H(668) research checkpoint v1.0.0 — exact obstructions, no matrix

H(668) research checkpoint v1.0.0

No Hadamard matrix of order 668 was found. This release pauses the
headline search and publishes three scoped, provisional, computer-assisted
papers together with their exact verification artifacts.

Papers

1. Exact repair obstructions around a 64-modular Hadamard matrix of
   order 668 — fixed-(q), the distance-(80/41) frontier, one certified
   long support exclusion, all nine short support exclusions, and the
   orientation cascade for twenty open long cases.
   • PDF SHA-256:
     0f203088cb77ab68b92424f0f59f4fa1dcc8d921fff4743d06a382aa010b1d39
2. Exact shell descent and finite-field norm gates in a
   prescribed-compression chart for Legendre pairs of length 333 — five
   shell-two orbits, eighteen dense orbits in the stated census scope,
   physical placement geometry, primitive units, and three pair-resultant
   norm keys.
   • PDF SHA-256:
     98df2842ae7ab6e3a505f6cb237e847b7ae837b0b662bde986c73125c8ca5da2
3. Semiregular (C_{37}) conference lifts at order 334: quotient
   classification, modular realizations, and low-rank obstructions —
   (625) quotient classes, the universal diagonal law, exact
   characteristic-two supports, and specified formal obstructions.
   • PDF SHA-256:
     c8193ce8485a2642044d11ead0e10bea15589e76bff20021218401a85153012e

Production ledgers

• h668-short-case-production-manifests-v1.0.0.tar.gz contains all 2,304
  retained production ranges behind the complete nine-short-case Eliahou
  certificate.
  • SHA-256:
    6e38f08e9c3c9798bc5ca87a46ae9ccffdb54c6f1ed3901d6ac9ce3fd5d69084
• h668-dense-shell-production-v2-v1.0.0.tar.gz contains all 729
  dense-shell production shards, the manifest, frozen aggregate, compiled
  helper, enumerator, and detached replay code.
  • SHA-256:
    493f73884ff5b5454f179b7754c0207178eeb70c70c27750daa610f3bda6c2df
  • A fresh extraction strictly reaggregates to
    3bccde87f456bfcd2f0c3da6ac8cf9cb3635538e831a95951003068ae87cae86.

Exact endpoint

The last continuation gate produced three exact pair-resultant norm
equalities in (\mathbb F_{167^3}^{*}), but every audited marginal image
is full and every triple image has affine rank three. A complete
margin-conditioned join still requires
5,091,993,547,996 invariant evaluations and approximately 81.5 TB under
the stated naive exact-key layout. The search is paused until a new
algebraic compression, proof-producing decomposition, broader construction
theorem, or materially different exact-computation architecture becomes
available.

No solver timeout is promoted as evidence of nonexistence. The release does
not construct or exclude an (H(668)), a Legendre pair of length 333, a
base sequence in BS(84,83), or a conference graph on 333 vertices.

Verification and responsibility

The paper-level SHA256SUMS, package READMEs, claim-to-artifact tables, and
the public research page give the shortest replay paths. Large historical
computations are published as inspectable ledgers but remain trusted
computer-assisted claims pending independent reimplementation.

The work was developed with heavy assistance from ChatGPT 5.6 Sol under
Alec Kriebel's direction. It has not been peer reviewed. Alec Kriebel is a
non-specialist and cannot independently certify every derivation,
implementation, citation, or novelty assessment.

Public checkpoint:
https://aleckriebel.github.io/Math/research/hadamard-matrix-order-668/