# Quantified “almost zero” results from the SWC investigation

6 September 2026. **Exact zero and RH remain unproved.** Two precise near-zero statements can nevertheless be established:

1. The limiting squared error of the approximation loop is at most **0.0061879318472**, in the original normalization ||1||₂² = 1. A support-preserving stochastic refinement loop can keep this small error at every future iteration with probability at least **0.999**.
2. A different, standard RH-equivalent quantity—the Balazard–Saias–Yor weighted logarithmic mean—satisfies **0 ≤ B < 10^−12**. This follows from published zero verification and zero-count estimates. It is a corollary of those results, not a new numerical verification record.

The two numbers measure different objects. Neither is a probability that RH is true, and neither establishes arbitrary accuracy.

The subsequent [direct induction attempt](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/B_INDUCTION.md) investigates whether these bounds can be bootstrapped to B = 0.

## 1. A proved bound on the infinite-loop limit

Let χ = 1 in the cell Hilbert space used in the [recursive investigation](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/RECURSIVE_INDUCTION.md). Let H be the closed span of the integer basis functions g_k. The previously constructed operator

\[
 T=\sum_{k=2}^{\infty}\frac{1}{k(k-1)}P_{g_k}
\]

has kernel H^⊥. For r_t = (I−T)^tχ, the spectral theorem gives

\[
 \lim_{t\to\infty}\|r_t\|_2^2
 =\ell:=\|P_{H^\perp}\chi\|_2^2.
\]

The saved [N = 2048 certificate](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/certificates/nb-N2048.json) supplies a finite F ∈ H with

\[
 \|\chi-F\|_2^2\le U,
 \qquad U<0.0061879318472.
\]

Projection gives the unconditional conclusion

\[
 \boxed{0\le\ell\le U<0.0061879318472.} \tag{50}
\]

Thus the limiting root-mean-square error is below 0.078664. This is a statement about the infinite limit, obtained from a finite verified approximation and the proved projection identity. It does not improve the earlier approximation bound or show that the limit is zero. For reference, the target's initial squared norm is 1; the normalization has not been changed to make the number small.

## 2. A support-preserving recursive SWC guarantee for all future iterations

We can use F as a learned, certified direction. Since U < 1, F ≠ 0. Set η = 1/1000 and define a distribution on unit directions by

\[
 u_*=F/\|F\|_2,\qquad \Pr(*)=1-\eta,
 \qquad u_k=g_k/\|g_k\|_2,\quad
 \Pr(k)=\frac{\eta}{k(k-1)}\quad(k\ge2).
\]

The probabilities sum to one, and every original basis direction retains positive mass. Starting with R₀ = χ, draw a direction and perform the orthogonal correction

\[
 R_{t+1}=R_t-\langle R_t,u\rangle u.
\]

For every direction and every state,

\[
 \|R_{t+1}\|_2^2
 =\|R_t\|_2^2-|\langle R_t,u\rangle|^2
 \le\|R_t\|_2^2. \tag{51}
\]

If the first direction is u_*, its projection is the best scalar multiple of F and therefore has error at most ||χ−F||₂² ≤ U. Equation (51) then preserves that bound at every later iteration, regardless of the later choices. Consequently

\[
 \boxed{\Pr\!\left(\forall t\ge1:\ \|R_t\|_2^2\le U\right)
 \ge\frac{999}{1000}.} \tag{52}
\]

This is an exact SWC mass argument. The first-draw event has known mass 999/1000; projection soundness and induction establish the all-time conclusion on that event. No Monte Carlo estimate or infinite product of success probabilities is used. Starting directly from the already verified residual χ−F instead gives the same bound for **every** subsequent projection path.

The error along any such path has a limit because it is nonnegative and nonincreasing. On the event in (52), that limit is at most U. Its being exactly zero is not established. The kernel component P_(H^⊥)χ is unchanged by every correction, so proving positive probability of convergence to zero would already force ℓ = 0 and hence RH.

The projections above are exact mathematical operations. A finite-precision implementation must certify its correction, or retain the old verified state when a proposed update cannot be checked. The theorem is not a claim that unverified floating-point updates satisfy (51).

