This is the complete geometric ladder from dimensional constraint through curvature sensitivity to metric stratification. The five chambers form a coherent progression: LI establishes interaction dimensionality, LII establishes boundary geometry responsiveness, LIII–LIV establish partition invariance, and LV proves hierarchical non-isometry.
The progression is not accidental. LI establishes geometric constraint, LII establishes curvature sensitivity, LIII–LIV establish partition invariance, and LV reveals that invariance is structural (partition-level) but not geometric (metric-level). Non-isometry is the natural culmination of pre-metric geometric asymmetry elevated to a metric theorem.
R1 failure alone (when R2, R3, R4 pass) indicates boundary thinness, not channel collapse. Structural stability requires R2 = R3 = 0. A framework conflating R1 failure with structural failure would incorrectly declare factorization broken when only projection geometry is stressed. This distinction is essential for correctly reading LV retention erosion in V2×V4.
The encoding hypothesis (Interpretation A) predicted that normalization maps dk_rel, B_rel, L_rel would restore V2×V4 retention to passing levels by absorbing curvature-induced scale drift. The prediction failed: Δ(passRate) = 0.0000 across all gates and all τ₂ levels — not approximately zero, exactly zero to six decimal places. This is a structural null: the normalization maps rescale thresholds proportionally with pool-level statistics, leaving pass/fail classification invariant. Interpretation A is falsified; Interpretation B (structural stratification) is confirmed.
The Tier-2 lift functor LFτ₂ on the witness-channel category ChanF (four channel objects with Lawvere-metric enrichment dτ₂(i,i) = 1 − ρi) is the identity on channel objects — R2 = R3 = 0, no cross-channel transport — but is not an isometry in the Lawvere-metric enriched sense: self-distances dτ₂(i,i) change as witnesses are lost. Object-class preservation and metric distortion are provably independent. Witness migration (R4 > 0) is compatible with Part 1: migrated witnesses change the weight on a channel object without changing the object itself.
Theorem 9.1 (Critical Erosion Law): under R2 = R3 = 0 and a critical retention model, migration onset τθmig ≥ τc + (θ/A(F))1/p(F). Theorem 9.2: the family Lipschitz constant LF is bounded by the curvature functional C(F) = maxi(BF · λi · Si), computed from kernel bias amplitude and boundary density prior to chamber execution. Hierarchical stratification is not merely observable — it is predictable.
Chamber LI ── What is interaction dimensionality?
↓ Pairwise (V-4 × V-5): Δ = 0.580, coverage 56%, Cohen's κ = 0.309
↓ Triple compositions: collapse to 83.3% coverage, residual −43%
↓ Theorem 3.1: Interaction geometry exists in dimension n ≤ 2 and collapses at n = 3.
│ Falsifier: stable triple-gate residual — not found.
│
Chamber LII ── Does curvature shift boundaries?
↓ SAIF Phase P₃ · curvature-responsive bifurcation · partition-preserving
↓ Theorem 3.2: Curvature deformation alters boundary position without altering
│ partition identity. Boundaries shift, channels do not.
│ Falsifier: partition change under deformation — not observed.
│
Chamber LIII ── Are there hidden mechanism classes?
↓ 56,877 mechanisms · adversarial relaxation · δmax=0.239 · basin unification at G3
↓ Theorem 4.1: No additional independent gate H within characterized domain.
│ Falsifier: mechanism class WH unlocked by fifth gate — not found.
│
Chamber LIV ── Does factorization survive perturbation?
↓ 4 perturbation families · R2 = R3 = 0 throughout · G3 parametric (P1 only)
↓ Theorem 5.1: Witness-separability and channel orthogonality preserved.
│ Falsifier: non-zero R2 or R3 — not observed.
│
Chamber LV ── Does factorization survive operator tier change?
3 Tier-2 families · dual-track · τ₂ sweep · R5-T slope stratification
Theorem 6.1: Partitions preserved (R2=R3=0). Geometry non-isometric (V2×V4).
Decision B: B1 triggered (Δ=0.0000) · B2 triggered (slopes −0.11 to −0.14).
Critical Erosion Law + kernel-predictive curvature functional C(F).
Falsifier: uniform R5-T slopes — not observed.
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Interaction geometry is dimensionally constrained. Curvature shifts boundaries.
Factorization exists and persists under perturbation.
Under operator tier change, it survives as structure while stratifying as geometry.
CHAMBER LV CERTIFIED. GEOMETRIC PROGRESSION LI → LII → LIII → LIV → LV CLOSED.
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The progression is not accidental. LI establishes geometric constraint, LII establishes curvature sensitivity, LIII–LIV establish partition invariance, and LV reveals that invariance is structural (partition-level) but not geometric (metric-level). Non-isometry is the natural culmination of pre-metric geometric asymmetry elevated to a metric theorem. Do not revise LV. Open projection-space reformulation (Option C) as a new chamber with its own preregistered protocol if warranted.