Abstract
We present the UNNS Substrate's first quantitative admissibility phase framework, emerging from a four-chamber seismic analysis suite (LXV) applied across three independent rupture systems. Prior to this work, the Rigid–Nonrigid Principle (RNP) was a structural separation criterion — a conceptual distinction between structures that descend and those that are representational artifacts. It is now a measurable phase theorem with computable descent conditions, boundary degeneracy bounds, and matching-theoretic inversion budgets.
The central empirical discovery: displacement fields admit a single global orientation unless invariance stability forces minimal decomposition. Kumamoto (k=2), Ridgecrest (k=1), and El Mayor (k=1) present both cases in clean contrast. Magnitude does not trigger splitting. Directional incompatibility under admissible operator nesting does. This is not rhetoric. It is a falsifiable, cross-validated pattern — and it is exactly what a substrate-level structural law looks like.
📐 Key Numerical Results
Before any narrative, the numbers speak:
🎯 The Central Question
In the UNNS framework, admissibility has long been conceptual: some structures descend under operator nesting; others collapse into representational artifacts. But where exactly is the boundary? When does a single global description suffice, and when is mechanism splitting forced by necessity?
Chamber LXV was designed to answer this on real seismic data — not synthetic grammars, not toy operators. Three independent earthquake rupture systems. GPS displacement fields. A deterministic preprocessing protocol. Preregistered station sets. And a single falsification criterion embedded before any run:
The Preregistered Criterion
A topology claim is only admissible if it survives the full smoothing sweep {1, 3, 7, 14, 21} days plus eligibility gates (SNR + amplitude thresholds). If k=2 is detected but collapses under any smoothing window, it is rejected as an artifact. Only filter-invariant structure counts.
This converts smoothing from a modeling choice into an adversarial invariance test.
🔬 The LXV Chamber Suite
Four chambers, three earthquake systems, two independent structural axes. Each chamber is a falsification instrument — engineered to reject the structural hypothesis if it can be broken, not to confirm it.
Kumamoto 2016 M7.0
Complex bilateral rupture with strong directional incompatibility across fault segments. GPS displacement hierarchy extremely coherent. Strong displacement magnitude gradients preserved across all smoothing windows.
Rank Stability: FULL · Topology: k = 2 (forced bifurcation) · Sep. Score: PASS
PASS — Two-lobe structure confirmedEl Mayor–Cucapah 2010 M7.2
Strong southern stations (P497, P493). Direction-stable across smoothing — but strict monotonic ordering fails in B2. A boundary degeneracy case: |V| = 2 vulnerable gaps but matching number ν(V) = 1.
Rank Stability: PARTIAL · Topology: k = 1 · Surprise: ν(V) = 1 despite |V| = 2
BOUNDARY — Structured degeneracy at marginRidgecrest 2019 M7.1
Expanded station pack including high-amplitude stations P595, CCCC. Despite large step magnitudes, directional field remains globally coherent. A definitive demonstration that magnitude alone cannot trigger topology splitting.
Rank Stability: PERFECT · Topology: k = 1 · Key: Large ≠ differentiated
PASS — Single global orientation survivesCross-System Topology
Applied across all three event systems. Computes orientation lobe count (k), ARI stability across smoothing, spatial separability score, and eligibility robustness. The universal test: can the field be globally oriented?
Protocol: Preregistered · Gates: SNR + amplitude · ARI: high across all PASS cases
BIFURCATION AS NECESSITY — not feature💡 The Central Discovery
Across three independent rupture systems with preregistered protocols, a single discriminating principle emerged with clarity that is rare in structural analysis:
This is precisely what a substrate-level law should look like: it defines when something happens and when it cannot. It presents both cases. And it explains the mechanism.
Why Kumamoto Splits and Ridgecrest Does Not
Kumamoto involves a complex bilateral rupture with directional competition between fault segments — two independently oriented displacement regimes that cannot be collapsed into a single global orientation without violating smoothing invariance gates. The substrate must differentiate.
Ridgecrest, despite slightly higher magnitude, involves a geometrically coherent linear strike-slip system. The displacement field remains directionally unified. Even with strong-amplitude stations (P595, CCCC) added in C2, the field holds. Large amplitude is not the trigger. Geometry is.
The Deeper Pattern
Very few structural frameworks can cleanly show: when splitting happens, when it doesn't, and why. The LXV suite presents all three. It does so with preregistered stations, deterministic preprocessing, and adversarial smoothing — the full apparatus of invariance-certified structural analysis.
