Chamber LI represents the completed operational execution layer following the successful validation of Axis V admissibility gates (V-3, V-4, V-5). This work establishes the first empirical demonstration that admissibility constraints compose non-additively in pairwise combinations but exhibit a dimensional constraint (n ≤ 2) — interaction geometry exists in 2D but vanishes in 3D compositions.
Phase P₁ (125,632 runs) established sharp single-gate boundaries and validated robust pairwise non-additive composition across V-4 × V-5 (Δ = 0.580, 56% parameter space, κ = 0.309). Triple-gate analysis (P₁-C) detected local signals but with sparse coverage (25.5%), failing preregistered falsification threshold F1 (80%). Phase P₂-A (259,200 runs) definitively resolved this ambiguity through systematic predicate relaxation: coverage improved 3-fold to 83.3%, but interaction structure systematically decreased — max residual dropped 43%, exclusion geometry disappeared entirely.
196884 derives from the Monster group's
first non-trivial representation dimension — a numerological anchor ensuring seeds are drawn from
high-entropy regions of parameter space. The block demonstrates:
Chamber LI executed a layered validation model where each phase built on validated results from previous stages while testing progressively more complex hypotheses. All four phases are now scientifically complete:
Phase P₁ (Baseline) — COMPLETE: Established single-gate behavior through comprehensive parameter sweeps (37 configs × 50 seeds = 1,850 runs). Outputs: calibrated phase diagrams showing cliff-like transitions (1.6–6.5% widths), critical threshold detection, utility-given-feasibility rates. All three gates (V-3, V-4, V-5) exhibited sharp, reproducible boundaries to within ±2%.
Phase P₁-B (Pairwise) — COMPLETE: Validated non-additive two-gate interaction without assuming compositional independence (225 configs × 50 seeds = 11,250 runs). Diagnostic focus: residual magnitude (Δ = 0.580), boundary topology (κ = 0.309), exclusion zones. Result: Pairwise gates compose geometrically, not additively. 56% of parameter space exhibits enhancement beyond independence prediction (p < 10⁻¹²).
Phase P₁-C (Triple) — COMPLETE (constrained-interaction null): Tested whether pairwise interactions extend to three-gate compositions (2,475 configs × 50 seeds = 123,750 runs). Detected local signal (Δ₃₄₅ = 0.227) but with sparse volumetric coverage (25.5%), failing preregistered falsification threshold F1. Result: Ambiguous under conservative predicates. Motivated follow-up refinement study to resolve artifact vs. genuine interaction question.
Phase P₂-A (Refinement) — COMPLETE: Definitively resolved P₁-C ambiguity through systematic predicate relaxation across three independent schedules (5,184 configs × 50 seeds = 259,200 runs). Coverage improved 3-fold (25.5% → 83.3%), but interaction structure systematically decreased: max residual dropped 43% (0.227 → 0.130), non-additive volume fell 47% (7.5% → 3.96%), exclusion geometry vanished entirely. All three preregistered success criteria failed. Result: Triple interaction is representational artifact from sparse sampling, not genuine structural feature. Dimensional constraint n ≤ 2 established.
Validated: Pairwise non-additivity (n = 2)
• Max residual: Δ = 0.580 (138% enhancement)
• Boundary curvature: κ = 0.309
• Non-additive volume: 56% of parameter space
• Statistical significance: p < 10⁻¹²
Bounded: Triple interaction (n = 3)
• Coverage improved: 25.5% → 83.3%
• Interaction diluted: Δ₃₄₅ 0.227 → 0.130 (43% drop)
• Exclusion disappeared: 1.82% → 0.0%
• All success criteria failed
DIMENSIONAL CONSTRAINT: n ≤ 2
Admissibility non-additivity is a lower-dimensional phenomenon.
Pairwise gates define complementary constraints that stabilize
utility regions. Triple gates over-constrain, fragmenting
feasibility windows.
First empirical evidence in the UNNS recursive substrate that feasibility
composition exhibits dimensional boundaries.
