A computer-assisted proof of $\Lambda\le0.1787854$

Status. This repository presents a computer-assisted proof of the bound below, developed through human–AI collaboration and published openly for independent verification. The argument uses no unproved conjecture: it rests on the published Polymath 15 criterion, the Platt–Trudgian verified height, and the fail-closed interval certificates in this repository, every one of which can be replayed from source. It has not yet been peer reviewed.

The primary paper is gomila_dbn_lambda_01787854_release.pdf.

The proposed conclusion is

$$ \boxed{\Lambda\le \frac{893927}{5000000}=0.1787854}. $$

The exact parameters are

$$ X=6000000185827,\qquad t_0=\frac{129}{800}=0.16125,\qquad y_0^2=\frac{87677}{2500000}=0.0350708, $$

where $y_0$ is the positive square root. Thus

$$ t_0+\frac{y_0^2}{2} =\frac{893927}{5000000}. $$

The logical form is unconditional: the argument does not assume the Riemann hypothesis or another unproved conjecture. It uses the published Platt--Trudgian finite verification of RH, the published Polymath Theorems 1.2 and 1.3, and the interval certificates and elementary lemmas in this repository. Everything is open for independent review: the cited-theorem applications, the new lemmas, the proof-to-code correspondence, and the computations, with the referee entry points recorded in OPEN_REVIEW_QUESTIONS.md.

This positive upper bound is not a proof of RH.

Which document should I read?

The package contains several prose documents. They are not interchangeable and only some are sealed. Sealed means covered by SHA256SUMS and therefore attested by verify.sh; unsealed material is working state that may change without a reseal and is not evidence for the proof.

Document	Sealed	Read it for
gomila_dbn_lambda_01787854_release.pdf	yes	The primary paper. The authored release manuscript (20 August 2026, bylined Jude Gomila, with thanks to Dan Romik and Max Atkin), including a human–AI collaboration disclosure; its program links are pinned to repository commit 6222740e, and its LaTeX source is in paper/release/. Start here.
output/pdf/dbn_lambda_01787854_candidate_audit.pdf	yes	The release-grade referee manuscript. Generated deterministically from paper/generate_paper.py. The package's own account of itself.
paper/external/gomila-proof-exposition.pdf	yes	A supplied theorem-style exposition, pinned by exact size and SHA-256 in verifiers/verify_external_exposition.py. Read paper/external/README.md first: it is the mandatory cross-check and records corrections required in a revised version.
dan-reworking/latex/exposition/	no	The referee's in-progress reworking of that exposition, with LaTeX source. Further developed than the sealed PDF, but not attested.
dan-reworking/latex/research-paper/	no	The same mathematics written as a research article — bylined Jude Gomila and [additional authors TBD], declarative (“We prove…”). Author list still to be finalized. Not attested.
PROOF_NOTE.md	yes	The complete theorem chain in the repository's own terms.

Authority runs top to bottom: where a reworked document disagrees with a sealed one, the sealed artifact is what the audit attests. Two files share the basename gomila-proof-exposition.pdf and have different contents; only the copy under paper/external/ is pinned.

Nothing in this table is a substitute for OPEN_REVIEW_QUESTIONS.md, which records the decisions actually being asked of a referee.

Background reading (to understand the setting)

Readers new to the de Bruijn--Newman constant $\Lambda$ and the heat-flow picture of the Riemann $\xi$ function will find Dan Romik, Orthogonal polynomial expansions for the Riemann xi function (arXiv:1902.06330, Advances in Mathematics) a helpful entry point. It develops Hermite, Meixner--Pollaczek, and continuous Hahn expansions of $\xi$, connects Turán's classical program on the location of the zeta zeros with the separate de Bruijn--Newman program, and gives an accessible account of the objects this proof manipulates: $\xi$, its heat-flow deformation $H_t$, and the reality-of-zeros criterion whose threshold is $\Lambda$. The exact reviewed PDF is identified by SHA-256 in references/README.md (arXiv non-exclusive-distribution license, so it is not redistributed here; author retains copyright); see BIBLIOGRAPHY.md entry 4.

The precise object used by the proof is the Polymath15 normalization $H_0(z)=\tfrac18,\xi(\tfrac12+\tfrac{iz}{2})$ and its heat flow $H_t$; the constant $\Lambda$ is the threshold at or above which all zeros of $H_t$ are real. The proof does not re-derive this theory — it instantiates the published Polymath15 upper-bound criterion (Theorem 1.2) at exact parameters.

Proof architecture

The package supplies the three hypotheses of Polymath Theorem 1.2 in its canonical order.

