[Matarener]

Requested by Mark · project thread

Change and mathematical scope

New package research/centre-block-coset-kernel: an exact finite certificate with a short written argument. It does not change the selected frontier, κ, or any supplier.

In the selected calls-0/4 block-repair model (Sx(g) = R x(g), Tx(d) = sum_p H[:,p] x_p(d rho_p), H = R/4), there is a nonzero common-kernel vector supported on 4080 owners:

• R_0 - R_6 - R_85 = 0.
• On the full 73-dim routes, K = <rho_0, rho_6, rho_85> has order 8160 and its even subgroup K+ has order 4080.
• x = (1, -1, -1) on ports (0, 6, 85) at every owner of K+, and zero elsewhere.
• Sx = 0 holds at all 4080 owners and Tx = 0 at all 4080 touched outputs, checked exactly.

This answers open question 1 of centre-block-moore-bound from above. Combined with its 15-owner floor, the minimum support is in [15, 4080].

This is a scalar-information statement only. The witness saves 1 coordinate out of 104·4080 = 424,320, so it is not a cost reduction. It also gives a census for single-triple witnesses: such supports are unions of K+ cosets. Among the 896 dependent triples, 56 have |K+| = 4080 and 840 have |K+| > 4080.

Remaining obligations: the exact minimum, mixed or higher-degree supports, the full kernel dimension on the coset, and all physical costs.

Task and overlap

• Card: centre-path-sharing.
• Scope: refined. The block-repair route of the copied-information method is pursued here.
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90.
• Builds on centre-block-moore-bound (PR3291), centre-block-repair-information (PR3071), PR3147, PR3097 and PR3168. Its authenticated loader is reused; nothing is vendored.
• No equivalent witness was found in the local packages or ledger. Live PR and Discussion search was not available from this environment.

Validation

python3.12 -B research/centre-block-coset-kernel/verify.py
python3.12 scripts/research_package.py --package research/centre-block-coset-kernel

• Both commands pass on Linux with CPython 3.12, standard library only, in about 1 minute.
• Expected output: PASS_CENTRE_BLOCK_COSET_KERNEL ... and PASS_REGISTERED_RESEARCH_PACKAGE.
• Behavioral controls:
  • Dropping one owner leaves exactly 3 nonzero outputs.
  • Changing one coefficient breaks Sx = 0.
  • The other K+ coset is also a valid support (positive control).
• Not run: the selected cold replay and a native common-frame refinement.

Credit

• Ross Richardson (PR3071): model and extraction.
• Sagar Thakrar (PR3291): 15-owner floor and loader.
• dcharb78 (PR3147): dependent-triple census.
• Gonzalo Mier (PR3097) and Sudarsh3301 (PR3168): route relations.
• Aaron Wang/navinor and ywz65: centre lineage.

New work by Matarener (session donor Mark), with substantial Claude (Anthropic) assistance. There has been no independent human review. Apache-2.0.

Open next questions

See the package README and TASK.json:

1. Close [15, 4080]. First step: a census of length-8 route relations.
2. Compute the full kernel dimension on the coset, via a representation reduction over K+.

🤖 Generated with Claude Code

https://claude.ai/code/session_01REJsJasf8uasZVDgPHMRGD