Structural Phase Diagnostics
Across Physical Domains:
Gravity Joins Seismology and Cosmology
Executive Summary
The UNNS cross-domain program tests whether structural admissibility signatures — patterns predicted by the substrate framework — appear consistently across physically unrelated systems. Two domains had already been examined: the cosmic microwave background (CMB multipole structure) and global earthquake distributions (seismic arc geometry). Both produced clear structural contrasts between real and synthetic systems.
This report introduces the third domain: planetary gravity fields. Chamber GRAV-I applies a spectral axis dominance diagnostic to spherical harmonic decompositions of Earth, Moon, and Mars gravity models, sweeping the harmonic degree from L = 2 to L = 300+. All three planetary bodies exhibit persistent distributed anisotropy — directional structure that survives spectral extension — while synthetic random fields behave qualitatively differently.
"The same structural diagnostic framework — built from admissibility geometry — produces meaningful, consistent output across seismology, cosmology, and planetary gravity. This is not a coincidence. It is the fingerprint of a substrate-level structural law."
1The Cross-Domain Program and Why Gravity Was Next
The UNNS substrate framework makes a specific structural prediction: admissible operator sequences preserve invariants. Not approximately. Not on average. Persistently — across the entire operator sweep, for any system that sits inside the stable interior of admissibility geometry.
That prediction is, on its own, a mathematical result. But what makes it extraordinary is its implication for the physical world: if real physical systems genuinely occupy interior regions of admissibility space, the same structural signatures should appear across completely unrelated physical domains. The mechanism changes entirely. The material changes entirely. The scale changes by many orders of magnitude. And yet the structural pattern should repeat.
Core Hypothesis of the Cross-Domain Program
Structural admissibility signatures are domain-transcending. They arise not from specific physical mechanisms, but from the placement of real systems within the admissibility manifold. Any physical system that sits in the interior of that manifold will exhibit the same class of structural behavior under operator extension — regardless of whether the underlying physics is tectonic stress, primordial density fluctuations, or planetary mass distribution.
The program therefore proceeds by selecting domains that are maximally independent in their generating physics and testing each with a purpose-built chamber that operationalizes the same diagnostic logic: vary representation scale, track structural invariants, compare real against synthetic. The pattern that appears — or fails to appear — directly constrains the geometry of the underlying substrate.
Why Gravity is the Right Third Domain
After cosmology and seismology, planetary gravity was selected for three reasons. First, independence of mechanism: gravity fields arise from mass distribution, entirely unlike radiation transport (CMB) or fault-stress dynamics (seismology). Second, canonical harmonic structure: gravity potential is naturally expressed in spherical harmonics, making spectral extension a well-defined and physically unambiguous operation. Third, multiplicity of bodies: Earth, Moon, and Mars offer three real-world gravity fields with sharply different internal structure, providing internal cross-checks within the domain before comparison across domains.
2What the Earlier Domains Already Showed
Before examining gravity, it is worth stating clearly what the earlier chambers established — because GRAV-I must be read as the continuation of a pattern, not as an isolated experiment.
CMB Multipole Alignment
The CMB chambers (CMB-I through CMB-III-FULL, CMB-SPECTRA-Σ) analyzed Planck 2018 TT/TE/EE data. The operator sequence was multipole truncation increasing from low ℓ upward. The invariant tracked was the quadrupole–octopole alignment angle θ₂₃.
Real sky: θ₂₃ stabilizes early and survives the entire spectral sweep. Under Monte Carlo null distributions, the observed alignment is statistically anomalous. Bootstrap-corrected effect sizes confirm the signal persists with high confidence.
Structural signature: confirmedEarthquake Arc Geometry
The LXV chamber series (A, B1, B2, C2, D) analyzed GPS displacement fields from three major earthquakes (Kumamoto 2016, Ridgecrest 2019, El Mayor–Cucapah 2010) using 13–16 stations per event.
The operator sequence was progressive smoothing of the displacement field. The invariant was the bilobe partition structure of station displacements. Real earthquake data: bilobe topology appears immediately and survives full smoothing sweep. Synthetic displacement fields: topology degrades under the same sweep.
