# Larger gradient optimization and exact finite error bounds

Private research continuation, completed 7 September 2026. The run improves finite approximation errors. It does not reach zero or prove RH.

## Result

The basis was expanded to g_2,...,g_4096, with 4,095 coefficients. Gradient-based optimization was run for both the original constant target and the clipped target h_3. The final rational coefficients were checked against the full infinite-cell error.

| Target | Previous certified squared-error upper bound | New certified squared-error upper bound |
|---|---:|---:|
| Original constant 1 | 0.006187931848 | 0.0054968440 |
| Clipped function h_3 | 0.010901797 | 0.007523824 |

The new upper bounds are below the previous candidates' exact lower error bounds, so these are verified improvements in the candidates themselves, not merely tighter estimates of unchanged coefficients. The previous constant candidate used N=2048; the previous h_3 candidate used N=256.

The new original-target candidate has squared error in

    [0.00543982938552..., 0.00549684398616...],

and the new h_3 candidate has squared error in

    [0.00731353400426..., 0.00752382337376...].

Those decimal intervals are descriptive displays of the exact rational endpoints in gradient-extension-results.json, not standalone outward-rounded certificates. The table above rounds its upper bounds conservatively.

## What the gradient run did

The optimizer used diagonally preconditioned conjugate gradients, a gradient-based method specialized to positive-definite quadratics. It optimized a numerical model consisting of the first 131,072 arithmetic cells plus the exact-form periodic-mean tail model evaluated in floating point. This tail model omits the periodic discrepancy, so its minimum is not automatically the minimum of the true full-domain energy.

The constant target started from the previously verified 2,047-coefficient approximation, extended by zeros. The h_3 target started from the previously verified 255-coefficient transfer, likewise extended. The constant run took 1,827 iterations; the h_3 run took 1,976. The stopping criterion was a relative recursive residual below 10^-11. Actual numerical gradient norms and modeled-loss traces were recorded as diagnostics.

The reported final modeled squared losses were approximately 0.005375663273 and 0.007241881483. Both are below the later verified whole-domain candidate-error intervals. This illustrates why the optimizer's loss was not used directly as an error certificate. The improvements also involve a larger basis and a different fitting objective, so they are not an experiment isolating the effect of additional iterations alone.

## Full-domain verification

Coefficients were rationalized as c_k=k b_k/2^44. The exact verifier used the existing integer residual sieve and outward rounding to a common 2^-80 grid. It checked 128 million cells for the constant target and 64 million for h_3. Periodic means, discrepancy bounds, and a separate amplitude bound enclosed the remaining infinite tails. The h_3 formulas are proved in CLIPPED_TRANSFER_ATTEMPT.md.

All acceptance inequalities use integers and rational numbers. NumPy proposes coefficients and computes diagnostic gradients; it does not decide the exact error claims. This is a computational certificate with written analytic justification, not a proof-assistant formalization.

## A stronger lower bound shows why more steps at this size cannot reach zero

A new finite-support dual certificate was constructed on 32,768 cells and then reloaded and checked exactly. It annihilates every allowed g_k for 2<=k<=4096. If its integer cell weights are alpha_m, Cauchy-Schwarz proves

    d_4096^2 >= (sum alpha_m)^2 / sum m(m+1) alpha_m^2.

The resulting exact lower bound exceeds 0.0045410457. Together with the new candidate,

    0.0045410457 < d_4096^2 < 0.0054968440.

This lower bound applies to every real coefficient choice in this finite basis and every optimization algorithm. In particular, no further gradient steps confined to this basis can reach zero or even squared error 1/256=0.00390625. A lower bound for one finite basis does not prevent the infimum over growing bases from being zero, and therefore does not disprove RH.

An earlier, weaker audit used the three-cell functional F_1+F_(p-1)-F_p, where p=4127 is prime and p=2 mod 3. It annihilates the same basis and has value one on both targets. Its squared norm as a functional is 2(p^2+1), giving the universal finite lower bound 1/[2(p^2+1)]. The larger dual substantially strengthens this for the constant target. An intermediate 8,192-cell dual gave approximately 0.0006546998; it was superseded by the saved 32,768-cell certificate.

## Reproduction and files

- `gradient-extension-candidates.json` contains both rational coefficient vectors, fitting metadata, and gradient traces.
- `gradient-extension-results.json` contains exact full-error bounds, the finite lower bounds, and file fingerprints.
- `gradient-extension-dual-N4096.json.gz` contains the stronger original-target dual witness.
- `gradient_extension.py` implements production and exact checking. Its large floating-point Gram cache is only an untrusted computational convenience and is not needed to verify the saved coefficients.

To recompute the saved candidates' full-domain bounds, run:

```bash
python3 research/swc-rh/gradient_extension.py --verify
```

To verify the saved stronger dual alone, use the existing standard-library checker:

```bash
python3 - <<'PY'
import gzip, json, sys
sys.path.insert(0, 'research/swc-rh')
from dual_verify import verify_dual
with gzip.open('research/swc-rh/gradient-extension-dual-N4096.json.gz', 'rt') as f:
    bound = verify_dual(json.load(f))
print('Verified lower bound:', float(bound))
PY
```

`--produce` regenerates the numerical candidates using NumPy; `--dual-cells 32768` regenerates and verifies the stronger dual and adds its result to the full-error report. Regenerating candidates may change floating-point details across runtimes. Full-error recomputation rewrites that report; run the dual step afterward if both kinds of result are wanted in the same JSON file.

The original PDF and ZIP remain older snapshots. No publication occurred. The conclusion is a measurable finite improvement and a certified finite limit on further optimization at N=4096, not an RH proof or a probability that RH is true.
