E8 lattice packing

E8 — the rank-4 Eisenstein module the eight-dimensional register closes on

LIBRARY        MATERIALS
IDENTITY_KIND  SPEC_NAME
IDENTITY       E8
SPEC_AUTHORITY The standard root lattice E8, as fixed in Conway and Sloane, Sphere Packings Lattices and Groups — a public mathematical object with a public name; Construction A over the Eisenstein integers with the ternary tetracode, SPLAG chapter 7; uniqueness of the even unimodular rank-8 lattice and of the 240-point spectrum, SPLAG chapter 4
TITLE          E8 as the rank-4 Eisenstein module over the tetracode — the manifold court for M^8 = S^4 x C^4, by exact enumeration
MEASURED       Kissing numbers by direct enumeration over the doubled window {-2..2}^8: Z^8 has 16 minimal vectors at doubled norm 4, E8 has 240 at doubled norm 8. Both lattices are unimodular at covolume 1 (REPORTED), so from the two counted minimal norms the packing density ratio is exactly 2^4 = 16 — an integer, not an estimate. Construction A over Z[omega] with the tetracode C_4, enumerated in integers: L = { x in Z[omega]^4 : x mod theta in C_4 } has index 9 in Z[omega]^4 (9 of 81 residue classes admitted), minimal Eisenstein norm 3, and exactly 240 minimal vectors — 216 lifts of the eight weight-3 codewords and 24 theta-multiples of unit basis vectors. All 240 stay admitted at norm 3 under x -> omega*x, so L is a Z[omega]-module, counted. The Gram spectrum of one E8 root against all 240, in doubled coordinates, is exactly (1, 56, 126, 56, 1) at inner products (+8, +4, 0, -4, -8), counted. Admission is a residue check in F_3: a + b*omega is 0 mod theta iff a + b is 0 mod 3, and a vector is admitted iff its residues form a tetracode word. The minimum distance is counted for every code named, never labelled: the tetracode's is 3; in its place a counted distance-1 code yields 12 minimal vectors at norm 1 and a counted distance-2 code yields 54 at norm 2, neither 240. The affine-linking rule of AffineLink.linking in apps/AffineNPHard (a Mac-only Jordan-bond surface, not the cell binary) applied to the 240 roots gives 21 distinct words, so the map roots -> Z^2 is not injective. Embedded as (q + r*omega, 0, 0, 0) — that embedding and no other — a linking word lifts into L without remainder iff q + r is 0 mod 3, which is 27 of 81 words over |q|, |r| <= 4.
PROGRAM        z8-vs-e8-lattice
FIGURE         E8  : 240
FIGURE         density ratio = (sqrt2)^8 = 2^4 = 16
FIGURE         minimal vectors of L, counted                : 240
FIGURE         lifts of the 8 weight-3 codewords (3^3 each): 216
FIGURE         27 of 81 words lift without remainder
FIGURE         arms run 10, broken 0
FIGURE         RUN_TERMINAL  COMPLETE
SEAL           811c21506b141b7f509f96e66cea21f036192753a9f3b3504386890b7aae9541
GRADE          MEASURED
WHERE_THE_LAW_LIVES  reproduce/z8-vs-e8-lattice.swift — the doubled-window enumeration of both lattices with their minimal norms counted, the Eisenstein construction with its tetracode admission test, and ten control arms in both directions: the tetracode's weight, theta dividing every theta-multiple, an injected non-codeword residue refused, a codeword lift admitted, the minimum distance of every named code counted (3, 1, 2), a distance-1 and a distance-2 code each failing to reproduce 240, the 16/240 enumeration with its minimal norms, omega-closure of all 240 minimal vectors, and the (1, 56, 126, 56, 1) Gram spectrum. A refused run prints RUN_TERMINAL REFUSED and emits no seal
SUPERSEDES     the 2026-09-07 entry's refusal "not a claim that E8 is the right lattice for any particular signal", retired 2026-09-18 as a category error, the amendment named by the founder LEDGER_SPEC_E8_JORDAN_PURGE: it imported the Shannon channel framing — a lattice as a modulation constellation judged by capacity and noise margin — into a court in which there is no signal. The 16, 240 and exactly-16x that entry measured stand unchanged; its other two refusals, "not a construction" and "the program prints no seal", are retired by the construction now enumerated and the seal now printed. Nothing counted on 2026-09-07 was withdrawn.
NOTE           INVARIANT E8_RANK4_EISENSTEIN_MODULE, graded: COUNTED here are 240 minimal vectors at norm 3 split 216 + 24, index 9, omega-closure, and the Gram spectrum; REPORTED (SPLAG ch. 4 and 7) is that a 240-point configuration with that spectrum is E8's root system and that L is a scaled E8 — unimodularity is not verified by this program. DEFINITION: the free Z[omega]-module of rank 4 over the ternary tetracode C_4. The kissing number 240 is the count of minimal norm-2 root links on the lattice, not a codebook size.
NOTE           DOCTRINE, not measured here: in UUM-8D, E8 is the topological constraint manifold for M^8 = S^4 x C^4 — the joint vQbit register closes on it, with the same six-fold unit symmetry as one vQbit in eight dimensions — and a state transition is admitted by residue or refused. The program prints this reading OUTSIDE its seal; the seal covers only the arithmetic the doctrine rests on.
NOTE           Two tiers, kept apart on purpose: the TRANSPORT tier is the 42-byte frame with Z^8 integer deltas frozen in Study 30's charter — no pod built, no tier run, and the only 42-byte frames in the substrate tree are ARP; the COURT tier is admission by tetracode congruence, measured here. Lifting that admission test into the MCP ingestion gate would be a cell change; no such gate or refusal token is shipped on any cell, and none is claimed by this entry.
NOTE           AMENDED 2026-09-18. The three 2026-09-07 REFUSED lines retired in place were, verbatim: "not a claim that E8 is the right lattice for any particular signal. It is the measured comparison at equal covolume and nothing more"; "not a construction. This entry names the object and what was counted about it; it carries no fabrication route and none is needed, because the object is a mathematical one"; "the program prints no seal, and this entry says NONE_PRINTED rather than leaving the field blank. Unsealed and sealed are different states". SEAL moved from NONE_PRINTED to the 64-hex above.
REFUSED        not a claim that any shipped frame carries E8 coordinates — the 42-byte Z^8-delta frame is a charter, not a deploy, and the court tier measured here is the admission test, not a wire
REFUSED        not a claim about a signal, a channel, a noise model or a decoder. None appears anywhere in the program; a state transition is admitted by residue or it is not, and that is the whole verdict
REFUSED        not a claim that the affine linking map is injective on the roots — 240 roots map to 21 words, measured — and not a claim that any embedding of a linking word other than (q + r*omega, 0, 0, 0) gives 27 of 81
FALSIFIER      an enumeration over the same two lattices returning a kissing number other than 16 or 240 or minimal doubled norms other than 4 and 8, a density ratio from those norms that is not exactly 16, a tetracode construction with a minimal-vector count other than 240 at minimal norm 3, a split other than 216 + 24, an index other than 9, a minimal vector whose omega-multiple is not admitted, a Gram spectrum other than (1, 56, 126, 56, 1), a code of counted minimum distance 1 or 2 reproducing 240, a counted minimum distance for the tetracode other than 3, or a vector with a non-codeword residue being admitted
REPRODUCE      cd reproduce && xcrun swiftc -O -swift-version 5 z8-vs-e8-lattice.swift -o /tmp/run && /tmp/run
NOT_ADVICE     Nothing in this entry is medical advice and it is not a recommendation to take anything.
ADDED          2026-09-07