# Positive-mass existence: a repaired SWC construction

Private follow-up working note. Nothing here is a publication or a claim that RH is proved, probably proved, or established in physical reality.

## 1. An unconditional fixed-tolerance result

Let F = sum(c_k g_k), k=2,...,2048, be the previously verified approximation. Its squared L2 error has the exact upper bound

    U = 183846930563526613052149167892727648239 /
        29710561639190402322382844850644900118528.

The new script checks the fingerprint of that certificate and reuses its earlier verification. It does not rerun its 128-million-cell calculation.

Set

    r = 393317043 / 5000000000,
    h = 1 / 20000,
    a = h / 2047 = 1 / 40940000.

Exact rational comparisons give U < r^2 and

    (r+h)^2 = 154895017335763849 / 25000000000000000000
            < 31/5000 = 0.0062.

For independent uniform signs xi_k, form

    F_xi = sum((c_k + a xi_k) g_k).

Since |g_k| <= 1 almost everywhere on an interval of length one, ||g_k||_2 <= 1. Therefore, for every seed,

    ||1 - F_xi||_2 <= ||1-F||_2 + a sum ||g_k||_2 < r+h.

Consequently every one of the 2^2047 seeds has squared error below 0.0062. The accepted mass is exactly one. The same bound holds for every coefficient perturbation in the closed cube [-a,a]^2047.

This proves existence and robustness at a specified tolerance. It improves the tolerance of the earlier sign-family guarantee, which was 1/128, by using a different perturbation law. It does not improve the already known center error U, nor establish an optimal error. Its mass-one statement is about this deliberately chosen neighbourhood, not a probability that RH holds.

## 2. A distribution covering every possible finite certificate

The [Báez-Duarte criterion](https://arxiv.org/abs/math/0202141), in the balanced normalization developed in the main notes, says that RH is equivalent to rational finite approximations of squared error below 2^-j for every integer j>=1. Probabilistic variants of this approximation problem also appear in [Darses and Hillion](https://arxiv.org/abs/1805.06733).

Here is a distribution with full support that does not assume such approximations exist. A sample is (N,D,b,M), with N>=2, D,M>=1, and b an integer vector of length N-1. Draw independently, conditional on N,

    P(N=n) = 2^{-(n-1)},
    P(D=d) = 2^-d,   P(M=m) = 2^-m,
    P(b_k=t) = 2^{-|t|}/3,  t an integer.

Each displayed law is normalized, and every tuple has strictly positive probability. Set c_k=k b_k/D. These parameterizations include every rational finite coefficient vector, possibly more than once.

A simple finite verifier computes the exact rational head

    H_M(c) = sum_{m=1}^M (1-sum_k c_k(m mod k)/k)^2 / [m(m+1)]

and the upper bound

    Q(c,M) = H_M(c) + (1+sum_k |c_k|)^2/(M+1).

The tail estimate follows from |g_k|<=1 and the telescoping cell weights. Thus E(c)<=Q(c,M), and Q(c,M) tends to E(c) as M increases. Every verifier call has finite running time, though the universal distribution is not an efficient search proposal.

Let p_j be the probability that Q(c,M)<2^-j. Soundness and the strictly positive tuple masses prove both directions:

    p_j > 0  iff  some rational finite c satisfies E(c)<2^-j.

For the reverse direction, strict inequality leaves enough margin for some finite cutoff M. It follows that

    RH  iff  p_j>0 for every j>=1.

Full support prevents an existing finite certificate from being assigned zero mass. It does not establish that a certificate exists. In particular the fixed-tolerance result above proves positivity for j=1,...,7 under this universal sampler, but does not supply the j=8 target 1/256 or all subsequent targets. Failure to supply such a certificate is not evidence that its mass is zero.

