# Goldbach exceptional-set preprint

Working title:

> A computer-assisted \(23/33+\varepsilon\) bound for the exceptional
> set in the binary Goldbach problem

The manuscript presents a computer-assisted proof of

\[
E(X)\ll_\varepsilon X^{23/33+\varepsilon},
\]

and the strict decimal corollary

\[
E(X)\ll X^{69697/100000}=X^{0.69697}.
\]

## Status

This is a circulation-quality research preprint, not yet an established
or peer-reviewed record. The exact finite branches close, but the following
independent checks are still requested before public theorem-level promotion:

1. the two-parameter, conductor-uniform reading of Zhao's packet theorem;
2. the common-modulus Pintz bridge and general-\(A\) use of Theorem K;
3. an independent source-level audit deriving the exported rational caps,
   count inequalities, and mass budgets from the cited Zhao--Pintz estimates
   (all finite optimization rows now have separate standard-library replay);
4. a stable archived code release with a commit/tag, license, and DOI.

The current Git repository has no commit yet, so the printed SHA-256 manifest,
not a Git revision, presently identifies the source.

## Contents

- `main.tex`: standalone academic manuscript.
- `build/main.html`: Pandoc/MathML preview used for structural validation.
- `pandoc-equation-refs.lua`: preview-build filter preserving tagged equation
  anchors and printed equation references.
- `preview.css`: narrow-screen containment for display equations, tables, and
  command transcripts.
- `requirements-certificate.txt`: pinned numerical-search dependency.
- `certificate-manifest.json`: exact rational constants, theorem-critical
  commands, expected row counts, and SHA-256 identifiers for the complete
  21-file certificate dependency closure.
- `verify_release.py`: single manifest-driven integrity and certificate runner.
- `references.bib`: citation database used to build the HTML preview.
- `near-siegel-witness.json`: machine-readable exact statement of the
  near-Siegel branch, including the fixed-\(c\) gap formula.
- `secondary-aligned-witness.json`: exact rational cap, mass, envelope, fill,
  and objective witnesses for all sixteen aligned secondary rows.
- `active-scalar-witness.json`: exact rational cap, mass, envelope, fill, and
  objective witnesses for all sixty-three active and four scalar rows.
- `MANIFEST.sha256`: source and verifier identifiers.

The proof scripts remain in the project-level `experiments/` directory.
The near-Siegel branch has a separate standard-library verifier,
`experiments/zhao_near_siegel_exact.py`; it uses neither NumPy nor
floating-point arithmetic.

The HTML preview is rebuilt from this directory with:

```sh
pandoc main.tex --standalone --mathml --citeproc \
  --bibliography references.bib \
  --metadata suppress-bibliography=true \
  --lua-filter pandoc-equation-refs.lua \
  --css preview.css --embed-resources \
  --output build/main.html
```

The filter is presentation-only. It does not alter any mathematical input.
It restores four pieces of visible structure that the Pandoc HTML pipeline
loses from the LaTeX source: the manuscript's manually printed equation tags
(rendered beside each display equation and used as the printed reference
text), the referent words of cleveref cross-references ("Section 5",
"Definition 4", "Table 3" instead of bare numbers), visible level-one
section numbers and table-caption numbers consistent with those references,
and a plain-text page title. Every printed number equals the number Pandoc
itself resolves for the corresponding reference. `preview.css` supplies the
reading measure, deterministic light/dark rendering, the centered title
block, and overflow containment for wide equations, tables, and command
transcripts.

