[petalas]

Summary

Scoped algebraic existence proof for Cinna Davis's norm-six kernel:

- explicit dyadic term upper bound: 19,648
+ dyadic existence upper bound:     19,600
- algebraic support-bill bound:     962,752
+ algebraic support-bill bound:     960,400

This saves 48 terms and 2,352 units in the algebraic separate-copy upper bound. The new contribution applies the existing dyadic rank-factorization lemma to the predecessor's rational shell-rank bound. It supplies no physical invoice or new kappa.

Only research/norm6-dyadic-compression/ is added. Produced by Nick Petalas. Credit Cinna Davis #2222 for the kernel, dyadic integrality and 49-bit chart; DreamingOfClouds #2244 for the corrected rational rank perspective; Dugongue #4494 for dyadic rank factorization; Nebe/Sloane and the retained Leech/Niemeier lineage for the geometry. No predecessor executable or result is audited.

Evidence

• Before: Research: dyadic norm-six lattice decoder and 49-bit chart (no new kappa) #2222 provides 19,648 explicit dyadic pairs and a rational rank bound of 19,600, without reducing the dyadic term count.
• After: the written proof gives at most 19,600 dyadic pairs for the same complete finite kernel. Every support pairing has binary rank at most 49, giving 960,400. This is an existence bound; no full-shell factor table is materialized.
• The exact verifier checks complete integer norm-six diagnostic shells with 12/48 ports and ranks 10/20, 120 further bounded dyadic matrices, an odd-pivot control, a zero-rank control and seven rejected mutations/invalid inputs. Every final product entry is checked, with separate rational rank elimination. Cubic indices are enumerated.
• Warning-strict normal and optimized checks each took 0.12 seconds locally on CPython 3.12.15. Both registered commands and the package runner pass. Generated receipts match exactly and bind all five package files.
• Committed-byte/source-history preflight, official export integrity and optimized replay from a read-only Git-free export with PATH=/nonexistent pass at 9d7f7805d147028fe5f0174e267ee05b73703f0b. The exported receipt equals the local receipt.

Exact reproduction from the repository root, using CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/norm6-dyadic-compression/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/norm6-dyadic-compression/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/norm6-dyadic-compression
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/norm6-dyadic-compression --check-source-history

Any CPython 3.12 installation can replace the displayed interpreter path. Expected markers: PASS_NORM6_DYADIC_COMPRESSION, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

Scientific sources stay pinned to main 4c0a359d88f22f05189d51f48387750b801a45d4. The pre-submission canonical refresh reached 81e9bd69e4a5b00ba9c713a1620e36d576d5522e; guidance, selected record and the relevant predecessors were unchanged. Live claims/open titles, targeted open bodies, main README terms and several distinctive merged searches were rechecked before the commit. Sparse notes, unseen diffs and unpublished work remain outside complete coverage.

Advance claim.

Limitations and unrun obligations

The full 48-dimensional shell, its factor table and exact kernel rank are not enumerated. Minimum six, the shell cardinality and the binary chart are cited predecessor dependencies. No efficient feature evaluation, coefficient-size bound or practical full-shell reduction is supplied. The 64-row/column diagnostic is not a full-shell generator.

Physical preparation, source access, side program, frames, arbitrary-dirty restoration, owners, precision, tape, compiler/caller admission, a matched bit supplier and outer costs remain unknown. The selected frontier is unchanged. The historical card's fully priced five-percent gate is not claimed; TASK.json records changed scope. No Lean/native build or selected cold replay was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Compress the symbolic features over Z[1/2] using the 48 norm-shell relations, with explicit coefficient and evaluation bounds, instead of enumerating the whole shell.
• Specify and price a complete chronological physical source/target contract before interpreting this algebraic bound as a supplier gain.

Related open work

No open PR studying this norm-six kernel or its term/support bounds was found in the final targeted norm-six/norm6 search and live overlap refresh. Merged #2222 and #2244 supply the kernel and rational rank bound; merged #4494 supplies the general dyadic factorization lemma. The contribution is their new compression application, rather than a competing general theorem or a replay of their evidence.

Completion refresh: this account's open PRs created since the CLAIM at 2026-10-23T08:12:04Z were listed, and their titles and full bodies were read.

• Sharpen degree-controlled cyclic inverse constants to pi (no new κ) #5080 studies degree-controlled cyclic-polynomial inverse constants. It computes no norm-six decoder term or support bound.
• Prove five-call optimum for three GL2 swaps at scalar-rank cap two #5083 studies scalar-rank-capped GL2 SWAP words. Its formal call tariff differs from this lattice-kernel factor count.
• Prove the exact two-transvection cutoff for partial Pauli maps #5085 studies partial Pauli-map transvection counts. It supplies no competing norm-six factorization or support bill.

No newly open account PR studies this kernel or computes the same quantity; no additional numerical computation was run.

CI notes

Both hosted checks passed on exact head 9d7f7805d147028fe5f0174e267ee05b73703f0b in run 39924592148: plan in 5m16s and the registered package replay in 1m22s. Run metadata confirms that exact head and conclusion success. The replay runs both warning-strict normal and optimized commands. Local registered checks, committed-byte/source-history preflight, export integrity and Git-free replay also passed. Hosted CI certifies execution of the bounded package checks; independent human and formal proof-kernel review of the scoped theorem remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Five new package files; shared code, frozen predecessors and selected-frontier metadata are unchanged.