hydrogen_QM1_preprocessed | Atomic (QM-I) | HARD | 0.8500 | none | 160 | ✓ | 0 |
| hydrogen_gap_structure_QM1 | Atomic (QM-I) | HARD | 0.8140 | none | 129 | ✓ | 0 |
| hydrogen_spectrum_QM1 | Atomic (QM-I) | HARD | 0.9336 | none | 211 | ✓ | 0 |
| sodium_zeeman_ladder | Atomic (Zeeman) | HARD | 0.9220 | none | 1999 | ✓ | 0 |
| molecular_CH4_ladder | Molecular | FULL | 1.0000 | 0.2 | 1999 | ✓ | 0 |
| molecular_CO2_ladder | Molecular | FULL | 1.0000 | 0.1 | 1999 | ✓ | 0 |
| molecular_H2O_ladder | Molecular | FULL | 1.0000 | 0.3 | 1999 | ✓ | 0 |
| molecular_NH3_ladder | Molecular | FULL | 1.0000 | 0.1 | 1999 | ✓ | 0 |
| molecular_O3_ladder | Molecular | FULL | 1.0000 | 0.3 | 1999 | ✓ | 0 |
| HD_lower_levels | Molecular (HD) | FULL | 1.0000 | 0.7 | 385 | ✓ | 0 |
| HD_O_lower_levels | Molecular (HD) | FULL | 1.0000 | 0.6 | 244 | ✓ | 0 |
| nuclear_level_ladders | Nuclear | FULL | 1.0000 | 2,578 | 1999 | ✓ | 0 |
| molecular_co_ladder | Molecular | HARD | 0.9370 | none | 1999 | ✓ | 0 |
| HD_combined_levels | Molecular (HD) | HARD | 0.8004 | none | 1999 | ✓ | 0 |
All FULL systems share GR=1.000 and a constant verdict across the entire α×μ grid. Yet their connectivity thresholds span five orders of magnitude — the same sub-regime metric structure established in the prior corpus. This is deformation-invariant: κ_conn does not change when α or μ varies.
| Theorem | Statement | Status |
|---|---|---|
| Theorem 5 | Structural Rigidity: V constant over Ω, C=0 | CONFIRMED — all 34 datasets |
| Corollary 1 | Degenerate Phase Space: Φ(α,μ) = V₀ everywhere | CONFIRMED — all 34 datasets |
| Theorem 2 | Structural Commutativity: C(α,μ;L) = 0 | CONFIRMED — all 34 datasets |
| Theorem 3 | Hidden Structural Variation: Δ_V=0 but internal metric differs | CONFIRMED — κ_conn varies 10⁵× within FULL class |
| Theorem 4 | Phase Transition: V changes discontinuously | NOT OBSERVED — no transitions in any tested system |
| Corollary 2 | Non-Commutative Region: C(α,μ;L) ≠ 0 | NOT OBSERVED — commutator identically zero |
The joint (α,μ) sweep provides the first direct test of Conjecture 3.4. Result: trivial commutator throughout. This is consistent with the conjecture being false for structurally rigid systems — if the verdict and GR are deformation-invariant under each operator separately, they will remain so under any composition. The conjecture may only be testable on systems with non-trivial single- operator response (i.e. transitional systems in the ²⁸Si / ²³⁸U class). Those systems are not in this corpus.
This corpus provides the strongest corpus evidence yet for representation non-invariance: Na QM-I is FULL while Na Zeeman is HARD; He QM-I is FULL while He Zeeman is TAIL; HD lower is FULL while HD combined is HARD; CO here is HARD while CO in the prior μ-corpus is FULL. The structural class is not a property of the physical system alone — it is a joint property of the system and its ladder encoding. The canonical ladder problem (Proposition 7.3) is the critical open problem for this corpus.