Structural signature: confirmedPlanetary Spectral Axis Dominance
Chamber GRAV-I applies a harmonic extension operator to spherical harmonic gravity potential coefficients. The operator sequence runs from degree L = 2 upward to full model resolution (L up to 720 for Earth, 300 for Moon, 85 for Mars).
The structural invariant measured is the axis dominance gap J₁ − J₂. This quantifies how strongly the gravity field is organized along a single spectral axis versus being distributed across multiple competing directions.
Structural signature: confirmed — see belowThe Pattern Common to Both Earlier Domains
In both cosmology and seismology, the key observation was the same: real systems lock their structural invariant early in the operator sweep, and hold it through the entire sequence. Synthetic systems start near the boundary of admissibility space and cross it quickly. This early-locking / late-degradation contrast is the operational signature of deep interior placement in the admissibility manifold.
3Why Gravity Fields are an Ideal Structural Test
Planetary gravity fields offer something neither the CMB nor seismic data can provide: a clean, physics-anchored preferred axis. Every rotating body has a rotation axis, and that axis produces the dominant J₂ (C₂₀) term in the spherical harmonic expansion. This physical axis is not inferred from the data — it is known a priori from orbital mechanics.
This makes gravity a higher-symmetry test case than either seismology or cosmology. Earthquake fields have no global symmetry axis — only local fault geometry. The CMB is statistically isotropic with relational (not absolute) directional invariants. Gravity sits at the high-symmetry end: it has a built-in physical preferred direction, and the question is simply whether the harmonic decomposition preserves that direction as L increases.
It is worth emphasizing what the test is not. It is not checking whether a preferred axis exists in the low-degree terms — that is trivially expected from physics. It is checking whether the dominance structure of the spectral decomposition is consistent with admissibility geometry as one extends to higher and higher harmonic degrees. Does the axis remain dominant, or does the growing weight of higher-degree terms erode its signature?
The Key Structural Question for Gravity
As harmonic degree L increases from 2 toward 300 or beyond, does the spectral axis dominance gap
J₁ − J₂ remain stable? Or does the accumulation of higher-order harmonic content
wash out the directional structure, producing a uniform distribution of spectral power across
all axes?
4Chamber GRAV-I: Structural Phase Diagnostics
Chamber GRAV-I v2.0.0 implements a single, precisely defined diagnostic: the spectral axis dominance gap. Given a spherical harmonic gravity field truncated at degree L, the chamber computes the power distribution across all possible spectral axes (test directions on the sphere), identifies the best-fitting axis (score J₁), the second-best (J₂), and reports the gap J₁ − J₂.
Axis Dominance Diagnostic
Admissibility Class Definitions
The gap value maps to a structural class, providing a categorical verdict for each body at each L:
| Class | Gap Range | Structural Interpretation | Expected for |
|---|---|---|---|
| I | Δ > threshold_I | Strong global axis — single direction dominates the spectral decomposition | Highly symmetric rotating bodies; low-degree models |
| II | threshold_II < Δ ≤ threshold_I | Moderate anisotropy — preferred direction present but with competition from secondary axes | Real planetary bodies with complex internal structure |
| III | Δ ≤ threshold_II | Distributed multi-directional structure — no single axis dominates; power is spread | High-degree models; bodies with strong competing structures |
| 0 | Δ ≈ 0 | Degenerate / random — spectral power is isotropically distributed; no preferred direction | Synthetic random fields; white-noise gravity models |
Important: Class III ≠ Structural Absence
A Class III result does not mean a planet lacks structure. It means that the structure is distributed across multiple competing directions — a richer anisotropy than a single-axis case. This is the physically expected outcome for bodies with multiple large-scale mass anomalies. The distinction between Class III (distributed anisotropy) and Class 0 (random) is the key discriminator for the cross-domain comparison.
