The case
Two perspectives, an authority bit, two record bits, two rule-integrity bits (audit and admission), a phase clock of period 2, and three maintenance modes (external reset, reciprocal production, feedback without production) in one disjoint-union kernel: 192 states (dmarx/PerspectiveLab@7e2edc2 experiments/finite_organization/). Seven 192 × 192 kernels, the ordinary one and six interventions, are exported entry by entry.
The chain is reducible: the mode never changes, and damaged feedback states can absorb. So no unique stationary law exists, and the lab declares one: an equal-weight mixture of the three intact classes, tested invariant. Every verdict below is relative to it.
Results, as stored (results/exact_profile.json)
- Informational closure. On the declared law the finite-history defect of the agents, and of the agents with phase, is 0 bits. For authority alone it is 0.147 bits at h = 1 and 0.132 at h = 6, and for authority with one record bit 0.103 at h = 1 and 0.092 at h = 5. These are finite h, not an infinite-history limit.
- Strong lumpability. The total-variation defect of the agents with phase is 0.42 globally and 0 on the intact support. The description that omits only the mode has defect 1.0 globally and 0 on the intact support. Closure on a support is not lumpability of the chain.
- Intervention deficiency. The declared mixture is ordinary 0.45, audit loss 0.15, admission loss 0.10, double loss 0.15, audit team off 0.10 and admission team off 0.05. A seventh intervention, both teams off, is enumerated with weight 0. The outcome is the next pair of rule integrities. Γ = I(S; Y^(I) | T(S), I) is 0.5051 bits for every description that omits the mode and 0 for the full state. It is exactly 0.55·H₂(1/3), since each non-ordinary intervention splits the three modes one against two.
- Γ depends on the source law. For the ordinary intervention alone it is 0 on the intact support and H₂(1/3) = 0.918 bits on the damaged support.
- Refinement. For this one law, mixture and outcome, Γ(authority) ≥ Γ(organization) ≥ Γ(full), CLAIM-409’s monotone half. In an ablation with the authority coupling g = 0, the global strong-lumpability defect is 0 for authority, 0.380 for authority with one record bit, and 0 for the full state. That pattern is the shape of CLAIM-409’s non-monotone half, but it is in strong lumpability, not in informational closure, which is what the claim’s three-bit chain is about.
Finite samples (v1.9)
dmarx/PerspectiveLab@9f82bfd experiments/closure_estimation/ and EXPERIMENT_REPORT_V19.md. Windows sampled from the declared law, 12 repetitions at each training size, held-out log-loss gain in bits:
| projection | exact defect | N = 300 | N = 19,200 |
|---|---|---|---|
| authority, h = 1 | 0.1474 | 0.0655 | 0.1468 |
| authority, h = 3 | 0.1323 | 0.0505 | 0.1307 |
| authority with one record bit, h = 2 | 0.0930 | −0.0082 | 0.0921 |
| agents with phase, h = 1 | 0 | −0.3673 | −0.0174 |
A negative gain is a difference of fitted risks, not a negative conditional information. At N = 300 the positive 0.0930 bits shows a positive gain in only 3 of 12 repetitions. For the exact zero all 12 gains stay negative at every N tested, because the larger model pays an estimation cost. So a non-positive gain is no evidence of closure. For Γ the held-out macro log-loss is 0.7718 bits at N = 300 and 0.5103 at 19,200, against the exact 0.5051.
In the separate 32-member population, a fixed one-step target, the next change of the mean record, has held-out MSE 6.27 × 10⁻⁶ from the current six-variable macrostate, 2.65 × 10⁻⁸ with four lags, and 3.8 × 10⁻³³ with the current mean local endorsement. The last is the update law recovered, not a discovery. None of this is closure of the macrostate.
What it can show
That on one exactly known kernel, closure on a support, global lumpability and sufficiency for a declared intervention mixture give different verdicts, and that the last is CLAIM-407’s Γ with its zero condition met once the mode is kept. That a finite-sample sign test of closure fails in both directions at the sample sizes tested.
What it cannot show
- Anything about real organizations. The parameters are chosen to give non-degenerate kernels, and the model is not a verified reduction of the 32-member population.
- A stationary verdict independent of the declared law, or a Γ on a scale shared with other intervention mixtures or outcomes.
- That reciprocal Boolean updates are closure of constraints (TERM-070), or agency.
Provenance
dmarx/PerspectiveLab@7e2edc2 experiments/finite_organization/ (model.py, diagnostics.py, run_exact.py, test_exact.py, results/; EXPERIMENT_REPORT_V18.md) and dmarx/PerspectiveLab@9f82bfd experiments/closure_estimation/ (EXPERIMENT_REPORT_V19.md). When filed, the finite-organization test suite was re-run at dmarx/PerspectiveLab@11c1495 and passed.