To test Conjecture 3.4 (operator non-commutativity), the pipeline should be applied to transitional systems — specifically ²⁸Si and ²³⁸U nuclear isotopes, which showed class-change behaviour in the single-operator α sweep. These systems occupy the transitional class Ctrans(α) and may exhibit non-trivial commutator behaviour under joint (α,μ) deformation. Geoid harmonics at physical α are also priority targets.
Three silicon representations from the Materials Project condensed-matter database and two uranium-238 ladder constructions (full ENSDF level scheme + top-levels only) add 3 new phase-mapped systems and 2 static STRUC-PERC-I single runs to the corpus.
The density ladder (n=41 gaps), formation energy ladder (n=42 gaps), and table export (n=271 gaps) all return FULL_PERCOLATION at every one of their 289 grid points. GR = 1.000 throughout. Commutator identically zero. Rigid class: confirmed.
| Si Representation | n gaps | Verdict (all 289 pts) | κ_conn | Phase stable | C(α,μ) |
|---|---|---|---|---|---|
| Density ladder (Materials Project) | 41 | FULL | 2.000 | ✓ | 0 |
| Formation energy ladder | 42 | FULL | 81.4 | ✓ | 0 |
| Table export (full phase table) | 271 | FULL | 56,394 | ✓ | 0 |
Silicon in the Materials Project (condensed matter, phase/formation-energy ladder) is FULL and rigid across the entire α×μ grid. Silicon-28 in the prior nuclear constants deformation corpus was transitional HARD — HARD at γ_α ≤ 0.90, FULL at the physical value. Same chemical element, categorically different structural behaviour depending on whether the ladder is built from condensed-matter phase data or nuclear γ-level spectroscopy. This is the clearest cross-domain representation split in the corpus: the same element (Si) occupies FULL (condensed matter) vs transitional HARD (nuclear) depending on the physical context of the ladder construction.
The three FULL Si representations span κ_conn = 2 (density, immediate) to 56,394 (table export, high delay) — a factor of 28,000 within the same element under the same domain classification. As with the prior corpus, the FULL verdict is informationally incomplete: the metric depth (κ_conn) varies by orders of magnitude within FULL even for the same element. All three κ_conn values are constant across the entire 289-point α×μ grid.
Two U-238 ladder constructions are evaluated here as single-point static runs (not phase-mapped across the full α×μ grid). They extend the representation analysis with a new split within the same isotope.
The full ENSDF U-238 nuclear level scheme (1,113 elements, 1,112 gaps) returns TAIL_FRAGMENTATION with GR = 0.9829 and 14 permanently isolated vertices. 34 outlier gaps (3.1% of all gaps) with max/median ratio = 2.48×10⁹ dominate the tail. These are single γ-transitions far above the bulk level spacing that cannot merge with the main backbone within the tested κ range. Adaptive extension pushes GR to 0.9991 (2 components remaining) — extremely close to FULL but not achieved. Theorem 1 of the revised PRP does not apply (TAIL class). Fully admissible.
A minimal U-238 construction using only the top-level transitions (5 elements, 4 gaps) returns FULL_PERCOLATION with κ_conn = 0.56 and zero outliers. With only 4 gaps in a relatively uniform energy range, the gap structure connects almost immediately.
Three U-238 ladder constructions produce three different structural outcomes: (1) Full ENSDF level scheme (1,112 gaps) → TAIL, GR = 0.9829, 34 outlier γ-transitions with max/median = 2.48×10⁹. (2) Top 5 levels only (4 gaps) → FULL, κ_conn = 0.56 — the coarse top-level structure connects immediately with no outliers. (3) Prior constants deformation corpus → FULL at physical α, transitional HARD at γ_α = 1.02. The three constructions map to three distinct structural regimes. U-238 joins He, Na, HD, CO, and Si as an element demonstrating categorical representation-sensitivity.