📏 What This Phase Gained — The New Invariants
Before this work, "rigid vs nonrigid" was a conceptual distinction at the level of RNP principles. Now it is measurable. Four new computable invariants convert phase talk into operator-computable quantities:
Before LXV
- RNP was a structural separation principle (conceptual)
- Inversion budget: "allow one adjacent swap at N=5" (heuristic)
- Rank, topology, degeneracy, smoothing drift: separate phenomena
- Stability was binary: admissible / inadmissible
- Certification thresholds were tuned
After LXV
- RNP has a measurable phase coordinate ℛ > (1,1)
- Inversion budget: k_min = ν(V) — structurally necessary
- All four phenomena are manifestations of bounded admissible deformation
- Admissibility has interior, boundary, and fragmentation regime
- Certification thresholds are derived, not tuned
The Four New Phase Invariants
| Invariant | Measures | What It Enables | Status Before |
|---|---|---|---|
| ℛ_mag | Magnitude rigidity modulus | Computable descent condition: ℛ > (1,1) | Not defined |
| ℛ_geo | Geometric rigidity modulus | Quantifies direction-space margin | Not defined |
| 𝒟 | Degeneracy index |V|/(N-1) | Predicts boundary regime size | Heuristic only |
| ν(V) | Matching number of gap graph | Derives inversion budget k_allowed | Empirical approximation |
| 𝒜 = min(ℛ_mag, ℛ_geo) | Admissibility margin | Universal law-detection filter | Entirely absent |
The El Mayor Surprise — A Structural Insight
The El Mayor B2 case revealed something that was not anticipated. With two vulnerable gaps (|V| = 2), one might expect two independent inversion degrees of freedom. Instead, the matching number ν(V) = 1.
Degeneracy Is Constrained by Path Matching Geometry
Vulnerable gaps are not independent. Their combinatorial structure — specifically, the maximum matching in their path graph — constrains the allowed inversion count. This was not obvious beforehand. Degeneracy is not arbitrary chaos at the boundary: it is combinatorially gated. Fragmentation does not occur immediately when the admissibility margin is breached. It is delayed by matching geometry. That is a substrate-level insight.
🔷 The Phase-Boundary Theorem Inside RNP
The central formal result emerging from this analysis is a computable descent theorem — the first time RNP has a quantitative boundary rather than a conceptual one:
If ℛ ≤ (1,1), rigidification fails by contradiction: no globally admissible nesting can maintain the single-field representation.
This is not empirical. It is a formal descent result inside admissibility geometry. RNP now has a computable phase coordinate.
The consequence for the UNNS program is significant: this converts RNP from a meta-principle to a phase theorem. The boundary is not inferred from data post-hoc — it is derivable from the operator structure and confirmed empirically across three independent earthquake systems.
Five Immediate Implications
(1) Certification thresholds can be derived, not tuned. (2) Structural signatures can be ranked by admissibility margin 𝒜. (3) Degeneracy can be predicted from baseline gap geometry. (4) Phase transitions can be located without model replacement. (5) Admissibility can serve as a universal law-detection filter.
🔗 Position Within UNNS — Cross-Branch Alignment
The LXV suite does not stand alone. It represents a convergence point for several prior UNNS branches, and it extends each of them in a distinct direction:
| UNNS Branch | Prior State | What LXV Adds | Relation |
|---|---|---|---|
| Axis I–II (Grammar / Utility) | Studied creation of structure | Studies persistence of structure | Complementary |
| Axis III (Observability / Interface) | Qualitative threshold | 𝒜 = min(ℛ_mag, ℛ_geo): computable threshold | Elevates |
| RNP (Rigid–Nonrigid Principle) | Meta-principle, no coordinate | Measurable coordinate ℛ > (1,1) | Transforms |
| Phase P₃ (Curvature Bifurcation) | Differentiation when single mechanism fails | Same motif, now on real GNSS fields | Strongest alignment |
| Chamber Work (LXV and before) | Empirical, diagnostic, falsification-oriented | Abstracted into envelope geometry + theory | Maturation |
Comparison to External Theoretical Traditions
The UNNS admissibility phase framework touches established traditions while differing in one critical respect: stability is defined relative to admissible operator nesting, not arbitrary perturbations. That distinction is what gives UNNS its unique character.
Structural Stability (Smale, Peixoto)
Classical structural stability studies persistence under arbitrary small perturbations. UNNS studies persistence under the admissible operator family — a constrained subset. The admissibility constraint is unique to UNNS. It is what makes the margin 𝒜 computable and meaningful.
Topological Data Analysis (Persistent Homology)
TDA studies topological features persisting across scale changes. LXV studies invariance-certified partition under resolution perturbation. Philosophically similar, but with an operator grammar constraint, a matching-theoretic degeneracy bound, and an RNP descent condition that TDA does not have. The LXV protocol is also more auditable and directly preregisterable.