Dataset Sources
GRAV-I processes spherical harmonic coefficient files in a standardized JSON schema derived from the official models. The chamber accepts both preloaded model files and the reference URLs below.
| Body | Model | L_max | Source File |
|---|---|---|---|
| Earth | EIGEN-6C4 | 720 | grav_i_eigen6c4_L720_subset.json |
| Moon | GL1500E | 300 | grav_i_moon_L300_subset.json |
| Mars | GMM-3 | 85 | grav_i_mars_L85_subset_chamber_schema.json |
| Synthetic | Mulberry32 seed 1337 | 300 | grav_i_synth_random_L300_seed1337.json |
5Results: What the Gravity Chamber Found
Primary Results Table
| Body | Model | L Range Tested | Axis Dominance Gap Δ | Class | Axis Stability |
|---|---|---|---|---|---|
| Earth | EIGEN-6C4 | 2 → 720 | ~10⁻³ | III | Persistent throughout sweep |
| Moon | GL1500E | 2 → 300 | ~10⁻³ | III | Persistent throughout sweep |
| Mars | GMM-3 | 2 → 85 | ~10⁻³ | III | Persistent throughout sweep |
| Synthetic | Random seed 1337 | 2 → 300 | ≈ 0 | 0 | No persistent structure |
The schematic above captures the key behavioral pattern. All three planetary bodies exhibit a similar trajectory: the axis dominance gap starts larger at low L (where the rotation-aligned J₂ term strongly dominates), then decreases as higher-degree terms add directional competition, and stabilizes within the Class III band — a persistent distributed anisotropy that survives the entire sweep. The synthetic random field shows no analogous structure: its gap remains near zero at all L, distinguishing it cleanly from all three real bodies.
The Real vs. Synthetic Contrast
Real Planetary Bodies
- Axis dominance gap stabilizes in Class III (Δ ~ 10⁻³)
- Structural locking occurs early in the sweep (L ~ 10–20)
- Gap is persistent: does not degrade at high L
- Same class across all three bodies despite very different internal structures
- Moon shows greatest variation but remains in Class III
Synthetic Random Field (seed 1337)
- Gap remains near zero at all L: Class 0
- No early structural lock — isotropic throughout
- Any apparent axis at low L is immediately erased by higher-degree noise
- Behavior consistent with boundary-regime placement in admissibility geometry
- No convergence to any stable directional structure
Why the Moon is Especially Significant
Earth and Mars both have obvious structural reasons for a preferred axis (rotation-induced oblateness, large-scale mass asymmetry). The Moon is more challenging: its mascon basins introduce concentrated mass anomalies distributed across the surface, the dominance ratio of the J₂ term drops substantially at higher degrees, and the overall field is rougher than either Earth or Mars.
Despite this, the Moon remains in Class III throughout the sweep. This is the most discriminating result within the gravity domain: it demonstrates that the persistent distributed anisotropy is not simply a consequence of overwhelming J₂ dominance. Something deeper anchors the directional structure, and the chamber resolves it cleanly.
6Structural Interpretation: Why Distributed Anisotropy?
The Class III result across all three bodies — distributed anisotropy rather than strong single-axis dominance — is not a failure of the diagnostic. It is the structurally correct outcome for real planetary bodies, and it is precisely what the admissibility framework predicts for interior-placed physical systems.
Real planetary gravity fields contain multiple competing large-scale structures, each contributing directional power to the spectral decomposition:
Multi-Scale Mass Anomalies
Equatorial bulge (J₂) provides primary axis. Continental masses, ocean basins, and mantle convection cells add competing secondary structures at higher L. The result is multi-directional spectral power that cannot be compressed to a single dominant axis.
Mascon Basins
Mascon basins (mass concentrations under large impact craters) produce concentrated gravity anomalies distributed across the surface. These compete strongly with the rotation axis at intermediate L values, producing the most variable Class III signal among the three bodies.
Tharsis & Crustal Dichotomy
The Tharsis volcanic plateau represents a uniquely large mass anomaly — essentially a continent-sized volcanic construct on one side of the planet. Combined with the crustal dichotomy (northern lowlands vs. southern highlands), Mars has strong competing spectral axes across many degree ranges.