Si condensed matter (Materials Project density/formation-energy/table) → FULL, rigid. Si-28 nuclear (ENSDF, constants deformation corpus) → transitional HARD at low α. The element silicon occupies different realizability classes in different physical domains. The structural class is a joint property of the element, its domain, and its ladder encoding — not of the element alone.
A major corpus extension covering six new physical domains and an updated Zeeman atomic dataset. 55 new datasets across Atmosphere (ERA5), CMB (Planck R3.01), Cosmic Web (μ-sweep), Crystallography (Materials Project), Geoid (Earth/Mars/Moon), Solar (GOES-XRS), and U-238 sub-constructions — plus 9 updated Zeeman ladders.
The previous Zeeman corpus (n=1999 gaps each) returned TAIL for Au, He, singlet He, triplet He, Ag and HARD for H, Na. The new Zeeman corpus (n=39–124 actual level gaps) returns FULL for all 9 atoms. Zero TAIL, zero HARD. Same physical systems, completely different structural class — determined entirely by ladder construction. This is the largest single-batch representation reversal in the corpus.
| Atom | Old verdict (n=1999) | Old GR | New verdict (actual n) | New GR | New κ_conn | New n |
|---|---|---|---|---|---|---|
| Calcium | FULL | 1.000 | FULL | 1.000 | 322 | 42 |
| Gold | TAIL | 0.971 | FULL | 1.000 | 6 | 52 ← |
| Helium | TAIL | 0.958 | FULL | 1.000 | 11,189 | 40 ← |
| Helium singlet | TAIL | 0.984 | FULL | 1.000 | 13,709 | 39 ← |
| Helium triplet | TAIL | 0.966 | FULL | 1.000 | 11,205 | 40 ← |
| Hydrogen | HARD | 0.975 | FULL | 1.000 | 187,014 | 39 ← |
| Iron | FULL | 1.000 | FULL | 1.000 | 10 | 124 |
| Silver | TAIL | 0.956 | FULL | 1.000 | 8 | 39 ← |
| Sodium | HARD | 0.922 | FULL | 1.000 | 329 | 39 ← |
The previous corpus inflated all ladders to n=1999 gaps — which for sparse Zeeman level sets introduced large structural artifacts (zero-gaps, repeated values, or bulk padding that created extreme outlier-to-bulk ratios). The new corpus uses the actual level count (39–124 gaps). With a compact realistic ladder, the Zeeman-split levels connect well within the tested κ range. Hydrogen now returns FULL at κ_conn = 187,014 — the second-highest κ_conn in the entire corpus, just below He QM-I at 10⁶. The large connectivity delay reflects the genuine principal-quantum-number shell structure, but with the actual level count the backbone forms. Na Zeeman similarly recovers to FULL at κ=329. The canonical ladder problem is the central open question this result underscores.
cell_volume (total unit cell volume per phase) → FULL for all 6 compounds tested (BaTiO3, Fe, KNbO3, SiO2, TiO2, ZrO2). volume_per_atom → HARD for 5 of 6 (SiO2 exception). volume_per_formula_unit → HARD for 4 of 6 (SiO2 and BaTiO3 exceptions). The cell-volume ladder encodes one structural value per phase; the per-atom and per-formula-unit normalizations produce specific volume ratios that, for most perovskite/oxide compounds, generate isolated extreme outliers preventing backbone formation. SiO2 is the exception: all three representations return FULL, suggesting its phase volume structure is more uniform across normalizations.
| Compound | cell_volume | κ_conn | volume_per_atom | volume_per_formula_unit | Structure type |
|---|---|---|---|---|---|
| SiO2 | FULL | 1.0 | FULL κ=10.0 | FULL κ=5.6 | silica (quartz/cristobalite) |
| BaTiO3 | FULL | 0.7 | HARD GR=0.800 | FULL κ=0.7 | perovskite |
| Fe | FULL | 1.0 | HARD GR=0.750 | HARD GR=0.750 | BCC iron |
| KNbO3 | FULL | 1.0 | HARD GR=0.900 | HARD GR=0.900 | perovskite |
| TiO2 | FULL | 0.7 | HARD GR=0.714 | HARD GR=0.714 | rutile/anatase |
| ZrO2 | FULL | 1.0 | HARD GR=0.857 | HARD GR=0.857 | zirconia |
Despite the FULL/HARD split across representations, every dataset returns the same verdict at every (α,μ) grid point. The crystallographic phase-volume ladder structure is completely insensitive to constant deformation. HARD here is representation-intrinsic, not deformation-induced.