Phase Transitions (Statistical Physics)
The tripartite structure — rigid interior, boundary degeneracy, fragmentation regime — looks like a phase diagram. But the phase coordinate here is structural margin, not a thermodynamic parameter. The transition condition is algebraic, not thermodynamic. The mechanism is operator geometry, not statistical fluctuation.
🌐 Why This Matters — The Architectural Shift
It is tempting to read the LXV suite as a seismological study. It is not. It is a demonstration that the UNNS Substrate framework — developed through synthetic operator grammars, spectral invariants, and recursive dynamics — produces falsifiable, cross-validated structural laws on real physical data.
UNNS Before LXV
UNNS diagnosed structural limits. It could identify when structures collapsed, when admissibility was breached, when recursive operators failed to produce utility. It was a diagnostic framework operating primarily on synthetic and calibration data.
UNNS After LXV
UNNS measures structural distance to limits. With computable margins, matching bounds, and a phase coordinate, it now predicts inversion counts, boundary degeneracy sizes, and topology flip conditions — on real-world displacement fields from independent earthquake systems.
The Methodological Shift Is Not Incremental — It Is Architectural
For the first time in the UNNS Substrate program, descent, degeneracy, geometry, operator nesting, and RNP are unified in one mathematical framework. With a computable admissibility phase diagram that covers the complete topology of structural stability — interior, boundary, and fragmentation regime — the framework has crossed from principle to theory.
The difference between principle and theory is the difference between "some structures survive admissible nesting" and "here is the exact margin by which they survive, and here is the exact combinatorial limit of degeneration at boundary."
The Substrate Is Not Binary — It Is Margin-Structured
The UNNS Substrate is not admissible/inadmissible. It has interior geometry, boundary geometry, and a fragmentation regime. Admissible operators behave like constrained contraction systems — they deform structure within a bounded envelope and induce structured degeneration near boundary. This generalises far beyond seismic displacement fields.
UNNS Is Now Predictive, Not Only Diagnostic
Previously, UNNS diagnosed collapse, instability, and emergence. Now it predicts: maximum inversion count (k_allowed = ν(V)), boundary degeneracy size (𝒟 from gap geometry), when topology will flip (ℛ crossing (1,1)), and when it structurally cannot flip. That is predictive structural mathematics.
📊 Smoothing as an Adversary — Invariance-Certified Results
A key methodological contribution of LXV is the treatment of smoothing as a falsification instrument rather than a modeling parameter. Standard practice in geophysics is to choose a smoothing window that produces clean results. LXV inverts this: a result is only valid if it survives all windows in the sweep.
Standard Approach
Pick a filter that looks good. Results are smoothing-sensitive. Parameter-tunable. Not preregistered.
LXV Approach
Filter-invariant structure only counts. Full sweep {1,3,7,15,21} days required. Only invariance-surviving claims advance.
The critical methodological point: the "direction weirdness" encountered in early El Mayor processing was traced to a CSV adapter path that introduced representation artifacts. The tenv3-native parsing path restored directional coherence. This is a concrete demonstration of what the RNP branch has always argued: representation artifacts exist; invariance filters detect them. The LXV suite supplied real empirical evidence for that abstract methodological claim.
📦 Resources — Data & Replication
All station data, topology profiles, and the interactive Chamber Array are available for independent verification. The entire pipeline is deterministic: same station set, same preprocessing protocol, same smoothing sweep — same results.
Paper & Interactive Chamber
Station Data Downloads
References & Prior Art
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Full Paper:
Perturbation-Admissible Structural Stability in the UNNS Substrate (PDF)
Complete formal treatment including phase invariants, descent theorem, matching-theoretic inversion budget, and full El Mayor analysis. -
Chamber Array LXV (Interactive):
chamber_array_lxv.html
Live interactive chamber suite. All four chambers (A, B2, C2, D) with full panel outputs, station eligibility maps, and smoothing-sweep stability visualisations. -
Prior UNNS Arc — Curvature Sign Preservation (LXI–LXIV):
chamber_array_lxi_lxii_lxiii_lxiv_analysis.html
The preceding arc established sign-preservation of the curvature scalar under admissible recursion across 7,400+ records. LXV extends the admissibility framework from operator space into real seismic displacement fields. -
UNNS Substrate — Phase P₃ (Curvature-Responsive Bifurcation):
chamber_lii_v1_3_2_CURV_RESP_ENCODING_FIXED
The Phase P₃ motif — differentiation appears when a single mechanism cannot satisfy constraints — is the direct theoretical precedent for the LXV empirical finding. LXV is the first real-world instantiation of P₃ bifurcation logic. -
Rigid–Nonrigid Principle (RNP):
The Rigid-Nonrigid Transition Principle
The foundational UNNS structural separation principle that LXV elevates from meta-principle to computable phase theorem via the descent condition ℛ > (1,1).