An assembled working-proof chain for Dittert's conjecture

Assembly scope (25 July 2026): The materials and cited sources below form a complete working proof chain for every integer dimension n >= 2. This is an assembly claim, not a claim that every link was proved independently in this repository. Original local proof notes cover n=4,5 and n=8 through n=16 (11 of the 13 dimensions in the finite gap 4 <= n <= 16); n=6,7 is an attributed reconstruction of Hongyuan Lu's pinned source argument; n >= 17 comes from Zhekai Pang's preprint; and n=2,3 come from the historical literature. None of the new local, reconstructed, or preprint portions is claimed to be formally verified, peer reviewed, or community-settled.

For a nonnegative n x n matrix A whose entries sum to n, define

$$ \Phi(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\mathrm{per}(A), $$

where the $r_i$ and $c_j$ are the row and column sums. Dittert's conjecture asserts that

$$ \Phi(A)\leq 2-\frac{n!}{n^n}, $$

with equality only at the matrix whose entries are all 1/n.

The standard conjecture is stated for n >= 2; the n=1 case is immediate from the definition. This repository contains proof notes, exact certificates, verification code, captured audit output, and checksums for the complete finite gap 4 <= n <= 16, together with a source map for the established endpoints.

Repository maintainer and author of the local proof/audit materials: Pedro Paulo Marques do Nascimento (pedromnasc).

Complete dimension coverage

Dimensions	Proof source	Review status in this chain
n=1	Direct substitution	Equality at [1], consistent with the uniqueness clause
n=2	Sinkhorn (1984)	Peer-reviewed journal result
n=3	Hwang (1987)	Peer-reviewed journal result
n=4,5	Exact local packages below	Reproducible working proofs; not yet formally peer reviewed
n=6,7	Lu's pinned source argument and the attributed reconstruction below	Two exact implementations pass; analytic reductions remain under review
n=8 through n=16	Exact local packages below	Reproducible working proofs; not yet formally peer reviewed
n >= 17	Pang, arXiv:2606.01531v1	Public preprint; not yet peer reviewed

Thus there is no uncovered integer dimension. Kafidov's independent dimension-16 preprint provides an additional cross-check at the join with Pang's range. See PROOF_STATUS.md for the audit conclusion, exact trust boundary, and citation guidance.

Unified proof architecture

The unified proof package reduces the finite-range chain to a single scalar inequality instead of one presentation per dimension. Its proof note shows that every boundary maximizer is excluded by one contradiction functional

Psi(u) = M (1-u)^n - gamma_n + Delta(u),

in which only the permanent floor M and the deficit bound Delta change. With the quadratic deficit bound this has a closed-form minimum eta_n(a,b) that is positive on 293 of the 295 Hall cuts of n=8..16. The two open rows are the marginal cuts of dimensions 8 and 9, closed by the same functional with a sharper Delta. Only those two rows and the honestly labeled n=4..7 base/residual packages remain exceptional.

This is the recommended entry point for reading the proof. The dimension-specific packages below remain the authoritative audit sources.

Sub-Dittert extension project

The subdittert package begins a rigorous extension of the same scaling architecture to the Cheon--Hwang sub-Dittert problem. It proves a new elementary-symmetric subset-deficit lemma and gives an exact Hall-cut scan for the generalized functional. A permanent lift bypasses the conjectural one-zero subpermanent minimizer at the first two new orders. At k=n-1, the inherited zero and the lift zero form the same two-zero permanent face used in the unified Dittert proof, giving the needed floor for 18 <= n <= 80. At k=n-2, exact tensor-Bernstein certificates control the lifted 2 x 2-zero-block face for 18 <= n <= 40. Converting the resulting boundary exclusions into full sub-Dittert theorems still requires an audit of the generalized structural reduction. Broader intermediate-order scans remain explicitly conditional.

