Coherence Collapse Analysis (CCA)
Coherence-based alignment, with effective dimensionality measurable on production traces.
The Coherence Collapse Analysis (CCA) is the mathematical foundation under CIRIS's coherence-based alignment thesis. It says when accumulating constraints actually defend against deception — and when they collapse into an echo chamber regardless of how many you add. The central tool is the Kish design effect, applied to AI alignment for the first time. The empirical anchor is the Constrained Reasoning Chains study on 6,465+ production agent traces.
The thesis
Coherence-Based Alignment.
Coherence-Based Alignment (CBA) is the position that an AI system stays aligned not by being trained on the right values once, but by being structurally required to maintain coherence across many independent constraints over time. The constraints accumulate. The space of trace shapes consistent with deception shrinks. Truth occupies the intersection of every constraint manifold; deception has to satisfy all of them simultaneously, which is a different and harder problem.
The catch: independence is load-bearing. Correlated constraints provide little additional security regardless of how many of them there are. CCA is the formal treatment of when constraints stop being independent and start being an echo chamber.
The central formula
The Kish design effect, applied to AI alignment.
The Kish design effect is a survey-statistics formula for converting a nominal sample size into the effective sample size when observations are correlated. CCA uses it to convert a nominal constraint count into the effective independent constraint count.
- ρ = 0: independent constraints. k_eff = k. Every constraint adds.
- ρ = 1: perfectly correlated constraints. k_eff = 1, no matter how many of them. The echo chamber.
- Critical rigidity: ρ_crit = 0.43. k_eff ≈ 2.3 at collapse — verified numerically in the Lean formalization.
- Healthy corridor: 0.2 < ρ < 0.7, below the singularity curve K_req · ρ = 1.
Source: FundamentalIdentity.lean (rigidity_collapse, ρ_crit, k_eff_at_collapse, k_eff_rho_crit_relation).
The empirical surface
Effective dimensionality on production AI traces.
The Constrained Reasoning Chains study built a 17-dimension constraint vector per trace from fields the lens already extracts at ingest, then computed two effective-dimensionality measures from the eigenvalue spectrum: participation ratio and entropy perplexity. Both are reported because they bound the answer from different sides.
Lifetime peak
N_eff_H = 9.51
Entropy-perplexity measure. Reached mid-April 2026, on a corpus going back to 2026-03-02.
Lifetime mean
N_eff_H = 7.20
Mean across rolling 500-trace windows over the full corpus. Includes mixed organic and QA-class traffic.
Operational floor
k_eff ≥ 9.2
Stability condition for operational autonomy in the Lean formalization (TruthManifold.is_stable).
These are not the same number. The 9.51 peak is the maximum measured organic codimension. The 7.20 mean is the corpus-wide average across mixed traffic. The 9.2 floor is the formal stability threshold the architecture targets. Reporting all three keeps the metric honest.
The boundary
The singularity boundary.
Above the singularity curve, time-to-truth diverges to infinity. Correlated constraints provide no additional security regardless of scale or time. The singularity is the deterministic edge of the healthy corridor: between the chaos regime (ρ < 0.2) and the rigidity regime (ρ > 0.7), below the curve K_req · ρ = 1.
- Singularity condition — K_req · ρ ≥ 1. Above this curve, T_truth → ∞.
- Time to truth — T_truth = K_req · (1 − ρ) / [α · (1 − K_req · ρ)].
- Required constraints — K_req = −ln(ε / V₀) / λ for safety target ε.
- Volume decay — V(k) = V₀ · exp(−λ · k_eff). Deceptive volume collapses exponentially in effective constraints.
Source: Coherence Collapse Analysis (Zenodo) and the Lean formalization in CIRISAI/RATCHET.
The honest ceiling
L-01: what no detector can do.
The CCA paper proves an information-theoretic limitation. Roughly 60% of emergent incoherence is detectable from marginal distributions; roughly 40% is fundamentally undetectable by any polynomial-time complete detector for compositional deception. The singularity is the deterministic boundary; L-01 is the information-theoretic ceiling on what any singularity-aware detector can see.
CCA is a topological cost-asymmetry argument over time, not a pointwise decision procedure. It says that running real ethical reasoning over weeks is structurally more expensive than producing single deceptive outputs that defeat any single detector — not that the deceptive output can always be caught.
Related work
Where this sits in the literature.
- Coherence-Based Alignment (CBA), PhilArchive. An independent paper using nearly identical framing. CCA provides the formal apparatus and empirical validation that the CBA philosophical framing calls for.
- Effective dimensionality in ML. Active term in the adversarial-robustness and generalization literature (Khachaturov & Mullins; Maddox / Wilson). CCA bridges this measurement to AI alignment governance with production data.
- Perspectival homogenization, FAccT 2025. Coined in "Value of Disagreement in AI Design, Evaluation, and Alignment." This is the failure mode CCA measures structurally.
- Constrained Reasoning Chains (CRC). The empirical telemetry study published April 2026 ( Zenodo) that validated the effective-dimensionality measurement on production traces.
Where this goes next
The math is one part of a larger architecture.
CCA grounds the measurement. The Coherence Ratchet is the operational mechanism. The Federation is the social and cryptographic structure that makes the measurement collective. Mission Driven Development is the engineering discipline that keeps the whole thing aligned with what it is for.