The Deep Point: Distributed Anisotropy IS the Structural Signature
A real planet with multiple large-scale mass anomalies should produce Class III. A planet producing Class 0 would be physically impossible — it would require perfect spherical symmetry in the mass distribution. The fact that all three real bodies settle in Class III, while the synthetic random field settles in Class 0, shows that the chamber is correctly discriminating between organized multi-directional anisotropy (physical) and disorganized random isotropy (synthetic). These are structurally distinct regimes of admissibility geometry.
7Cross-Domain Comparison: Three Systems, One Framework
The value of GRAV-I is not visible in isolation. It becomes visible only when placed beside the seismology and cosmology results and read as the third independent probe of the same structural geometry.
The table version of the same comparison makes the structural regularity explicit:
| Domain | Physical Mechanism | Operator Family | Structural Invariant | Symmetry Class | Result |
|---|---|---|---|---|---|
| Cosmology | Primordial density fluctuations | Spectral truncation ℓ↑ | Quadrupole–octopole angle θ₂₃ | Intermediate · relational | Persistent ✓ |
| Seismology | Tectonic fault-stress dynamics | Displacement smoothing ↑ | Bilobe partition topology | Low · topological | Persistent ✓ |
| Gravity | Planetary mass distribution | Harmonic extension L↑ | Axis dominance gap Δ | High · distributed axis | Persistent ✓ |
| Synthetic (all) | Random / noise generation | Same as above | Same diagnostic applied | None · boundary-like | Breaks ✗ |
8The Emerging Phase Geometry of the Admissibility Manifold
The cross-domain pattern becomes most meaningful when interpreted geometrically. The UNNS framework proposes that operator space has a phase structure: an interior region of stability where invariants persist, and a boundary region where they break. Each experimental domain tests the boundary from a different operator direction.
The geometry above is schematic but structurally precise: each experimental domain tests the interior-versus-boundary distinction from a different operator direction. The striking result is that all three tests locate real systems in the same region — the stable interior — while synthetic systems cluster near the boundary, regardless of which direction the probe comes from.
A Constraint on the Geometry: The Interior Must Be Large
If the interior stability region were narrow, different operator directions would produce inconsistent results — some domains finding stability, others finding breakdown, depending on whether the particular operator direction happens to cross the boundary before or after the real systems. That is not what the data shows. All three domains show stability for all real systems throughout their full operator sweeps. This implies the interior basin is broad, with real physical systems well inside it — not marginally inside.
Early Locking: The Signature of Deep Interior Placement
Across all three domains, the structural invariant locks in early in the operator sweep. The CMB θ₂₃ angle stabilizes at low ℓ. Seismic bilobe topology appears at small smoothing windows. The gravity axis dominance gap settles in Class III by L ~ 10–20. After this early lock, all three invariants are essentially stationary throughout the remaining sweep.
This is the quantitative signature expected for systems with deep interior placement. A system near the boundary would show late locking (if at all) and early degradation of the invariant under small perturbations. A system far inside the interior locks immediately, because the structure is so redundantly encoded in the representation that no reasonable operator sweep can dislodge it.
9Implications for the UNNS Substrate
Three independent experimental campaigns, each targeting a completely different physical domain, have now returned the same structural verdict. The significance of this convergence cannot be assessed within any single domain — it only becomes visible at the cross-domain level.
What the Three Domains Together Establish
1. Structural Admissibility Signatures are Not Domain-Specific
The same diagnostic framework — vary representation scale, track invariants, compare real against synthetic — produces coherent, consistent results across seismology, cosmology, and planetary gravity. The structural contrast between real and synthetic systems is reproducible across mechanisms, scales, and invariant types. This rules out explanations based on domain-specific physics.
2. Symmetry Class Determines Invariant Carrier, Not Stability Itself
The type of invariant that persists (topological, relational, distributed anisotropy) is determined by the physical symmetry class of the domain. But the fact of persistence is independent of symmetry class. This suggests that the stability phenomenon is governed by the admissibility geometry of the substrate, while the specific invariant carrier is a secondary projection determined by domain-level symmetry.
