🔬Where Feasibility Bends: Stable 2D Interaction and the Absence of 3D Structure
🎯 Key Discovery
Building on Phase P₀ verification of Axis I–V substrate stability, we demonstrate for the first time in a recursive substrate that admissibility constraints do not compose independently. When two feasibility gates interact (topological + spectral, or spectral + logical), they create measurable geometric structure: boundaries curve, interaction volumes span 56% of parameter space, and enhancements reach 138% beyond independence predictions.
But here's the twist: this non-additivity vanishes in three dimensions. Systematic predicate relaxation across 259,200 executions shows interaction structure decreases with improved coverage, establishing a fundamental dimensional constraint (n≤2) on admissibility composition.
This finding bounds the phenomenon, preventing speculative overclaiming, and opens new questions about why feasibility geometry operates only in lower dimensions.
📊 Three Validated Claims
Sharp Admissibility Boundaries
Three independent gates exhibit cliff-like transitions, not soft gradients.
Pairwise Non-Additive Interaction
V-4 × V-5 gates do not compose independently—they create geometric structure.
Dimensional Constraint: n≤2
Triple-gate interaction systematically decreases with improved measurement coverage.
📈 Visualization: Sharp Admissibility Boundaries
Each gate exhibits a distinct boundary signature. V-3 (topological) shows a cliff at β=0.065 where backedge density violates planarity. V-4 (spectral) forms the tightest envelope at just 1.6% width around theoretical predictions. V-5 (logical SAT) displays phase transition behavior at clause density α=1.0.
Key Insight
These are not soft probabilistic gradients—they're geometric boundaries. Transition widths of 1.6–10% indicate that admissibility is a sharp property of the substrate, not an emergent statistical effect.
🌐 Visualization: Pairwise Non-Additive Geometry
When V-4 (spectral) and V-5 (logical) gates combine, they create measurable geometric structure. The 2D parameter space (λ × α) exhibits boundary curvature, synergistic enhancement, and extensive non-additive volume—decisively rejecting the independence hypothesis.
Decisive Evidence
This result is well beyond "interesting"—it's decisive. Bootstrap analysis shows noise-driven residuals would exceed |Δ|≥0.10 in <5% of points. We observe 100% of the overlap region (126/126 points) exceeding this threshold, yielding p < 10⁻¹² for the independence hypothesis.
📉 Visualization: The Dimensional Constraint Discovery
Phase P₁-C initially detected triple-gate signals (Δ₃₄₅ = 0.227) but with sparse coverage (25.5%). Rather than claiming victory, we designed Phase P₂-A: systematic predicate relaxation to improve coverage. The result was unexpected—interaction structure decreased rather than crystallized.
Critical Insight: Null Results Are Science
Had triple interaction been genuine, relaxation should have preserved or enhanced the structure. Instead, we observe systematic dilution—the hallmark of a sampling artifact that attenuates with denser measurements. This null result is publishable and valuable: it establishes a dimensional bound (n≤2) on non-additive admissibility composition.
🧱 Phase P₀: Axis I–V Verification (The Foundation)
Before Phase P₁ began, the entire Axis I–V chamber stack was re-executed and verified under controlled conditions. This verification phase established that admissibility geometry results do not depend on implementation artifacts, chamber drift, or representational choices.
What Phase P₀ Verified
Phase P₀ was not exploratory research—it was substrate stability validation. All five axes (Ancestral Correlation, Path Ensemble, DAG Embeddability, Spectral Invariants, XOR-SAT Feasibility) were re-executed with locked parameters and fixed seeds to confirm:
- Deterministic JSON reproducibility across all chambers
- Seed block integrity (196884–196933) locked and cross-validated
- Utility thresholds consistent across mechanism classes
- Cross-chamber consistency (no contradictions in shared seeds)
- No mechanism drift from prior validation phases
- Projection invariance (τ-level outcomes independent of Ω-level display)
Step 1: Canonical Reference
Fixed-seed runs with locked parameters across all V-1 through V-5 chambers
Step 2: Projection Invariance
Confirmed τ-level outcomes invariant under Ω-level representation and visualization choices
Complete Archive
Full JSON output archive from all Axis I–V chambers with deterministic reproduction
Why Phase P₀ Matters
Phase P₁ was not exploratory—it was a second-order investigation built on verified substrate baseline.
