⟨ Chamber XLII — Observer-Grammar Probe ⟩

Irreversibility, Relevance, and the Emergence of Utility | v1.2 | UNNS Framework

v1.2 Changelog (Operationalizing Mode Collapse):
• Accumulated multi-bin modes: Modes now persist across steps with 0.98 decay (not recomputed one-hot)
• Warmup threshold: ω fires only after ≥25 steps (prevents premature collapse at step 0)
• Genuine plurality: Multiple phase bins can be non-zero simultaneously before ω collapses
• Meaningful entropy: Entropy reduction now measures actual commitment, not degenerate zero→zero

v1.1 Fixes: ω fires at most once (not 400×), utility no longer hard-gated to zero
v1.0 and v1.1 results invalid — v1.2 is first valid test of H₇/H₈

Chamber Identity

Chamber ID: XLII
Parent Chambers: XLI v1.1 (Structural Permission), XLI-B (Grammar Extension)
Status: Exploratory, hypothesis-opening
Schema: unns.xlii.operator_program.v1.0

Core Question

Does explicit irreversibility—mode commitment via collapse operator ω—unlock utility islands in recursive operator grammars?

Empirical Motivation

XLI and XLI-B established that:

Admissibility saturates at ~30-35% under all enrichments
Projection improves with asymmetry but remains subcritical
Utility correlates with selection (r≈0.72) but caps below islands

This rules out: pure selection, kernel length alone, resonance amplification alone.

New Primitive: ω (Collapse Operator)

ω is a grammar operator that: • Evaluates phase-binned mode distribution (K=8 bins over [0,2π)) • Selects exactly one mode according to its rule • Suppresses all other modes irreversibly • Projects state onto selected mode • Continues recursion (suppressed modes cannot re-enter) ωᵣ (Random): Uniform random mode selection ωᵍ (Guided): Argmax amplitude, tie-break on lowest entropy

Hypotheses

H₇ — Irreversibility Hypothesis
Claim: Utility islands require explicit irreversible collapse steps. Reversible grammars remain subcritical.
Prediction: Variants with ω show discontinuous increase in median G∘, potentially exceeding 15%.
Falsifier: No ω-containing variant reaches utility island criteria.

H₈ — Relevance Asymmetry Hypothesis
Claim: Collapse must be relevance-guided to unlock utility. Random irreversibility destabilizes or remains futile.
Prediction: ωᵍ variants outperform ωᵣ in both projection stability and G∘.
Falsifier: ωᵍ shows no systematic advantage over ωᵣ.

Execution Controls

Variant

Grammar: [M₂→M₁] → (ρ?) → S₃ → (ω?)

Execution Parameters

Depth: 400 steps

Seeds: 300

Mode Bins: K=8 over [0,2π)

Ready to execute

Technical Specifications

Metrics & Thresholds (Inherited)

Metric	Threshold	Purpose
Projection Stability	≥ 70%	Stable trajectory criterion
Utility Island (G∘)	≥ 15%	Actionable structure emergence
Survival Depth	≥ 200 steps	Adequate recursion exposure
Admissibility (Gτ, G₂)	Non-explosive	Substrate permission bound

Late Stabilization Index (LSI) — Diagnostic Only

LSI = projection stability measured over final 50% of recursion steps

LSI does not replace projection criterion | Not used for hypothesis acceptance | Exists to interpret path-dependence revealed in XLI-B

Variant Space (Frozen)

Axis	Values	Count
Resonance (ρ)	OFF, ON	2
Collapse (ω)	NONE, ωᵣ, ωᵍ	3
Total	6 variants