Structural theorems on the Fermat quartic fourfold lattice, from an attempt on the dimension-128 sphere-packing record
This repository collects the theorems proven during Operación Glotón, an attempt to beat the densest known sphere packing in dimension 128 — the Mordell–Weil lattice MW128 of Elkies, unbeaten since 2001 (centre density δ = 2⁹⁷·⁴⁰). The attempt works entirely inside one highly symmetric object: V(4,4)-prim, the primitive middle-cohomology lattice of the Fermat quartic fourfold x₀⁴ + ··· + x₅⁴ = 0 — positive definite, even, rank 141, determinant 2¹²⁸. The target is a rank-128 sublattice dense enough to give δ ≥ 2⁹⁸·⁴⁴.
The record is open. The work fully controls one half of the problem — the price, a 2-adic quantity understood down to a proven floor — and runs into a precise, identified wall on the other half — the lattice minimum, an archimedean condition that no structural invariant determines. That wall is now mapped in closed form: the norm-12 vectors obstructing the minimum are classified, counted, and their symmetric attack routes are sealed with certificates — yet the minimum itself remains archimedean and open. The full accounting is in THE_OPEN_PROBLEM.md.
Independent of the record, what this repository offers is the rigorous structural mathematics the attempt produced: a connected body of theorems about V(4,4)-prim, each established in exact integer (or Gaussian-integer) arithmetic on a single 8 GB laptop and independently re-checkable. The keystone is the Bridge Theorem — the identity proving that the lattice's arithmetic (Hodge) norm and its geometric intersection form are one and the same. Several of these results are of interest beyond the packing problem.
New to the subject? Start with the Guide for everyone.
Author: Rafael Amichis Luengo · Madrid · github.com/tretoef-estrella Hardware: MacBook Air M2 (2022), 8 GB RAM, single thread, throttled to 25% CPU. No swap, no cluster, no cloud. Exact arithmetic throughout.
Fix the Fermat quartic fourfold X : x₀⁴ + ··· + x₅⁴ = 0 ⊂ P⁵. Its 960 linear planes (15 perfect matchings of six coordinates × 64 phases) span a rank-142 lattice; the primitive part V(4,4)-prim = ⟨h⟩^⊥ is the rank-141 sublattice orthogonal to the hyperplane class, even and positive definite with det = 2¹²⁸.
A section is a primitive S ⊆ V(4,4)-prim of rank 128 (codimension 13). With min(S) its squared shortest length and pr(S) = det(S)/2¹²⁸ its price, the centre density obeys log₂ δ = 64·log₂(min/4) − 64 − ½·log₂ pr. The problem is whether a section exists with min ≥ 24 and pr < 2⁸·⁰⁸ (which yields δ ≥ 2⁹⁸·⁴⁴, beating MW128). The two conditions oppose — price pulls the section together, the minimum pushes its vectors apart — and a record section is their equilibrium. Status: open (THE_OPEN_PROBLEM.md).
Two forms live on the plane lattice: the geometric intersection form Γ (plane self-intersection 7; pairwise incidences +1 / 0 / −2 for planes meeting in a point, disjoint, or sharing a line) and the Hodge norm computed from Movasati–Villaflor periods as a sum of eighth roots of unity. The theorem proves they are one form: G = 4Γ − J (with J the all-ones matrix), hence ‖v‖²_Hodge = (v·v)_geom − N(v)²/4, coinciding exactly on the primitive sublattice (N = 0). As a corollary the minimum is closed: positive-definiteness makes the vectors below any bound a finite, explicit set, replacing an infinite eighth-root-of-unity minimisation with finite integer linear algebra. This identity is what makes the whole structural route computable.
These are the results original to this work, all about V(4,4)-prim, all byte-exact.
| Theorem | What it establishes |
|---|---|
| The Spectrum Theorem | A complete diagonalisation of the intersection form by K₆ symmetry: exactly four nonzero eigenvalues 16·{15,6,3,2} = {240, 96, 48, 32}, multiplicities 2·{1,10,15,45}, governed by the eigenvalue law λ(a) = 16·N_a (N_a = matchings carrying character a). The exact shape of the form, with a structural law for it. |
| The Parity Theorem | The span of any set of plane families has odd primitive rank — complex conjugation acts on each support with a single fixed point (the impostor χ* = (2,2,2,2,2,2)), forcing rank = 1 + 2·(#conjugate pairs). Hence no rank-128 family span exists: the section must be cut, not assembled. An entire construction route closed by proof. |
| The Star Theorem | A two-faced reading of the lattice: every lattice-theoretic invariant is 2-adic (arithmetic face), while the geometry of its minimal vectors is governed by S₆ acting on K₆ (geometric face), the two being independent constraints on the cut. It finitizes the price side but proves the minimum side archimedean and irreducible (Remark 5.3′) — the open core of the record. |
| The Steel Theorem | The dual of V(4,4)-prim integerizes at exactly 2⁵, and its cheap functionals carry a 2-adic denominator ladder capped at 2⁴ (the Wild Tooth tower depth at 2). The price of a cut is a determinant over that ladder, with floor 2⁵·⁵⁸⁵ — read from structure, not stumbled on by search. |
| The Closure Theorem | The catalogue of chorus support-types of lattice vectors is finite and closed — no new type is born at any norm, the same finite set across every stratum (rooted in the ten 3+3 bipartitions of K₆). This turns an intractable ~10¹⁷·⁵-node enumeration into finite, type-by-type bookkeeping. |
| The Katana Steel Method | The generation recipe underlying Steel: reduce the scaled integer dual (32·G⁻¹) by LLL then BKZ-24 in coordinate form, read short vectors and their pair-combinations, weigh each exact dual-norm. A method (φ-free, byte-exact), promoted to structure by the Steel Theorem. |
V(4,4)-prim's determinant 2¹²⁸ and the Steel ladder's 2⁴ cap come from a family of discriminant theorems proven in the sister campaign on Fermat surfaces (the Hodge–Fermat campaign). They are included here, with attribution, because Operación Glotón rests on them; their native home is the surface campaign.