1. Verified-height hypothesis (i). Platt--Trudgian verify RH through $T_{\rm PT}=3000175332800$. The criterion needs only $X/2=3000000092913.5$, leaving the exact margin $350479773/2$.
2. Final-time right-half-line hypothesis (ii). The proof establishes that, at $t=t_0$, the finite Triangle certificates cover all windows $N=690988,\ldots,3840000$, and the standalone tail lemma covers every $N\ge3840000$. The two lanes overlap on the complete $N=3840000$ window. Together they prove nonvanishing for $$ x\ge X+\sqrt{1-y_0^2},\qquad y_0\le y\le\sqrt{1-2t_0}. $$
3. Intermediate-time barrier hypothesis (iii). The proof uses a fail-closed Arb certificate proving $$ H_t(z)\ne0 \quad\text{on}\quad [X,X+1]+i[0.1809,1],\quad 0\le t\le t_0. $$ This closed rectangle contains the complete curved barrier required by Theorem 1.2.

Together with the theorem weld, Theorem 1.2 gives $\Lambda\le t_0+y_0^2/2=0.1787854$; each supplier and the weld are open to independent re-derivation.

Some historical notes under vendor/ state older conditional implications that did not themselves certify hypotheses (ii) and (iii). They are retained as provenance; the new finite, tail, and barrier arguments are the suppliers of the three hypotheses.

Decisive certified margins

• The $3,149,013$-row finite scan has $$ T_{\min}=0.000000791366,\qquad E_{\max}\le0.000000233494905212337849, $$ hence $$ T_{\min}-E_{\max} \ge0.000000557871094787>0. $$ The $E_{\max}$ budget and this margin are certified authoritatively by the standalone FLINT/Arb program verifiers/verify_prop410_arb.c, run at 256 and 512 bits in the pinned container and strictly parsed (assembly prerequisite P17). The original mpmath.iv computation and its derived copy under independent/prop410/ remain as same-backend corroboration only.
• One standalone FLINT/Arb implementation, run at both 256 and 512 bits, proves $D<0.999721$ and a normalized post-error margin greater than $0.00017352$. A separate Python interval implementation supplies cross-implementation corroboration.
• The barrier uses an approximation allowance $0.00125$. A separate 256-bit Arb calculation bounds the complete Theorem 1.3 error, using the conservative Proposition 6.6(vi) corollary, by $$ 0.000356523011600040<0.00125. $$
• All 883 consecutive closed time prisms pass. The independently parsed minimum prism margin has certified lower bound $0.519849894613872543374989997$, and the final coverage endpoint contains $129/800$. Complete Linux/GCC/FLINT 3.0.1 and macOS/Clang/FLINT 3.6.0 replays both pass the strict 54-check parser.

These margins are rigorous interval inequalities in the recorded certificates; the rounded decimals above are for orientation.

New theorem-level bridges in this package

• NATIVE_BINDING.md proves by exact Dirichlet convolution that each positive stored Triangle floor is already a lower bound for the paper's normalized $|f_t|$. No extra Euler-factor conversion is needed.
• ERROR_CONSTANT_WELD.md derives the conservative $10.50$ effective-error constant directly from Proposition 6.6(vi), avoiding reliance on the displayed $10.44$ in equation (24).
• WINDOW_FREEZE_THEOREM.md proves the conservative directions of the per-window $x$-freeze and the exact half-open endpoint coverage.
• TAIL_LEMMA.md states and proves the all-$N$, all-$y$ tail contraction theorem consumed by a standalone FLINT/Arb implementation.
• DERIVATIVE_BOX_LEMMA.md proves the uniform spatial-envelope, discrete-sum, and holomorphic-quadrature contracts behind the barrier derivative bounds.
• BARRIER_CERTIFICATE.md gives the closed-prism, interpolation, winding, coefficient-provenance, Taylor-remainder, effective-error, and $t=0$-continuity arguments for the barrier.

Historical standard-majorant, site-glue, and winding artifacts under vendor/ are retained for provenance only. They are not proof inputs for the new native binding, exact site coverage, or closed barrier certificate. The target error paths use $x-6.66$ for $e_A+e_B$ and the conservative Proposition 6.6(vi) corollary with $x-12$ and $10.50$ for $e_{C,0}$.

Supplied theorem-style exposition

A mathematician supplied a separate 22-page theorem-style exposition of the proof. It is valuable because it puts the complete three-hypothesis weld and the load-bearing review points into conventional mathematical paper form.

The supplied draft also contains known numerical, logical, attribution, and supplement-provenance defects. It is therefore preserved as a supplemental draft, not substituted for the repository proof record and not treated as external sign-off. Read paper/external/README.md first, then paper/external/gomila-proof-exposition.pdf. The cross-check binds the PDF to its exact SHA-256, the v3 baseline commit, the controlling repository values, and a page-by-page correction list.