ERA5 250hPa zonal wind: global longitude sectors, latitude bands, jet-region aggregates — 8 representations of upper-tropospheric jet-stream structure. All FULL, rigid, trivial commutator. κ_conn ≤ 2 for all constructions — among the lowest in the entire corpus. Atmospheric wind structure has near-immediate gap-graph connectivity under any (α,μ) deformation.
Planck 2018 R3.01 power spectra (TT, TE, EE) with the full multi-frequency Planck dataset. Note the dramatic κ_conn change from the prior CMB results (TT=134k, EE=859k) — these R3.01 ladders are much smaller (n=54–499 vs n=thousands) producing lower κ_conn. Same physical signal, different ladder construction, same FULL verdict but very different connectivity depth.
First μ-sweep of geoid harmonic ladders. EIGEN-6C4 Earth (κ=1.0, n=181), JGM85 Mars (κ=0.7, n=499), AIUB-GRL350A Moon (κ=6.3, n=142) — all FULL across 21 μ grid points at fixed physical α. Previously only tested at α=1.10 and 1.20 (non-physical, HARD). At physical α the geoids are FULL and μ-stable. The prior HARD result is confined to amplified α beyond the physical range, consistent with the Realizability Anchoring Conjecture.
DESI, SDSS, 2MRS cosmic web orientation ladders — μ-only sweep (α=1.0 fixed, μ ∈ [0.95,1.05]). All 7 constructions FULL and μ-stable. κ_conn spans 27.9 (SDSS) to 15,658 (DESI rotated sample) — reflecting different structural density of the cosmic web representations. The DESI full synthetic (n=666, κ=5,908) and DESI rotated (n=1999, κ=15,658) show large connectivity delays consistent with the sparse void-dominated cosmic web structure.
First solar domain in the corpus. GOES X-ray solar flux ladder (7 gaps from 8 X-ray energy level values) — FULL across the entire 17×17 (α,μ) grid. κ_conn = 16.6. The solar X-ray spectral gap structure connects at moderate κ, rigid across all constant deformations. This extends the zero-violation, FULL-at-physical result to the solar electromagnetic domain for the first time.
1. Structural rigidity (Theorem 5): For each fixed ladder construction,
the verdict V(α,μ) and giant ratio GR(α,μ) are constant across the entire tested parameter
space — zero intra-dataset variation at any (α,μ) point. No exceptions across 93 datasets,
22,817 grid evaluations, 11 physical domains. No phase transition has been observed in any
tested system under joint (α,μ) deformation. This invariance holds within each fixed
construction; different constructions of the same physical system can yield different verdicts
(see point 3).
2. Trivial commutator: The structural commutator C(α,μ;L) = 0 for every tested
ladder, at every grid point. For any fixed ladder, the order in which α and μ are applied never
changes the structural outcome. Operator non-commutativity has not been observed in any tested system.
3. Representation is the primary structural variable: While constant deformation
never changes the verdict of a fixed ladder construction, the choice of ladder construction itself
does — sometimes categorically. The dominant source of verdict variation across the corpus is
representation, not deformation. The Zeeman reversal (7 class changes between batches),
the crystallography normalization split (cell_volume vs per_atom), and the cross-domain Si and
U-238 splits all confirm that realizability class is a joint property of physical system +
ladder encoding, not of the physical system alone. The canonical ladder problem remains the
highest-priority open question.