Structural Completeness, Robustness,
and Hierarchical Non-Isometry
Executive Summary
Five computational chambers — each a falsification instrument — answer a single question: What is the geometry of admissibility under recursive operator dynamics?
The answer arrived in stages. Each chamber had explicit falsifiers that could have terminated the arc. None did. Instead each result forced the next question, and the questions form a chain that closes on a structural theorem.
The chain: Interaction geometry is dimensionally bounded. Curvature shifts admissibility boundaries without breaking partition identity. No hidden mechanism classes exist. Factorization survives adversarial perturbation. Under operator tier change, factorization survives — but metric geometry stratifies.
The Structural Arc: Five Chambers, One Progression
The progression is not accidental. LI establishes geometric constraint. LII establishes curvature sensitivity. LIII–LIV establish partition invariance. LV reveals that invariance is structural but metric geometry is not. Non-isometry is the natural culmination of pre-metric geometric asymmetry elevated to a metric theorem.
Dimensional Interaction Constraint
Pairwise admissibility gates produce non-additive interaction geometry (Δ = 0.580, κ = 0.309). Triple compositions do not sustain independent interaction structure — exclusion geometry vanishes at n = 3.
n ≤ 2 CONSTRAINT CONFIRMEDCurvature-Responsive Bifurcation
Curvature deformation shifts admissibility boundary location. Identity partitions remain stable throughout. Geometry is not rigid — it is structurally anchored. This is pre-metric symmetry.
BOUNDARY MOTION CERTIFIEDChannel Completeness
Adversarial relaxation across 56,877 mechanisms reveals no hidden mechanism classes. All residual structure collapses into a single unified basin at the G3 bifurcation boundary. No fifth gate exists.
4-GATE BASIS CLOSEDFactorization Robustness
Four perturbation families, identity-preserving tracking. R2 = R3 = 0 throughout all perturbation types and strengths. Factorization is structural, not a tuning artifact.
R2 = R3 = 0.0000 CERTIFIEDTier-2 Selective Transfer
Channels commute under lift (R2 = R3 = 0 throughout). Retention geometry does not. V2×V4 exhibits systematic erosion 2–3× steeper than stable families. Normalization null rules out encoding artifacts.
HIERARCHICAL NON-ISOMETRYChamber LIV v1.1.0 — Witness-Separability Robustness
The question Chamber LIV asked is simple and important: Is the factorization structure we discovered a fundamental property, or was it an artifact of the specific domain we tested it in? The answer required putting the structure under systematic adversarial stress.
Protocol: Four Perturbation Families
Chamber LIV applied four distinct identity-preserving perturbation families to the validated Tier-1 witness pool (N = 2000 mechanisms per family). "Identity-preserving" means the same mechanism IDs are tracked throughout — changes in witness status reflect genuine structural change, not sampling noise.
The Four Robustness Metrics
Theorem 3.1 — Factorization Robustness (Certified)
Under all four perturbation families within the characterized domain, witness-separability and channel orthogonality are preserved. R2 = R3 = 0 under all perturbation types at all tested strengths. G3's R1 drop under P1 (encoding perturbation) is classified as parametric sensitivity — boundary thinning — not structural failure. The distinction matters: structural stability (R2, R3, R4 jointly) is separate from parametric robustness (R1).
Structural vs. Parametric: A Key Distinction
Chamber LIV introduced a separation that carries forward into all subsequent analysis. A failure of R1 alone, when R2–R4 pass, indicates boundary thinness — the witnesses for that gate sit close to the admissibility threshold, so perturbations can displace them. It does not indicate channel collapse — the gate's conceptual independence is intact. This separation proved critical for interpreting Chamber LV.
Chamber LV v1.1.0 — Tier-2 Selective Transfer Diagnostic
The question was no longer whether factorization survives perturbation within a tier. The question became: what happens when the operator tier itself changes? Does the factorization structure transfer, or does tier-change break it?
The answer was unexpected in its precision: factorization transfers completely — but geometry does not transfer uniformly.
Three Operator Families Under Test
Three Independent Structural Signals
Signal I — Channel Orthogonality Invariance (R2, R3)
Across all three families, all five τ₂ levels, and both evaluation tracks: R2 = 0.0000 and R3 = 0.0000 throughout. Not approximately zero. Exactly zero. Channel orthogonality is not a Tier-1 artifact. Witness-separability and factorization survive operator tier change completely.