Without P₀ verification, critics could argue that admissibility geometry results depend on chamber implementation details, parameter drift, or seed selection artifacts. Phase P₀ closes this critique by demonstrating that:
- All chambers reproduce identically with fixed seeds
- Cross-chamber consistency matches Axis-V theory predictions exactly (e.g., seed 196884: V-1,V-2,V-3,V-4 → utility TRUE; V-5 → utility FALSE)
- No mechanism changes occurred between validation and measurement phases
This makes Phase P₁ findings robust to substrate implementation and establishes the dimensional constraint (n≤2) as a mathematical property, not a computational artifact.
🗓️ Research Timeline: Foundation → Discovery → Resolution
This research exemplifies cumulative experimental methodology. Phase P₀ verified substrate stability, then each subsequent phase either validated, bounded, or extended prior claims without retroactive reinterpretation.
Phase P₀: Axis I–V Verification
Purpose: Verify substrate stability and establish baseline reproducibility
Status: ✓ VERIFIED
Outcome: All 5 chambers (V-1 through V-5) reproduce identically with locked seeds. Cross-chamber consistency confirmed (seed 196884: V-1,V-2,V-3,V-4 utility TRUE; V-5 utility FALSE). No mechanism drift detected.
Documents: Step 1, Step 2, JSON Archive
Phase P₁-A: Sharp Boundaries
Finding: All three gates exhibit sharp admissibility transitions (1.6–10% width)
Status: ✓ VALIDATED
Executions: 1,850 runs
Phase P₁-B: Pairwise Interaction
Finding: V-4 × V-5 interact non-additively (56% volume, Δ=0.580, κ=0.309)
Status: ✓ VALIDATED
Executions: 11,250 runs
Phase P₁-C: Triple Interaction
Finding: Local signals (Δ₃₄₅=0.227) but sparse coverage (25.5%)
Status: ⚠ AMBIGUOUS
Action: Failed F1 criterion (80% threshold) → Designed P₂-A follow-up
Executions: 123,750 runs
Phase P₂-A: Systematic Relaxation
Finding: Coverage improved to 83.3%, interaction decreased to 0.130 (-43%)
Status: ⊞ BOUNDED (n≤2)
Interpretation: Dimensional constraint established—triple interaction not robust
Executions: 259,200 runs across 3 schedules
Preregistration Success
All phases followed preregistered protocols with success criteria and falsification conditions specified before data collection. Phase P₁-C correctly triggered falsifier F1 (geometry integrity). Phase P₂-A was explicitly authorized as a controlled follow-up, not retroactive patching. This discipline prevented overinterpretation and exemplifies reproducible computational physics.
🧪 Chamber Implementations
All results derive from operational chambers in the UNNS Laboratory. Each chamber is self-contained, browser-executable, and requires no external dependencies.
Chamber LI-P₁-A
Independent sweeps of V-3, V-4, V-5 to establish individual boundary sharpness and location.
Chamber LI-P₁-B
Joint 2D sweep (V-4 × V-5) detecting non-additive composition and boundary curvature.
Chamber LI-P₁-C
Full 3D sweep (V-3 × V-4 × V-5) revealing sparse overlap and ambiguous signals.
Chamber LI-P₂-A
Systematic relaxation across 3 schedules to resolve P₁-C ambiguity with controlled expansion.
🌟 What This Means
For UNNS Theory
Admissibility gates define geometric boundaries in history space, but these boundaries exhibit dimensional constraints. Pairwise constraints can mutually stabilize utility regions through non-additive coupling, but triple and higher-order gates over-constrain the system, causing intersection regions to collapse.