P₁ (Single Gate) ✓ LOCKED
├─ V-3: Topological sweep [11 configs]
├─ V-4: Spectral sweep [15 configs]
└─ V-5: Logical sweep [11 configs]
↓
↓ [Sharp boundaries validated: 1.6–6.5% transition widths]
↓
P₁-B (Pairwise) ✓ LOCKED
└─ V-4 × V-5 interaction [15×15 = 225 grid]
│
│ [Non-additivity confirmed: Δ = 0.580, κ = 0.309, 56% volume]
↓
P₁-C (Triple) ✓ LOCKED (null)
└─ V-3 × V-4 × V-5 composition [11×15×15 = 2,475 volume]
│
│ [Sparse overlap: 25.5%, local Δ₃₄₅ = 0.227, F1 failed]
│ [Ambiguity: signal vs. artifact?]
↓
P₂-A (Refinement) ✓ LOCKED (dimensional constraint)
└─ Controlled predicate relaxation [3 schedules × 1,728 configs]
│
│ [Coverage: 25.5% → 83.3%]
│ [Interaction: Δ₃₄₅ 0.227 → 0.130 (43% drop)]
│ [Exclusion: disappeared entirely]
↓
DIMENSIONAL CONSTRAINT ESTABLISHED: n ≤ 2
Total Corpus: 384,832 independent executions
Publication Status: Ready for submission
All Chamber LI phases adhered to strict falsification standards preventing post-hoc rationalization. The P₂-A null result exemplifies this discipline:
Preregistration: Sweep tables, refinement rules, and success criteria documented before execution. No parameter tuning post-observation. Grid resolutions and statistical thresholds locked in chamber configuration. P₂-A success criteria preregistered before observing systematic interaction dilution.
Null-result acceptance: When P₁-C detected ambiguous signals, rather than claiming discovery, the result was classified as constrained-interaction null. P₂-A was designed as explicit follow-up with authorized predicate relaxation. When all three P₂-A success criteria failed, the null result was reported transparently without retroactive justification. Absence of triple interaction is scientifically meaningful: it establishes dimensional constraint n ≤ 2.
Frozen infrastructure: Seed block 196884–196933 immutable across all phases. Same 50 seeds ensure direct comparability between P₁ single-gate, P₁-B pairwise, P₁-C triple-gate, and P₂-A refinement results. Prevents selection bias in seed choice. Total corpus: 384,832 independent executions with identical seeds, reproducible to floating-point precision.
Cumulative interpretation discipline: P₁-C did not invalidate P₁-B. P₂-A did not invalidate P₁-C's measurement accuracy (sparse overlap was correctly measured). Each phase either validates or bounds prior claims without retroactive modification. This cumulative approach—where phases build on validated results while testing progressively more complex hypotheses—exemplifies reproducible experimental physics methodology applied to computational research.
Independent replication: All chambers export JSON with complete metadata. Third-party validation requires only: (1) seed block, (2) preregistered parameters, (3) gate implementations from Axis V. No hidden degrees of freedom.
Chamber LI occupies a strategic position in the UNNS research architecture:
Upstream from Axis V: Depends on validated gate implementations (V-3 DAG embeddability, V-4 spectral feasibility, V-5 XOR-SAT constraints). Chamber LI is not validating gates — it assumes Axis V correctness and explores compositional behavior.
Downstream to Phase G: Provides empirical boundary maps required for experimental bridge development. P₂-A projection diagnostics directly inform which gate combinations produce detectable observables versus admissible-but-dark configurations.
Parallel to Chambers XXXII–XXXIX: While earlier chambers explored individual mechanism validation, Chamber LI focuses on systematic composition under frozen protocols. Represents maturation from exploratory testing to production execution.
Foundation for Chamber XL+: Interaction geometry discovered in P₁-B/C informs design of future chambers targeting phase-exposure diagnostics and higher-dimensional admissibility manifolds.