Signal II — Retention Erosion: Monotone and Family-Specific
| Family | Gate | τ=0.1 | τ=0.3 | τ=0.5 | τ=0.7 | τ=0.9 | R5-T Slope | Class |
|---|---|---|---|---|---|---|---|---|
| V2×V3 | G1 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | +0.0000 | Stable |
| V2×V3 | G2 | 1.000 | 0.994 | 0.944 | 0.926 | 0.889 | −0.0290 | Stable |
| V2×V3 | G3 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | +0.0000 | Stable |
| V2×V3 | G4 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | +0.0000 | Stable |
| V2×V4 | G1 | 0.975 | 0.924 | 0.847 | 0.856 | 0.763 | −0.0492 | Mild erosion |
| V2×V4 | G2 | 0.883 | 0.722 | 0.586 | 0.377 | 0.340 | −0.1432 | Stratification |
| V2×V4 | G3 | 0.923 | 0.772 | 0.691 | 0.599 | 0.448 | −0.1122 | Stratification |
| V2×V4 | G4 | 0.967 | 0.874 | 0.742 | 0.642 | 0.517 | −0.1132 | Stratification |
| V6×V7 | G1 | 0.983 | 0.878 | 0.843 | 0.826 | 0.748 | −0.0522 | Mild erosion |
| V6×V7 | G2 | 0.980 | 0.932 | 0.816 | 0.803 | 0.769 | −0.0551 | Mild erosion |
| V6×V7 | G3 | 0.979 | 0.951 | 0.888 | 0.828 | 0.825 | −0.0432 | Stable |
| V6×V7 | G4 | 0.965 | 0.908 | 0.844 | 0.674 | 0.645 | −0.0872 | Mild erosion |
Signal III — The Normalization Null (Decision Criterion B1)
The critical falsification test. If V2×V4's retention erosion were caused by encoding artifacts — curvature-induced scale drift in the gate input distributions — then encoding-invariant normalization should restore retention. Three normalization maps were applied: dk_rel (G2), B_rel (G3), L_rel (G4).
The Prediction That Failed
The encoding artifact hypothesis predicted: normalization would restore V2×V4 retention to passing levels. The prediction failed at every gate, every τ₂ level, to six decimal places. Δ(passRate) = 0.0000000 everywhere. Not approximately zero. Structurally zero — because the normalization maps rescale thresholds proportionally with population statistics, absorbing global shifts but leaving witness-specific boundary erosion untouched. This is a structural null, not a null of insufficient statistical power.
| Family | Gate | Raw passRate | Norm passRate | Δ | Interpretation |
|---|---|---|---|---|---|
| V2×V3 | G1–G4 | 1.000 | 1.000 | 0.000 | Control |
| V2×V4 | G2 | 0.400 | 0.400 | 0.000 | Normalization insufficient → B |
| V2×V4 | G3 | 0.400 | 0.400 | 0.000 | Normalization insufficient → B |
| V2×V4 | G4 | 0.600 | 0.600 | 0.000 | Normalization insufficient → B |
| V6×V7 | G1–G4 | 1.000 | 1.000 | 0.000 | Control |
| V6×V7 | G4 | 0.600 | 0.600 | 0.000 | Control (mild erosion) |
Staged Degradation: A Geometric Signature
The migration metric (R4) reveals a particularly important ordering. Under V2×V4, the degradation proceeds in distinct stages — and the ordering of those stages carries structural information.
The ordering is the discovery: R1 fails first. R4 crosses threshold later. R2 and R3 never move from zero. This means witnesses lose single-gate identity before the partition structure breaks down, and migration follows erosion rather than preceding it. This is the signature of geometric boundary thinning, not channel coupling.
Hierarchical Stratification: Three Geometric Strata
Decision B — Confirmed: Both Criteria Triggered
- B1 (Normalization Null): V2×V4 fails after normalization. Δ(passRate) = 0.000000 across all gates. Encoding hypothesis falsified.
- B2 (Slope Stratification): V2×V4 normalized slopes (−0.11 to −0.14) are 2–3× steeper than stable families. Structural stratification confirmed.
Conclusion: Factorization survives operator tier change. Admissibility geometry does not lift uniformly. That is a stronger statement than universal transfer.
Factorization Inevitability: From Discovery to Necessity
The paper "Factorization Inevitability in Recursive DAG Admissibility" answers the deeper question: why must admissibility factorize? The shift from "we found four gates" to "factorization is inevitable" is the decisive compression moment.
Theorem 1 — Factorization Inevitability
If a gate set is witness-separable, admissibility irreducibly factorizes into k independent constraint channels.
- Irreducibility: No gate is implied by the conjunction of the remaining gates. Each has a witness violating only it.
- Channel-specific unlock: Each gate's witnesses become viable only when that specific gate is relaxed — not when any other is relaxed.
- Orthogonal failure modes: The failure channels are pairwise distinguishable by witnesses. No monolithic constraint can reproduce this structure — channel information is structurally irreducible.
Theorem 2 — The DAG Correspondence
Any genuinely independent new gate discovered in an extended domain must correspond either to refinement of existing DAG-level invariant classes, or to extension of the substrate model. In neither case does it falsify the factorization principle. The theory is refineable, not brittle.
The Four Invariant Channels
| Gate | Name | DAG Invariant | Predicate |
|---|---|---|---|
| G1 | Geometric Curvature | Path geometry | curvature_method = turning_angle |
| G2 | Baseline Separability | Node distinguishability | Δκ > 0.05 |
| G3 | Bifurcation Capability | Branch structure | bifurcation ∧ sharpness > 0.10 |
| G4 | Locality Consistency | Causal ordering | locality_score < 0.10 |
Why This Matters
Before: "We found four gates through chamber validation." After: "Admissibility must factorize; current domain reveals four channels corresponding to four DAG invariant classes." The four are not a list. They are a theorem. The 72% contraction of mechanism-class space is now understood as the joint restriction of four independent projection channels — each eliminating a distinct mechanism family.