Start here

For mathematical review, read in this order:

1. Primary paper (PDF)
2. Referee manuscript (PDF)
3. PROOF_NOTE.md
4. OPEN_REVIEW_QUESTIONS.md
5. CANDIDATE_PARAMETERS.md
6. NATIVE_BINDING.md
7. ERROR_CONSTANT_WELD.md
8. WINDOW_FREEZE_THEOREM.md
9. TAIL_LEMMA.md
10. DERIVATIVE_BOX_LEMMA.md
11. BARRIER_CERTIFICATE.md
12. the Polymath and Platt--Trudgian papers listed in BIBLIOGRAPHY.md

REVIEW_SCOPE.md gives a referee workflow and separates theorem review from software replay. MAXIMUM_CHECKS.md records the strongest checks performed before handoff and the final-seal checks that must remain green. Reviewers should also read ADVERSARIAL_REVIEW_PROTOCOL.md for the dependency blueprint, boundary matrix, mutation catalogue, and independent recomputation standard. They should also read CITATION.md, REVIEW_TERMS.md, and THIRD_PARTY.md. SECURITY.md states the security reporting, execution, integrity, and secret-handling boundaries. CONTAINER_IMAGE.md specifies the pinned local build and release-attested OCI digest verification.

The release-grade referee manuscript is generated deterministically from paper/generate_paper.py. See paper/README.md for generation and visual-QA commands.

Quick stored verification

python3 -m venv .venv
. .venv/bin/activate
python3 -m pip install --require-hashes -r requirements.txt
./verify.sh

The assembly transcript must include

RESULT: UNCONDITIONAL PROOF ASSEMBLY PASS
STATUS: computer-assisted unconditional proof; not yet peer reviewed.

“PASS” means that the encoded predicates were certified; it is not a claim of peer review. The phrase UNCONDITIONAL in the machine transcript names the logical form of the argument (no unproved conjecture is assumed), and the STATUS line is the sealed verifier's fixed output string, quoted here verbatim so the transcript match is exact.

Fresh replays

Portable container replay:

docker build --platform linux/amd64 -t dbn-lambda-01787854-review .
mkdir -p replay/container-review
docker run --rm --platform linux/amd64 --network none --read-only \
  --cap-drop ALL --security-opt no-new-privileges \
  --user "$(id -u):$(id -g)" \
  --tmpfs /tmp:rw,exec,nosuid,size=4g \
  -e REVIEW_OUTPUT=/review-output/evidence \
  -v "$PWD:/work:ro" \
  -v "$PWD/replay/container-review:/review-output" \
  -w /work dbn-lambda-01787854-review

Fresh coefficient regeneration and all 883 barrier prisms:

./scripts/run_barrier_replay.sh replay/barrier

Fresh 256- and 512-bit runs of the standalone Arb tail checker:

./scripts/run_tail_arb.sh replay/tail_arb

Fresh 256- and 512-bit runs of the authoritative Arb Proposition 4.10 error-budget checker:

./scripts/run_prop410_arb.sh replay/prop410_arb

Expensive regeneration of every finite row:

docker build --platform linux/amd64 -t dbn-lambda-01787854-review .
review_image_id=$(docker image inspect --format '{{.Id}}' \
  dbn-lambda-01787854-review)
IMAGE=dbn-lambda-01787854-review \
EXPECTED_IMAGE_ID="$review_image_id" \
  ./scripts/run_full_sweep.sh replay/full_sweep

The output path must not already exist. The full sweep regenerates all $3,149,013$ tuples and compares them with the 15 sealed certificate shards.

Package map

• gomila_dbn_lambda_01787854_release.pdf: the primary paper.
• certificates/: the complete compressed finite certificate.
• barrier/: closed-prism sources, coefficient data, provenance evidence, and the sealed 883-prism run.
• logs/: finite, Dini, normalizer, Python-tail, Arb-tail, Arb-prop410, and assembly transcripts.
• src/: finite Triangle producers.
• verifiers/: fail-closed parsers and independent interval checkers.
• provenance/: direct all-height proofs and historical audit notes.
• vendor/: version-locked upstream material and historical provenance.
• scripts/: stored, container, barrier, tail, and full-sweep replay paths.
• scripts/fetch_release_image_oci.py and scripts/verify_release_image_archive.py: complete offline-image reconstruction and fail-closed verification.
• references/: the versioned upstream proof record.
• ADVERSARIAL_REVIEW_PROTOCOL.md: falsification and independent-sign-off protocol.
• SECURITY.md: security reporting and execution policy.
• CONTAINER_IMAGE.md: canonical container and exact verified image binding.
• paper/: deterministic source for the referee manuscript.
• paper/external/: supplied theorem-style exposition and its mandatory repository cross-check.
• output/pdf/: release-grade referee manuscript.

License

• Source code original to this repository: MIT.
• Documentation, data, certificates, and transcripts original to this repository: CC BY 4.0.
• Third-party material under vendor/, references/, and paper/external/ remains under its own recorded terms; see THIRD_PARTY.md.
• Exceptions (all rights reserved pending author agreements): the authored manuscripts gomila_dbn_lambda_01787854_release.pdf (with its source under paper/release/) and dan-reworking/, and the independent/ programs authored by Dan Romik. See LICENSE-DOCS for details.

Cite via CITATION.md.

This repository presents a computer-assisted proof with a complete, replayable evidence chain. Independent mathematical and computational review is invited; OPEN_REVIEW_QUESTIONS.md records the questions referees are asked to decide.