## Certificate commands

The preferred release command, run from
`research/goldbach-boundary-layer`, is:

```sh
python3 -u preprints/goldbach-exception-069697/verify_release.py
```

It refuses optimized Python, requires the pinned NumPy version, verifies every
declared source hash and exact rational bridge identity, runs all three
packet commands, the independent near-Siegel command, and the independent
aligned-secondary replay with the flags below, and checks their row counts and
success markers, together with the independent active/scalar replay. It also
checks the SHA-256 digest of each exact stdout
transcript against the frozen digest in `certificate-manifest.json`.  A fast
integrity-only check is available as:

```sh
python3 -u preprints/goldbach-exception-069697/verify_release.py --hash-only
```

The equivalent individual commands are:

Run from `research/goldbach-boundary-layer` without `python -O`:

```sh
PYTHONPATH=experiments PYTHONPYCACHEPREFIX=/tmp/goldbach-pyc \
python3 -u experiments/zhao_069697_full_active_audit.py \
  --A 33/10 --cutoff 1.466 --tolerance 1/10000 \
  --step 1/200 --end 1.24 \
  --positional-boundaries .92,1,1.05,1.10,1.15,1.20,1.25,1.30,1.40,1.466
```

```sh
PYTHONPATH=experiments PYTHONPYCACHEPREFIX=/tmp/goldbach-pyc \
python3 -u experiments/zhao_secondary_aligned_trial.py \
  --scope all --cap-bins 0 --tolerance 1/10000
```

```sh
PYTHONPATH=experiments PYTHONPYCACHEPREFIX=/tmp/goldbach-pyc \
python3 -u experiments/zhao_secondary_scalar_audit.py
```

```sh
PYTHONPATH=experiments PYTHONPYCACHEPREFIX=/tmp/goldbach-pyc \
python3 -u experiments/zhao_near_siegel_exact.py \
  --witness preprints/goldbach-exception-069697/near-siegel-witness.json
```

The last command certifies, with outward exact rational arithmetic,

\[
\mathcal C(29/1000,83/1000,13/250,26/5,0)<99.729,
\qquad \varepsilon_z\leq 1/1000,
\]

the exact identity \(100(2+1/50)=202\), and

\[
\kappa_{\rm NS}(33/10,c)\geq
\min\!\left\{
\frac{54811}{10000}\min\!\left(c,\frac1{200}\right),
\frac{6329235347}{200000000000}
\right\}>0
\]

for every fixed \(c>0\) in the near-Siegel branch.

The secondary flags shown above are theorem-critical.  The current release
defaults match them, but they are written explicitly so an archived command
cannot silently drift to a smaller diagnostic scope.

The sixteen finite aligned secondary optimizations can also be replayed
independently, without NumPy or any import from the proof implementation:

```sh
PYTHONPYCACHEPREFIX=/tmp/goldbach-pyc \
python3 -u experiments/check_secondary_aligned_witness.py \
  preprints/goldbach-exception-069697/secondary-aligned-witness.json
```

This checks the exact theorem row set and parameters, the class-count
recurrence, common class coordinates, raw caps and repeated tails,
suffix-maximum envelopes, mass budgets, greedy fills, objectives, and strict
publication ceiling.  Its explicit trusted boundary consists of the exported
analytic caps, integer count inequalities, and mass budgets; their derivation
from Zhao's estimates remains a source-level analytic obligation.

The sixty-three active and four scalar rows have the same independent,
standard-library replay:

```sh
PYTHONPYCACHEPREFIX=/tmp/goldbach-pyc \
python3 -u experiments/check_active_scalar_witness.py \
  preprints/goldbach-exception-069697/active-scalar-witness.json
```

This checker verifies the exact row cover, count recurrences, common
coordinates, repeated-tail suffix majorants, greedy mass fills, objectives,
and strict \(983/1000\) ceiling. It deliberately accepts the exported analytic
caps, count inequalities, and mass budgets as its source-level trust boundary.

## Before an arXiv or journal upload

- Replace or confirm the author name, affiliation, email, acknowledgements,
  and any required AI-assistance disclosure.
- Obtain at least one specialist analytic audit and one independent code audit.
- Add an explicit license selected by the author.
- Create a Git commit and signed release tag, archive it, and update the paper
  with the permanent URL/DOI.
- Compile with a full TeX Live installation and inspect every page.

The local bundled Tectonic runtime currently crashes on its own smoke test,
and no TeX Live installation is present. The TeX source parses cleanly through
Pandoc, but the authoritative PDF build is pending explicit approval to install
the managed TeX Live runtime.