Local exact proof and audit material

Dimensions	Proof note	Audit and reproduction notes
n=4	PDF · LaTeX	Independent audit report · bundle README
n=5	PDF · LaTeX	exact verifier and reproduction guide
n=6,7 (attributed reconstruction)	PDF · LaTeX	two exact implementations and status · attribution and source pin
n=8	PDF · LaTeX	two exact verifiers and reproduction guide · source notes
n=9	PDF · LaTeX	two exact verifiers and reproduction guide · source notes
n=10	PDF · LaTeX	two exact verifiers and reproduction guide · source notes
n=11,12,13	PDF · LaTeX	exact verifier and independent-audit guide
n=14,15,16	PDF · LaTeX	reproduction guide · source and citation notes

The separate n6 directory contains the repository's earlier, independently developed proof-search material rather than the attributed reconstruction above. It includes an exactly closed seven-parameter two-zero family, its exact SOS verifier, exact certificates for all three complete three-zero pattern types, the earlier fully symmetric subcase, and a spectral stationarity obstruction. Consequently any hypothetical boundary maximizer must have at least four zeroes. The precise remaining gaps and negative search results are kept in the strategy notes.

Reproduce the exact checks

Python 3.10 or newer is sufficient for the standard-library verifiers. NumPy is used by two additional n=4 checks and the exact SOS verifiers for the closed n=6 families. SymPy is used by the closed n=6 symmetric-subcase verifier and the independent audits for n=6,7 and n=11 through 16. Exact dependency versions used in automated verification are recorded in requirements-audit.txt. On distributions that mark system Python as externally managed under PEP 668, install these dependencies in a virtual environment as shown below; do not use --break-system-packages.

git clone https://github.com/pedromnasc/dittert-conjecture-proof.git
cd dittert-conjecture-proof

# Dependencies for every available verifier
python3 -m venv .venv
. .venv/bin/activate
python -m pip install -r requirements-audit.txt

# Unified finite-range reductions: exact verifier and integration tests
cd unified
sha256sum -c SHA256SUMS
python3 -I verify_unified_reductions.py
python3 -O -I verify_unified_reductions.py
python3 -I test_unified_package.py

# Sub-Dittert research reductions and penultimate-order lift certificates
cd ../subdittert
sha256sum -c SHA256SUMS
python3 -I verify_subdittert_reductions.py
python3 -O -I verify_subdittert_reductions.py
python3 -I test_subdittert_package.py

# n=4: check file integrity and run three independent certificate readers
cd ../n4
sha256sum -c SHA256SUMS
python3 verify_primary.py dittert_n4_exact_certificate.npz
python3 verify_literal_square_numpy.py dittert_n4_exact_certificate.npz
python3 verify_literal_square_stdlib.py dittert_n4_exact_certificate.npz

# n=5: check file integrity and the exact quintic certificate
cd ../n5
sha256sum -c SHA256SUMS
python3 verify_primary.py dittert_n5_exact_certificate.npz

# n=6: verify the closed subcases (not the full conjecture)
cd ../n6
sha256sum -c SHA256SUMS
python3 -I verify_seven_parameter.py
python3 -I test_rejects_corrupt_seven_parameter.py
python3 -I verify_three_zero_equalized.py
python3 -I test_rejects_corrupt_three_zero.py
python3 -I verify_three_zero_independent.py
python3 -I test_rejects_corrupt_three_zero_independent.py
python3 -I test_integral_gram.py
python3 -I verify_symmetric_subcase.py

# n=6,7: attributed reconstruction of Lu's candidate and independent audit
cd ../n6-n7
sha256sum -c SHA256SUMS
python3 -I verify_dittert_n6_n7.py
python3 -O -I verify_dittert_n6_n7.py
python3 -I test_n6_n7_package.py
python3 -I audit_dittert_n6_n7_sympy.py

# n=8: check integrity, run two independent exact verifiers, and test them
cd ../n8
sha256sum -c SHA256SUMS
python3 -I verify_dittert_n8.py
python3 -I audit_dittert_n8_stdlib.py
python3 -I test_n8_package.py

# n=9: check integrity, run two independent exact verifiers, and test them
cd ../n9
sha256sum -c SHA256SUMS
python3 -I verify_dittert_n9.py
python3 -I audit_dittert_n9_stdlib.py
python3 -I test_n9_package.py

# n=10: check integrity, run two independent exact verifiers, and test them
cd ../n10
sha256sum -c SHA256SUMS
python3 -I verify_dittert_n10.py
python3 -I audit_dittert_n10_stdlib.py
python3 -I test_n10_package.py

# n=11,12,13: check file integrity and run both exact audits
cd ../n11-n13
sha256sum -c CHECKSUMS.sha256
python3 -I verify_dittert_n11_n13.py
python3 -I audit_dittert_n11_n13_sympy.py

# n=14,15,16: check file integrity and run both exact audits
cd ../n14-n16
sha256sum -c MANIFEST.sha256
python3 verify_dittert_n14_n16.py
python3 audit_dittert_n14_n16_sympy.py

Run the n=4 and n=5 programs without Python's -O option. Each assertion-based verifier rejects optimized mode explicitly so that proof checks cannot be silently disabled. The same commands run automatically on every push and pull request through GitHub Actions.

