Which String Vibrations Become Which Generations on the Dwork Quintic — aomart research revision

Which String Vibrations Become Which Generations on the Dwork Quintic

No vacuum satisfying all nine GSC conditions has been constructed. The manuscript retains its explicit family calculations.

This aomart research revision is not a proof of all eight requested closures. No journal acceptance or new human review is claimed. The package has been compiled and visually checked; all eight PDFs have no unresolved references, overfull boxes or off-page text in the recorded check. The archival record for this exact release version has been confirmed.

Files committed with this release target:

• 07-DworkGenerations-aomart.pdf
• 07-DworkGenerations-aomart-source.zip
• 07-DworkGenerations-scope.md

The repository has an active Zenodo release webhook. No guessed or inherited DOI is assigned to this revision.

Verified Zenodo version DOI: 10.5281/zenodo.23601924. The DOI identifies this archived research revision; the scientific scope stated above is retained.

Which String Vibrations Become Which Generations on the Dwork Quintic, release 1.2.0

This release adds the subsection "Free quotients of the one-parameter families" and sets the paper in a single column for submission.

• For a simply connected Calabi–Yau threefold with Picard group Z, an automorphism of prime order p without fixed points forces p to divide chi(X, kL) for every k, by the holomorphic Lefschetz formula of Atiyah and Bott.
• For the thirteen hypergeometric one-parameter complete intersections the greatest common divisor g_X of these values is 5, 1, 2, 1, 3, 2, 1, 8, 4, 1, 1, 1, 1 (X_5, X_6, X_8, X_10, X_{3,3}, X_{4,2}, X_{3,2,2}, X_{2,2,2,2}, X_{4,4}, X_{6,6}, X_{4,3}, X_{6,4}, X_{6,2}).
• Every prime dividing the order of a group acting freely divides g_X, so seven types, among them the sextic and the dectic, admit no free action at all, and no free quotient of any of the thirteen carries three families with the standard embedding.

The archive now holds 215 checks (94 new ones in code/checks/free_quotients.py, in exact arithmetic) and 14 Lean 4 theorems, each axiom-free or depending on propext only. The engine is cited at Anaxia release 3.2.1 (doi:10.5281/zenodo.23524945). paper/paper1/ holds the compiled paper and paper/paper2/ its LaTeX source as one zip. The paper cites this release by its release page and the concept DOI, 10.5281/zenodo.23480310.

Assets: the PDF, the LaTeX source with the figures as PNG (tex.zip), and the arXiv package (tar.gz) with the checks, the engine record and the Lean file in anc/.

Which String Vibrations Become Which Generations on the Dwork Quintic, release 1.1.0

This release adds Section VIII, Theorem E and Appendices D and E: the three other one-parameter pencils of Fermat type, the sextic in P(1,1,1,1,2), the octic in P(1,1,1,1,4) and the dectic in P(1,1,1,2,5).

• Each member with psi^d prod w_i^{w_i} ≠ 1 is a smooth Calabi–Yau threefold with h^{1,1} = 1 and h^{2,1} = 103, 149 and 145.
• The generations are the monomials x^b with 0 ≤ b_i ≤ n_i − 2 of weighted degree d, but they carry only 96, 122 and 100 distinct characters of the symmetry groups of order 108, 128 and 100: 7, 27 and 42 classes of two or three generations share a character, found by hand through the wrapping lemma.
• The socle functional is psi^r prod w_i^{k_i} in sector r and zero outside the sectors, so every coupling allowed by the symmetry is nonzero for psi ≠ 0.
• The mass map of the invariant direction has rank 99, 125 and 109 at every member with psi ≠ 0, leaving 4, 24 and 36 generations massless, and 44, 85 and 72 at psi = 0; the blocks are listed in Tables IV and V.
• No member and no free quotient of a member carries three families, by Riemann–Roch on the descended norm line bundle.

The quintic is the only one of the four pencils on which the characters label every generation and the invariant deformation gives every generation a mass.

The archive now holds 121 checks (Python and Macaulay2 checks of the weighted pencils added) and 14 Lean 4 theorems, each axiom-free or depending on propext only, and the engine record covers five targets, run on Anaxia release 3.1.0 (doi:10.5281/zenodo.23484160). The paper cites the archive by its concept DOI, 10.5281/zenodo.23480310.

Assets: the PDF, the LaTeX source with the figures as PNG (tex.zip), and the arXiv package (tar.gz) with the checks, the engine record and the Lean file in anc/.

Which String Vibrations Become Which Generations on the Dwork Quintic, release 1.0.2

The paper now cites its archive by the Zenodo version DOI of release 1.0.1, 10.5281/zenodo.23480492, together with the concept DOI 10.5281/zenodo.23480310 of all versions, and the citation file and the README list both. The master zip DworkGenerations_masterzipfinal2.zip is rebuilt from this release.

The mathematics, the 64 checks and the 11 Lean 4 theorems are those of release 1.0.0.

Assets: the PDF, the LaTeX source with the figures as PNG (tex.zip), and the arXiv package (tar.gz) with the checks, the engine record and the Lean file in anc/.

Which String Vibrations Become Which Generations on the Dwork Quintic, release 1.0.1

The paper now cites its archive by the Zenodo version DOI of release 1.0.0, 10.5281/zenodo.23480311, and the citation file and the README list that DOI and the concept DOI 10.5281/zenodo.23480310 of all versions. The repository also holds DworkGenerations_masterzipfinal2.zip, which collects the paper, its LaTeX source, the arXiv package and the full archive in one file.

The mathematics, the 64 checks and the 11 Lean 4 theorems are those of release 1.0.0.

Assets: the PDF, the LaTeX source with the figures as PNG (tex.zip), and the arXiv package (tar.gz) with the checks, the engine record and the Lean file in anc/.

Which String Vibrations Become Which Generations on the Dwork Quintic

First release of the paper "Which String Vibrations Become Which Generations on the Dwork Quintic" by Deep Bhattacharjee and Ushashi Bhattacharya, with its checks and the record of the Anaxia engine.

• The 101 generations of the heterotic string on the quintic pencil carry 101 distinct characters of the symmetry group (Z/5)^3; the 24 characters that do not occur are the classes of the permutations of (0,1,2,3,4).
• Every Yukawa coupling allowed by the symmetry is a power ψ^r of the modulus, with r = 0, 1, 2, 3 on 566, 425, 280 and 160 unordered triples, 1431 in all, and every other coupling vanishes.
• The invariant direction pairs each generation with exactly one partner; the mass matrix has determinant ψ^70, full rank for ψ ≠ 0, and rank 51 at the Fermat point, where the masses are given by the Gamma function.
• The Geometric Selection Conjecture is equivalent to the existence of one viable vacuum fixed by mirror symmetry; no smooth quintic and no free quotient of one carries three families with the standard embedding. The existence of a viable vacuum is open.

Every result is proved by hand in the paper. code/run_checks.sh runs 64 independent checks in C, C++, Python, Julia, Macaulay2, Lean 4 and shell, and lean/Strings.lean proves 11 Lean 4 theorems, each axiom-free or depending on propext only. code/bllm/ holds the failure, change and retry record of the four targets on the Anaxia engine (creelie/eBLLM, release 3.0.5).

Assets: the PDF, the LaTeX source with the figures as PNG (tex.zip), and the arXiv package (tar.gz) with the checks, the engine record and the Lean file in anc/.