This suggests a fundamental principle: constraint complexity saturates at n=2. Beyond pairwise interactions, feasibility becomes additive or degenerates into sparsity.
Critically, Phase P₀ verification established that this dimensional constraint is a mathematical property of the substrate, not an artifact of chamber implementation, seed selection, or parameter choices. The same seed (196884) produces consistent admissibility patterns across all five verified chambers, confirming that the geometry reflects intrinsic structure.
For Broader Physics
If admissibility geometry proves generic across recursive substrates, it offers a new lens for understanding:
- Constraint satisfaction phase transitions: Why solution spaces undergo percolation transitions as constraint density increases
- Emergence of effective laws: How lower-dimensional structure might generate higher-dimensional behavior without infinite tower of corrections
- Selection principles: Why certain theories are "realized" while others remain mathematically consistent but physically absent
- Dimensional reduction: Why effective theories often involve 2-body interactions (pairwise potentials, bilinear couplings)
Methodological Impact
This work demonstrates that null results can strengthen research programs:
- Preregistered protocols with falsification criteria prevent motivated reasoning
- Systematic follow-ups resolve ambiguities without retroactive reinterpretation
- Transparent reporting of null results bounds phenomena and prevents overclaiming
- Dimensional constraints are more valuable than finding arbitrary n-dimensional structure
🔬 Improving Methodology through Preregistration: Preregistration Done Right
This research exemplifies how preregistration protects scientific integrity without sacrificing discovery:
- Phase P₀ substrate verification: Before any admissibility geometry measurements, all Axis I–V chambers were re-executed with locked parameters to verify reproducibility, cross-chamber consistency, and absence of mechanism drift
- Success criteria specified a priori: All thresholds, grid resolutions, and validation metrics defined before data collection
- Falsification conditions stated upfront: F1 (geometry integrity), F2 (non-degeneracy), F0 (mechanism independence)
- Null results treated as informative: P₁-C F1 failure triggered Phase P₂-A, not dismissal or rationalization
- Phase boundaries clearly defined: P₂-A explicitly authorized as refinement phase, not retroactive patch
- Comprehensive reporting: All 384,832 executions documented, including negative results
Result: A research program where Phase P₀ verified the substrate, then each subsequent phase validated, bounded, or extended prior claims through cumulative evidence.
📋 Quantitative Summary
| Phase | Finding | Key Metric | Executions | Status |
|---|---|---|---|---|
| P₁-A | Sharp boundaries exist | 1.6–10% transition width | 1,850 | ✓ VALIDATED |
| P₁-B | Pairwise non-additivity | Δ=0.580, 56% volume, κ=0.309 | 11,250 | ✓ VALIDATED |
| P₁-C | Triple interaction ambiguous | Δ₃₄₅=0.227, 25.5% coverage | 123,750 | ⚠ F1 FAIL |
| P₂-A Sched A | Conservative relaxation | Δ₃₄₅=0.149, 76.4% coverage | 86,400 | ↓ DILUTION |
| P₂-A Sched B | Moderate relaxation | Δ₃₄₅=0.136, 76.4% coverage | 86,400 | ↓ DILUTION |
| P₂-A Sched C | Aggressive relaxation | Δ₃₄₅=0.130, 83.3% coverage | 86,400 | ⊞ n≤2 |
| TOTAL EXPERIMENTAL CORPUS | 384,832 | ⊞ BOUNDED | ||
📄 Full Research Paper & Laboratory Access
Explore the complete methodology, validation frameworks, and chamber implementations.
📖 Read Full Paper (PDF) 🧪 Explore Live Chambers
UNNS Research Collective
Investigating the geometric structure of feasibility in recursive substrates
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Phase P₀ verification documents, chamber implementations, complete data archives, and preregistration protocols are publicly available.
Full research program: Axis I–V verification (P₀) → Admissibility geometry (P₁–P₂) → Dimensional constraints established.
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