The expected final status is CERTIFIED for each n=4 verifier, the n=5 verifier, and the primary n=8, n=9, and n=10 verifiers; SEVEN-PARAMETER TWO-ZERO FAMILY CERTIFIED, TWO EQUALIZED THREE-ZERO FAMILIES CERTIFIED, INDEPENDENT THREE-ZERO FAMILY CERTIFIED, and SYMMETRIC TWO-ZERO SUBCASE CERTIFIED for the n=6 subcase checks; EXACT N=6,7 RECONSTRUCTION CERTIFIED and INDEPENDENT SYMPY/NUMPY N=6,7 AUDIT CERTIFIED for the attributed package; INDEPENDENT AUDIT CERTIFIED for the independent n=8, n=9, and n=10 verifiers and each SymPy audit; and ALL CASES CERTIFIED for each multi-dimension primary verifier. No floating-point value is used in a correctness decision. The n=8, n=9, and n=10 packages use only the Python standard library and test identical behavior under Python's optimized mode.

Verification scope

The programs verify the finite algebraic parts of the arguments: certificate identities, polynomial calculations, exact rational bounds, and final strict inequalities. They do not reprove the published structural theorems used in the reductions. Those dependencies and citations are identified in the proof notes, the n=4 audit report, the n=5 proof note, the n=6,7 reconstruction and attribution record, the n=8 proof note, the n=9 proof note, the n=10 proof note, the n=11,12,13 proof note, and the n=14,15,16 source notes.

Passing all programs establishes the exact finite certificate layer, not the entire human proof by itself. The assembled all-dimensions conclusion also uses the historical n=2,3 theorems and Pang's n >= 17 preprint. The status and evidence for each link are recorded in PROOF_STATUS.md.

Within the local original arguments, the n=8 and n=9 analytic reductions are the principal human-review boundary. Their exact programs verify the resulting polynomial identities, interval bounds, and case enumerations, but do not independently derive the stationarity, support, and cut-optimization reductions that produce those obligations. A modeling error in that analytic layer would therefore not be detected merely by rerunning the certificates.

Independent scrutiny of both the mathematical reductions and the software is welcome. Please use GitHub Issues for errors, questions, or independently reproduced results.

Citation

Repository citation metadata is provided in CITATION.cff. Please cite the original range-specific authors identified above and retain the working-proof and peer-review status when discussing the assembled result.

License

The papers, LaTeX sources, documentation, certificates, audit material, and other non-software research content are licensed under CC BY 4.0. The verification code and supporting software infrastructure are licensed under the MIT License. See LICENSE.md for the precise scope, including archived bundle contents.

Bundle provenance

The repository was assembled on 23 July 2026 from the three original bundles below; the n=5 certificate plus the n=8, n=9, and n=10 proof packages were subsequently developed in this repository on the same date. The untouched archives are retained under original-bundles/; the reviewed working directories have their own current checksum manifests. Provenance hashes for the initially supplied n=10 verifier and output are recorded in n10/SOURCE_NOTES.md.

The attributed n=6,7 reconstruction was added on 25 July 2026 from Hongyuan Lu's immutable source commit; its separate discovery, licensing, and adaptation record is n6-n7/ATTRIBUTION.md.

• n=4 original bundle
• n=11,12,13 original bundle
• n=14,15,16 original bundle

6b42ad7ebe73fc1686f7442c093ef872c1b25f30dfab1510da27be6000260e01  original-bundles/dittert_n4_audited_bundle_2026-07-23.zip
bfe6f67cd2eb5a6428bd3476236332dfb83ece6b6950db7ef238e96c7a1c0270  original-bundles/dittert_n11_n13_exact_proof_bundle_2026-07-23.zip
3536bf1bcb6ab1b141b1bea74c7ad9ab41b307b3dfa6beb13f883e3a5739149a  original-bundles/dittert_n14_n16_exact_proof_bundle_2026-07-23.zip