3. Gravity and CMB May Share an Admissibility Surface
Both gravity and CMB use spherical harmonic decompositions as their representation, and both extend spectrally by increasing the maximum degree L. This structural similarity suggests they may lie on the same admissibility surface — harmonic directional rigidity under nested spectral extension — with different symmetry realizations. Gravity produces an absolute axis invariant; CMB produces a relational angle invariant. But the stability mechanism may be the same.
4. The Chambers are Mapping the Admissibility Manifold
Each experimental domain provides a slice of the admissibility manifold — a probe of its phase structure from a specific operator direction. With three slices now aligned, the global geometry of the manifold becomes increasingly constrained. The evidence points to a structure with a large stable interior basin and a sharp instability boundary, with real physical systems consistently occupying deep interior positions.
What Remains Open
The experimental evidence establishes the pattern. Several important theoretical questions remain open:
- Analytic derivation of the stability thresholds (C*, μ-margin, and the Class I/II/III boundaries) from first principles of admissibility geometry, rather than empirical fitting.
- Formal characterization of the admissibility surface shared by gravity and CMB — whether it has a precise algebraic definition within the UNNS framework.
- Extension to additional physical domains: if the pattern is truly substrate-level, it should appear in further domains (fluid dynamics, quantum systems, electromagnetic fields) under analogously constructed operator families.
- The operator chamber arc (Axis IV–V) provides a fourth probe direction (mechanism complexity rather than representation scale), and integrating it with the three physical domains may reveal the full 2-dimensional structure of the admissibility plane.
10Conclusion
Chamber GRAV-I adds planetary gravity fields as the third independent domain in the UNNS cross-domain certification program. Earth, Moon, and Mars all exhibit persistent Class III distributed anisotropy across their full harmonic sweeps, contrasting cleanly with synthetic random gravity fields that show no persistent directional structure.
Taken in isolation, this is a clean result about planetary gravity. Taken as the third member of a cross-domain series, it is something considerably larger: a third independent confirmation that real physical systems occupy structurally stable interior regions of admissibility geometry, while synthetic and random systems do not.
Chamber GRAV-I v2.0.0 implements the spectral axis dominance diagnostic with preregistered parameters, frozen datasets, and explicit falsification criteria. All results are available for independent reproduction via the chamber interface and data links below. The companion manuscript provides full derivations, dataset specifications, and formal admissibility class definitions.
References and Data Links
- CHAMBER GRAV-I v2.0.0 — Structural Phase Diagnostics (interactive chamber)
- Directional Rigidity of Planetary Gravity Fields Under Harmonic Extension (manuscript PDF)
- Earth — EIGEN-6C4, L_max = 720 (grav_i_eigen6c4_L720_subset.json)
- Moon — GL1500E, L_max = 300 (grav_i_moon_L300_subset.json)
- Mars — GMM-3, L_max = 85 (grav_i_mars_L85_subset_chamber_schema.json)
- Synthetic Random — Mulberry32 seed 1337, L = 300 (control dataset)
- Seismology deformation data archive (LXV series) — Kumamoto, Ridgecrest, El Mayor–Cucapah
- Planetary gravity fields archive — Earth, Moon, Mars models
- Cosmology / CMB archive — Planck 2018 TT/TE/EE data
- Core theoretical papers and certification reports of the UNNS Substrate program (v1 archive .zip) — includes Admissibility Depth & Submultiplicativity, Convex Sign Preservation, Curvature Sign Preservation, Directional Rigidity Under Multipole Truncation, Factorization Inevitability in Recursive DAG Admissibility, LV Certification Report, Structural Position of the Structural Lawhood Framework, Operator–Manifold Admissibility Geometry, Perturbation-Admissible Structural Stability, Structural Completeness/Robustness/Hierarchical Non-Isometry, Structural Lawhood as Interior Admissibility Phase Geometry, UNNS Observability–Admissibility Duality Theorem, UNNS Chambers Certification Report.