# Can the available SWC results justify near-certainty about RH?

Private working note. The attempted combination does not establish a numerical probability that RH is true.

## What was combined

1. The original sign sampler has certified positive mass at squared-error target 1/128. Independent repetition raises its probability of finding a witness at that target.
2. The later bounded perturbation family has mass one at squared-error target 0.0062. This follows from a neighbourhood of a previously verified center; it is not a new experiment testing RH.
3. Nine finite-basis contractions are certified. Their extension to all sufficiently large basis indices remains unproved. These basis indices are not nine independent statistical trials of RH.
4. The bound B<10^-12 is a consequence of established zero verification and zero-count estimates. In particular it reuses the published zero-verification input; it must not be counted as independent evidence in addition to that input without a joint statistical model.

These statements are compatible, but they are about different events or deterministic quantities. There is no justified rule that multiplies their success probabilities into a posterior probability of RH. A quantitative smallness bound on B is not a probability bound on the event B>0.

## A precise statistical obstruction

The two illustrative models in POSITIVE_MASS_EXISTENCE.md have errors E_0(n)=2^-n and E_b(n)=b+2^-n, where n has mass 2^-n and b=2^{-(J+2)}. The models have different limiting errors, yet exactly the same acceptance sets at targets 2^-j for every j<=J.

Consequently any data consisting of independent pass/fail trials at those levels has identical likelihood under the two models. The Bayes factor is one. If a prior assigns weight r to the zero-limit model, the posterior after such data remains r:

    posterior = r L / [r L + (1-r)L] = r, for observed data with L>0.

This holds for every r strictly between zero and one. Thus arbitrarily high confidence can be put into the prior without being learned from these observations. The script checks exact likelihood equality for a specified illustrative dataset and several prior weights. These are toy models of approximation errors, not models established for the actual zeta function. They demonstrate the insufficiency of the generic finite-acceptance inference; they do not prove that all available arithmetic evidence is uninformative about RH.

## Why more random searches alone do not fix calibration

Suppose a hypothetical independent test detects an exception with probability q when the tested alternative holds. The probability of no detection in n trials is (1-q)^n. To bound this uniformly over a class of alternatives using the same formula requires a justified positive lower bound q>=q_min for every alternative in that class.

If q can be arbitrarily small, the supremum of (1-q)^n is one for every finite n. By Bernoulli's inequality, (1-q)^n>=1-nq. For illustration, q=10^-12 and n=10^6 give a no-detection probability at least 0.999999. This is a hypothetical calculation, not a measurement of actual RH counterexample-detection probability.

Our work has not established any such uniform q_min for detecting an RH counterexample. The lower bound on finding a good finite approximant is a different probability and cannot be substituted for it. Absence of observed counterexamples can inform beliefs under additional models; it does not supply a calibrated universal false-certification risk without those assumptions.

## Check against additional literature

- [Báez-Duarte's approximation criterion](https://arxiv.org/abs/math/0202141) requires approximation at arbitrarily small error, not merely a high success rate at one fixed error.
- [Darses and Hillion](https://arxiv.org/abs/1805.06733) develop probabilistic approximation criteria equivalent to RH. The probabilistic formulation does not by itself establish the required approximation properties.
- [Platt and Trudgian](https://arxiv.org/abs/2004.09765) rigorously verify the critical-line statement through height 3*10^12. This supplies a finite-height theorem, not a random sample of an infinite uniform distribution on zero ordinates.
- A [July 2026 revision of a Colombeau--Beurling preprint](https://arxiv.org/abs/2606.22562v2) states an equivalence using association and uniform L2 boundedness of a specified net. Its abstract gives the construction under RH and a converse conditional on the stated properties. This was an abstract-level scope check; its full argument was not independently validated and is not a premise here. It does not supply the missing unconditional bound.
- The abstract of [Hayou's neural-network reformulation](https://arxiv.org/abs/2309.09171) describes an approximation/minimization reformulation. It does not claim a solution or calibrated RH confidence level. This was likewise a scope check, not validation of its full argument.

The publisher page for a separate 2026 Friedrichs-angle article could not be retrieved (HTTP 403); no conclusion is drawn from that inaccessible article.

## Outcome and next mathematical target

The statistical combination supplies no justified near-100% probability for RH. It preserves the fixed-accuracy existence statements and the conditional all-accuracy search theorem. The smallest unfulfilled dyadic accuracy target after the certified 1/128 level is 1/256=0.00390625. Reaching that target would extend the finite evidence, while an argument for every accuracy would address RH itself. No new 1/256 certificate was obtained in this audit.

Run `python3 research/swc-rh/statistical_confidence_audit.py` to reproduce the exact illustrative probability comparisons. The generated JSON explicitly leaves the actual RH probability unset. The original paper and ZIP remain earlier snapshots.
