# Reality Revision 1.39 — Anytime evidence for a declared Markov sign model

**Date:** 20 September 2026  
**Lab:** Rope Tomography Lab 0.14.0

## Result

The v1.26 dependence counterexample is resolved only within one explicit model
class. Conditional on the first nonzero sign, the nonzero projected-residual
signs must form a fixed first-order homogeneous binary Markov chain. For a
declared transition matrix (P_0), a past-only Krichevsky--Trofimov predictor
(widehat Q_t) gives

\[
L_n=\prod_{t=2}^{n}
\frac{\widehat Q_t(X_t\mid X_{t-1})}{P_0(X_t\mid X_{t-1})}.
\]

Under the null, each factor has conditional expectation one. Thus (L_n) is a
nonnegative martingale and Ville's inequality bounds ever crossing 100 by 1%.
The proof conditions on the first sign; its distribution need not be balanced.

Fresh synthetic validation fixed (P_0) to 90% sign persistence, 5,000
histories, and 400 looks:

- Declared Markov null: **39/5,000** crossings.
- Same paths with every sign inverted: the identical **39/5,000** decisions.
- Same transition law with a fixed positive initial sign: **36/5,000**.
- 97% persistence: **4,629/5,000**.
- 75% persistence: **4,999/5,000**.
- Balanced iid signs: **5,000/5,000**.

The old iid conditional-sign process crossed on **4,134/5,000** declared
Markov-null histories, confirming that marginal balance does not satisfy its
conditional-balance assumption. A separate implementation regenerated the
chains and reproduced **24,000,000** decisions. All **110 tests passed**.

These are written mathematics and finite synthetic computations, not measured
physical evidence. The construction follows Fields, Javidi, and Shekhar,
*Sequential One-Sided Hypothesis Testing of Markov Chains* (22 January 2025):
https://arxiv.org/abs/2501.13187

## Limitations and retained corrections

The guarantee requires the exact fixed transition matrix and a first-order
homogeneous law in nonzero-sign event time. It does not cover a null matrix
estimated from scored data, changing persistence, higher-order dependence, or
an average transition frequency. Exact zeros are skipped, so the declared law
must remain true after that time change. Magnitudes, scale, tail thickness,
physical mechanism, and theological cause are outside this detector.

The engine is standalone and does not share its 1% budget with another rule.
The v1.14 closure-idempotence correction, all v1.15 corrections, and the exact
model-class assumptions on the (2K)-sample recurrence result remain in force.

Next test: derive a composite-null e-process valid uniformly for every
prespecified persistence in ([0.88,0.92]), then challenge it at both boundaries
and on non-Markov dependent signs before changing the point-null engine.