| Theorem | What it establishes (regime) |
|---|---|
| The Sweet Lie Theorem | The rank of any alliance of plane families of V(4,d) is a finger-count over K₆; telescopes to rank V(4,d) = DS₄(d)+1. (fourfold — the engine the structural theorems above run on.) |
| The Watermark Theorem | ` |
| The Double Ladder Theorem | The discriminant group (Z/m)^a × (Z/m²)^b for prime degree — the complete elementary-divisor profile. (surface) |
| The Bend Theorem | The discriminant shift at small primes: b(3)=7, b(5)=2, flat for every prime ≥ 7, by a power-sum channel criterion. (surface) |
| The Wild Tooth Theorem | The p = 2 even kingdom — b(2)=14 (squarefree), b(2)=−6 (deep towers) — and the universal ledger b(p) = 36 − corank(σ) − radical. (surface; supplies the Steel 2⁴ cap.) |
| The Three-Ring Theorem | The composite-degree discriminant group b_p = (3m−16) + b(p), with a proven non-separability of the bend. (surface) |
The Orbital Map — how the theorems connect, and which regime (surface n=2 vs fourfold n=4) each one speaks to. Read it to see the whole ecosystem at once and to keep the surface/fourfold scope straight.
The price side is solved (a cut at price 2⁷·⁹⁰⁶⁹ < 2⁸·⁰⁸ exists, byte-exact); the minimum side (min ≥ 24) is archimedean and open. The obstruction to the minimum — the norm-12 wall — is now understood as an object: classified in closed form, censused exactly, and its most symmetric attack routes sealed with certificates; the minimum itself nonetheless remains archimedean, decided only by exact computation, not by any structural invariant. Full accounting: THE_OPEN_PROBLEM.md.
- Numbers from files only. No quantity is claimed that is not read from an exact-arithmetic frame file.
- Byte-exact, atom by atom. Every determinant, every eigenvalue, every dual-norm in exact integer or Gaussian-integer /
flint-rational arithmetic — no floating point in any load-bearing claim. - Proven, and scoped honestly. Each document states what it proves and what it does not; the record itself is never claimed.
- Commodity hardware. A single 8 GB laptop, one thread, throttled, no swap. The constraint is part of the result.
| File | |
|---|---|
README.md |
this overview |
GUIDE_FOR_EVERYONE.md |
plain-language introduction |
THE_OPEN_PROBLEM.md |
the honest status of the record |
THE_BRIDGE_THEOREM.md |
keystone — Hodge norm = geometric form |
THE_SPECTRUM_THEOREM.md |
the form, fully diagonalised |
THE_PARITY_THEOREM.md |
odd-rank obstruction |
THE_STAR_THEOREM.md |
the two-faced reading and the open core |
THE_STEEL_THEOREM.md |
the dual 2-adic ladder and the price floor |
THE_CLOSURE_THEOREM.md |
finite, closed support-type catalogue |
THE_KATANA_STEEL_METHOD.md |
the dual-functional generation method |
THE_SWEET_LIE_THEOREM.md |
the fourfold rank engine (shared) |
THE_WATERMARK_THEOREM.md |
surface discriminant order (shared) |
THE_DOUBLE_LADDER_THEOREM.md |
surface discriminant group (shared) |
THE_BEND_THEOREM.md |
small-prime discriminant shift (shared) |
THE_WILD_TOOTH_THEOREM.md |
the p=2 kingdom (shared) |
THE_THREE_RING_THEOREM.md |
composite-degree group (shared) |
THE_ORBITAL_MAP.md |
how the theorems connect |
NOMORESQUIRRELS2.py |
a verification / search script |
Released under the MIT License (see LICENSE).
- A. Degtyarev, I. Shimada, On the topology of projective subspaces in complex Fermat varieties. J. Math. Soc. Japan 68:3 (2016), 975–996. arXiv:1405.4683.
- N. Elkies, Mordell–Weil lattices in characteristic 2. (the MW128 record packing.)
- N. Aoki, T. Shioda, Generators of the Néron–Severi group of a Fermat surface. Progress in Mathematics 35, Birkhäuser (1983), 1–12.
- H. Movasati, R. Villaflor, Periods of linear algebraic cycles. (the period formula behind the Bridge.)
- T. Shioda, Some observations on Jacobi sums. Advanced Studies in Pure Mathematics 12 (1987), 119–135.