### The deterministic averaged loop also has a uniform small-error bound

Define

\[
 K=(1-\eta)P_F+\eta T,
 \qquad\bar r_t=(I-K)^t\chi.
\]

This is a deterministic averaged operator, distinct from the individual random trajectories. It is a positive contraction, and ker K = H^⊥ because F ∈ H. Choose a = 79/1000, so the exact certificate establishes √U < a. Then

\[
 \|\bar r_1\|_2
 \le(1-\eta)\|(I-P_F)\chi\|_2
       +\eta\|(I-T)\chi\|_2
 <(1-\eta)a+\eta=\frac{79921}{10^6}.
\]

Contractivity gives, for every t ≥ 1,

\[
 \boxed{\|\bar r_t\|_2^2
 <\left(\frac{79921}{10^6}\right)^2
 =0.006387366241<0.0064.} \tag{53}
\]

Its limiting squared error is the same ℓ bounded in (50). These guarantees deliberately reuse the known good direction F. They provide a small-error invariant and a convergence-to-a-small-limit statement, not a better approximant or a new RH theorem.

## 3. Almost-sure discovery of a fixed small-error witness

The existing Rademacher SWC certificate gives accepted mass greater than 1/5 for its exact error target 1/128. Independent repetitions therefore satisfy

\[
 \Pr(\text{no accepted witness in }m\text{ trials})\le(4/5)^m.
\]

Exact rational arithmetic gives:

| Trials | Guaranteed probability of finding a verified witness with squared error below 1/128 |
|---:|---:|
| 21 | Greater than 99% |
| 62 | Greater than 99.9999% |

The probability of never finding a witness is zero. This is an **almost-sure fixed-tolerance search theorem**. It is not almost-sure convergence to zero and is not a statistical confidence level for RH. An explicit witness is already available, so repeated search is an illustration of the SWC guarantee rather than a necessary way to obtain it.

## 4. A sharper near-zero result for a different RH-equivalent quantity

Define the fixed, normalized quantity

\[
 B=\frac{1}{2\pi}\int_{-\infty}^{\infty}
 \frac{\log|\zeta(\tfrac12+it)|}{\tfrac14+t^2}\,dt.
 \tag{54}
\]

The weight is the probability density of a Cauchy random variable of scale 1/2. Thus B is a weighted logarithmic mean, not an approximation error or a proportion of zeros.

The following established results are the inputs.

The [Balazard–Saias–Yor identity, stated as equation (1.1) by Bui, Lester and Milinovich](https://arxiv.org/html/1306.0856), gives

\[
 B=\sum_{\Re\rho>1/2}\log\left|\frac{\rho}{1-\rho}\right|,
 \tag{55}
\]

Here the nontrivial zeros are counted with multiplicity. Every summand is positive, so B ≥ 0 and B = 0 is equivalent to RH. No RH-conditional asymptotic from that paper is used here.

[Platt and Trudgian](https://arxiv.org/abs/2004.09765) rigorously verified that all zeros with 0 < Im(ρ) ≤ H = 3·10^12 lie on the critical line.

[Trudgian's Corollary 1](https://arxiv.org/html/1208.5846) provides, for t ≥ H,

\[
 \left|N(t)-\frac{t}{2\pi}\log\frac{t}{2\pi e}-\frac78\right|
 \le\frac{111}{1000}\log t+\frac{275}{1000}\log\log t
       +\frac{2450}{1000}+\frac{1}{5H}. \tag{56}
\]

Here N counts all positive-ordinate nontrivial zeros with multiplicity. Values at zero ordinates can be handled by right limits. The published verification of zeros was not independently rerun in this project.