## 3. Repairing the infinite random-search argument

Assume, explicitly, that p_j>0 for every j. Take an infinite IID sample sequence from the universal distribution. For a fixed j, the probability that its first t samples all fail is (1-p_j)^t, which tends to zero. Hence the probability of never finding a witness for this j is zero. Taking the countable union over j proves

    P(for every j, some finite sample index certifies error < 2^-j) = 1.

No common positive lower bound on the p_j is necessary. No independence between the events for different j is necessary. For an operational implementation, after sample t check it for j=1,...,t; each required level is tried infinitely often, and each check terminates. Every level is reached after a finite, random amount of work almost surely. This does not mean that infinitely many levels finish at one finite time, nor give a finite stopping certificate for RH.

If explicit lower bounds delta_j>0 are known, allocate failure budget eta*2^-j to level j. Choose a finite number r_j of independent trials satisfying

    (1-delta_j)^{r_j} <= eta*2^-j.

The union bound gives probability at least 1-eta that all levels succeed within their respective budgets. This is a valid conditional positive-mass existence argument for an entire approximation sequence. The missing premise is still positivity at every actual accuracy level. The script's numerical budget examples use hypothetical delta_j and are labeled accordingly.

The incorrect construction would demand that one fixed finite coefficient vector succeed at every tolerance. If L=lcm(2,...,N), all its basis functions vanish on cell I_L, so E(c)>=1/[L(L+1)]>0. No finite vector is an exact witness. Thus the acceptance events for a single sampled tuple decrease to the empty set and p_j tends to zero, even if RH is true. Different tolerances require different approximations; this is compatible with the almost-sure sequence construction.

## 4. Why finite mass measurements cannot settle the remaining premise

The following are logical countermodels, not claims about zeta zeros. Sample n>=1 with probability 2^-n. Compare errors

    E_0(n)=2^-n,       E_b(n)=b+2^-n,
    b=2^{-(J+2)}.

For every j<=J, both models accept exactly the same samples n>=j+1 at target E<2^-j. Their exact success masses are both 2^-j. All pass/fail observations restricted to those levels are therefore identical, even with unlimited repetitions. Nevertheless their limiting errors are respectively zero and b>0. At j=J+2, the second model has zero accepted mass.

This example concerns acceptance observations at finitely many targets, not equality of the underlying exact error values. The script checks the endpoint inequalities defining these acceptance sets and their geometric tail masses for J=7,20,100. It demonstrates why extrapolating observed positive masses to every level needs an additional justified assumption.

One useful qualification: for a distribution on the integer accuracy levels with strictly positive mass at each level, an *established* probability-one statement that the sampled level is solvable would imply every level is solvable. There are no nonempty null subsets of such a countable full-support space. But a confidence level strictly below one can still hide a failure at a sufficiently low-weight level. Neither that exact probability-one premise nor a justified statistical estimate of it has been established here.

## 5. Empirical and ground-truth interpretation

The supplied SWC v6 manuscript states under “Mathematical and physical reality” that empirical false-certification guarantees are conditional on explicit assumptions. Its mathematical existence rule separately requires verifier soundness and positive accepted mass. This note follows those two endpoints.

We have no calibrated likelihood model for the collected finite certificates under RH and under every relevant alternative, and no justified prior yielding a numerical posterior probability of RH. Repeated random searches measure properties of their stated sampling laws. A Bayesian belief about RH could be explored after choosing additional assumptions, but those choices would not turn into a measured probability of mathematical or physical ground truth. No such numerical truth probability is assigned here.

The successful claim is: **valid approximants exist in an explicitly described positive-mass family at squared-error tolerance 0.0062.** The repaired infinite-search claim is conditional. The all-accuracy existence claim and B=0 remain unresolved. The previous certified obstruction to the restricted Möbius update remains valid; this construction does not delete or overturn it.

## Reproduction

Run `python3 research/swc-rh/positive_mass_existence.py` from the workspace root. It writes `positive-mass-existence-results.json`. These follow-up files are separate from the previously generated paper and ZIP snapshot.