The Unified Structural Theory
The paper "Structural Completeness, Robustness, and Hierarchical Non-Isometry" consolidates five chambers into a coherent theory with seven formal results.
1. Dimensional Interaction Constraint (LI)
Admissibility interaction geometry exists in n ≤ 2 compositions and collapses at n = 3. Interaction geometry is inherently low-dimensional.
2. Curvature-Responsive Bifurcation (LII)
Curvature deformation alters admissibility boundary position without altering partition identity. Geometry moves. Partitions hold. Pre-metric symmetry.
3. Local Structural Completeness (LIII)
No fifth independent gate exists within the characterized domain. The four-gate basis is closed and complete.
4. Factorization Robustness (LIV)
R2 = R3 = 0 under all four perturbation families at all tested strengths. Partition invariance is structural.
5. Projection Non-Isometry (LV)
Admissibility partitions are preserved under Tier-2 lift (R2 = R3 = 0), but retention geometry ρᵢ(τ₂) is not uniformly metric-preserving across operator families.
6. Critical Erosion Law
Migration onset satisfies τ_mig ≥ τc + (θ/A(F))^{1/p(F)}. Staged degradation has a predictable onset derivable from kernel parameters.
7. Categorical Formulation — The Cleanest Statement
The Tier-2 lift functor L is the identity on channel objects — empirically: R2 = R3 = 0 — but fails to be an isometry in the Lawvere-metric enriched category ChanF — empirically: self-distances d(i,i) = 1 − ρᵢ change as witnesses are lost.
Channel objects commute under lift. Metric geometry does not. That is the categorical core of hierarchical non-isometry. This is not collapse. This is stratification.
The Curvature Functional: Prediction Before Execution
The non-isometry constant LF — governing how steeply retention erodes — can now be predicted before running the chamber, from kernel bias parameters:
What Has Been Gained: A Structural Theory of Operator Stratification
The Vertical Progression
This is not incremental work. Five chambers answer five structurally ordered questions that form a complete arc:
| Chamber | Question | Result | Significance |
|---|---|---|---|
| LI | Does interaction geometry extend freely? | No. 2D-bounded. | Dimensional cap law |
| LII | Can curvature deform without breaking identity? | Yes. Geometry moves, partitions remain. | Deformation-without-collapse principle |
| LIII | Are there hidden admissibility channels? | No. 4-gate basis complete. | Completeness theorem |
| LIV | Is factorization robust under perturbation? | Yes. R2=R3=0. | Robustness theorem |
| LV | Does operator lift preserve admissibility geometry? | No. Partitions commute, metric stratifies. | Non-isometry theorem |
The Decisive Conceptual Contribution
Most frameworks collapse two distinct properties: partition structure (whether witnesses remain single-gate) and retention geometry (whether witness density is preserved under lift). These are independent. The data show partition structure is preserved and retention geometry is family-dependent. A framework that tests only partition structure would miss the hierarchical stratification entirely; a framework that conflates geometry failure with partition failure would incorrectly declare factorization broken. The LI–LV arc separates them rigorously.
Methodological Significance
This is not philosophical structuralism. Each chamber has explicit falsifiers, specified in advance. The normalization null was allowed to kill the hypothesis — it didn't. The metric upgrade did not overwrite earlier results. No step relies on reinterpretation of prior failure. That is methodological integrity, and it is rare.
Implications for Operator Theory and Physics
Operator Theory
Operator families form geometric strata. Not all lifts are equivalent. Hierarchy exists in operator geometry with direct implications for multi-scale systems, recursive constraint composition, and stratified admissibility spaces.
Systems Theory
Constraint composition is not freely extensible across dimensions. Curvature sensitivity produces stratified stability classes — resembling phase transitions and renormalization-type phenomena, but derived from a discrete admissibility substrate without assuming physics.
Computational
A kernel → curvature functional → non-isometry constant → migration onset pipeline now exists. Hierarchical stratification is predictable before chamber execution. Discovery becomes computation.
Key Numbers
The Clean Final Statement
No hype. No inflation. Each chamber had explicit falsifiers. No step relies on reinterpretation of prior failure. The normalization null was allowed to kill the hypothesis — it didn't. That is methodological integrity.
Chamber LV: CERTIFIED. Vertical Arc: CLOSED. Discovery first. Reformulation later.
References and Instrumentation
- Chamber Instrumentation Array: LI–LV Structural Arc — Interactive computational suite for all five chambers
- Structural Completeness, Robustness, and Hierarchical Non-Isometry — Unified structural theory, UNNS Collaboration, February 2026
- Factorization Inevitability in Recursive DAG Admissibility — Derivation and current classification, UNNS Collaboration, February 2026
- Chamber LV Certification Report — Tier-2 Selective Transfer, theorem certification
All chambers operate as self-contained falsification instruments with preregistered protocols. UNNS Research Collective, February 2026.