### Bounding the complete infinite sum

Write ρ = β+iγ with β > 1/2. Then

\[
 \log\left|\frac{\rho}{1-\rho}\right|
 =\frac12\log\left(1+\frac{2\beta-1}{(1-\beta)^2+\gamma^2}\right)
 \le\frac{1}{2\gamma^2}.
\]

The verified height excludes off-line zeros with |γ| ≤ H. There are no nontrivial real zeros. Conjugation gives equal contributions at ±γ, and reflection across the critical line pairs each right-hand zero at positive ordinate with a distinct left-hand zero of equal multiplicity. Therefore

\[
 B\le\frac12\sum_{\gamma>H}\frac1{\gamma^2}, \tag{57}
\]

where the sum now runs over **all** positive-ordinate zeros, with multiplicity. Integration by parts gives

\[
 \sum_{\gamma>H}\frac1{\gamma^2}
 =-\frac{N(H)}{H^2}+2\int_H^{\infty}\frac{N(t)}{t^3}\,dt.
 \tag{58}
\]

Let α = 1/(2π). The explicit estimate (56) implies the deliberately coarse bounds

\[
 N(t)\le\alpha t\log t\quad(t\ge H),\qquad
 N(H)\ge\alpha H(\log H-3)-\log H-3. \tag{59}
\]

Here are the constant checks, so the simplification does not hide an asymptotic assumption. Use 1 < log(2πe) < 3, log log t ≤ log t, and the error coefficient 0.111+0.275 = 0.386. The upper side of (56) is at most

\[
 \alpha t\log t-t/8+0.386\log t+4.
\]

For t ≥ 16, log t ≤ t/4; and 57t/2000 ≥ 4 holds throughout t ≥ H. The remainder above is consequently nonpositive. On the lower side, drop the positive 7/8 and bound the error by log t+3, obtaining the second inequality in (59).

Substituting (59) into (57)–(58) yields

\[
 B\le\frac{\log H+5}{4\pi H}
       +\frac{\log H+3}{2H^2}.
\]

The exact positive partial sum Σ_(k=0)^33 29^k/k! is greater than H, proving log H < 29. Using π > 3 then gives

\[
 \boxed{0\le B<\frac{17}{6H}+\frac{16}{H^2}
 =\frac{531250000001}{562500000000000000000000}
 <10^{-12}.} \tag{60}
\]

All final scalar comparisons are checked using rational arithmetic. The constants 1 < log(2πe) < 3 can be obtained from the elementary inequalities 3 < π < 22/7 and e > 8/3: in particular 2π < 44/7 < 64/9 < e².

This is an unconditional near-zero corollary of the cited theorems. It does **not** show B = 0. A possible zero at very high ordinate, or very near the critical line, would contribute an arbitrarily small positive amount to (55). There is no established positive gap that converts (60) into exact zero.

Nor does (60) improve the bound (50) on the integer-basis approximation error: B and ℓ are different quantities. No quantitative relation transferring the tiny B bound into a comparably tiny ℓ bound is asserted here.

## 5. Status, provenance and reproduction

| Statement | Status |
|---|---|
| Infinite approximation-loop error has a limit ℓ ≤ 0.0061879318472 | Proved from the prior certificate and projection theorem |
| Stochastic refinement remains within that bound for every later iteration with probability at least 0.999 | Proved for the specified mixture |
| Deterministic averaged refinement has squared error below 0.0064 after its first step and forever thereafter | Proved analytically |
| Independent SWC search eventually finds a witness below the fixed target 1/128 with probability one | Proved; exact repetition bounds checked |
| Fixed BSY logarithmic mean satisfies 0 ≤ B < 10^−12 | Proved using the cited zero-verification and zero-count theorems |
| ℓ = 0, B = 0, arbitrary-accuracy approximation, or RH | **Not proved** |

The [arithmetic checker](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/near_zero.py) binds its reused approximation certificate to the previously recorded fingerprint and checks the new scalar inequalities. It does not repeat the earlier 128-million-cell computation, verify the external computation of zeta zeros, or formalize the cited analytic theorems. Those inputs and the displayed derivations form the mathematical proof dependencies. No novelty priority is claimed.

From `/Users/kt/Documents/ChatGPT/mathstuff`:

```bash
python3 research/swc-rh/near_zero.py
python3 -m unittest discover -s research/swc-rh -p 'test_*.py' -v
```

The [recorded near-zero results](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/near-zero-results.json) retain the exact rational bounds, source links, certificate fingerprint, probability scopes and trial counts.

The combined suite has **41 passing tests**, including exact repetition thresholds, the logarithm certificate, the final BSY bound, and a zero-count tail identity with multiplicities. These checks validate the numerical deductions; the analytic proofs and external theorem inputs are identified above.
