[petalas]

Summary

Scoped written proof with an exact finite matrix certificate, produced by Nick Petalas. Construct an explicit dyadic multiplicity operator for the historical A_1 dihedral component W:

216-dimensional pinned orbit space
    -> C=[V,RV], with rank C=128 and image C=W
    -> C^-1 (A_1|W) C = diag(T,T)
    -> T=K/8, K an explicit 64-by-64 integer matrix

K has 668 nonzero entries in [-64,96]. Each column of V has two or three nonzero entries. The least common denominator of T in this basis is exactly eight.

This is a partial constructive answer to unpaidmoon #7065's Open next question 2. That question asks for an integer matrix of T. The new rational matrix and its actual intertwiner are explicit; an invariant lattice giving an integer T remains open. K represents 8A_1, so its factor of eight is retained. The predecessor's polynomial is a cited premise, not a replay target. No new kappa or selected-frontier change is claimed.

Credit unpaidmoon #7065, head ee0202a8eafa4c0b84bef97ecfb5691f6affc9c6, for W, its generators and the explicit-matrix question; #6964 for the pinned orbit basis and B=4A_1; #6944/#7008/#7023 for the splittings; JoyboyBrian for the handoff; aliromman #4958/#4720 for the query/Bell action; krzysztofwojt for the one-slot polynomial; and the retained #1719 lineage. Related Nick Petalas annihilator/width results are background, not steps of this construction.

Advance CLAIM. Canonical base e54523517d0f03e58352e647c403a5af3e525a57; the formal-interface-lemma card changes scope to this constructive algebraic side result.

Evidence

Before, #7065 supplies the polynomial square and leaves the multiplicity matrix unconstructed. After, the new deterministic construction emits all entries of K and V and checks their complete operator and dihedral identities.

Local CPython 3.12.15, macOS arm64, standard library only:

• Both warning-strict normal/optimized direct checks and both registered VERIFY commands pass. Each registered command was also timed separately at 0.214 and 0.215 seconds, below its 120-second limit. These are timeout checks, not performance comparisons.
• A separate dense integer product checks all 27,648 entries of 2BC=C diag(K,K). Independent dense modular elimination proves rank C=128. Exact scaled-projector, rotation and reflection identities identify its image with W and expose the same T on both copies. Nine semantic controls pass.
• Metadata/source pins, in-memory Python compilation, function-size inspection, diff whitespace, committed-byte and declared source-history checks pass. The publication preflight checks nine package/input files at bf5e56f3d62d13f632f6e9934112367831539600.
• The generated matrix/basis certificate has canonical SHA256 98ceb6e760b187a30e185e3b5dcd4041018801bca9ea0d60c94631001e86e7f7. Receipts bind all six package files and all three live inputs. No receipt is handwritten. No predecessor executable is imported.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/a1-dihedral-multiplicity-matrix/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/a1-dihedral-multiplicity-matrix/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/a1-dihedral-multiplicity-matrix
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/a1-dihedral-multiplicity-matrix --check-source-history

Another CPython 3.12 path may replace that executable. Add --matrix to a direct command to emit the complete new matrix and basis. Optional --emit /absolute/new/scratch/result.json creates a source-bound receipt outside the checkout and refuses to overwrite an existing file. Expected marker: PASS_A1_DIHEDRAL_MULTIPLICITY_MATRIX.

Limitations and unrun obligations

This is exact linear algebra on one historical pinned operator. T is dyadic, not integral in this basis. The polynomial consequence chi_T=q uses #7065 as a cited premise; neither q nor its degree-128 square is recomputed. No new nullity, full stabilizer census, physical decoder, basis-conversion tariff, native/rounded execution, dirty restoration, ownership/storage/tape bill, compiler closure, supplier admission or multiplication theorem is supplied. No selected cold replay, predecessor degree-216/881 replay, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. No local package obligation is deferred to CI alone.

Open next questions

• Construct an explicit invariant lattice giving an integer T with a sparse intertwiner. Begin with the emitted K/8 and retain the conversion denominators.
• Bind this decomposition to an actual physical consumer and price source/target access, basis conversion, restoration and the full recursive invoice. The complete physical cost is unknown.

Related open work

These titles and bodies were read without additional computations to check their results. The new result constructs T and compares no new spectral multiplicity or projector denominator.

• Record the characteristic polynomial of A_1 on the 128-dimensional dihedral isotypic component #7065 supplies W, its polynomial and this matrix question; the computed dimensions 128 and 64 agree with its stated dimensions.
• Record the characteristic polynomial of A_1 on the 80-dimensional isotypic summand #7061 computes the polynomial of the different 80-dimensional isotypic summand.
• Record that the 8-dimensional c=1 isotypic summand is reducible under… #7023 computes the polynomial of the different eight-dimensional isotypic summand.
• Prove the c=1 ++ 216-space is reducible under its signed normalizer. #7008 supplies the dihedral normalizer decomposition; this makes its two-dimensional representation's multiplicity operator concrete.
• Record the c=1 stabilizer action outside the {g,t} normalizer. #7022 studies additional preservers of the 216-space outside the Klein normalizer.
• Record that c=1 movers miss the 8-dimensional dihedral summand. #7059 studies movers of the eight-dimensional isotypic summand.
• Record geometric multiplicity 24 for eigenvalue 0 on the pinned 216-by-216 matrix #7018 computes zero multiplicity on the full pinned 216-space; no such multiplicity is computed here.
• Record geometric multiplicity 12 for eigenvalue 2 on the pinned 216-by-216 matrix #7064 computes eigenvalue-two multiplicity on that 216-space.
• Record geometric multiplicity 24 for eigenvalue 0 on the other three 216 joint sign spaces #7066 computes zero multiplicities on the other three joint sign spaces.
• Record the characteristic polynomial of one 8-dimensional c=1 joint eigenspace #7019 studies separate eight-dimensional joint spaces in a nontrivial sign character.
• Record a c=1 stabilizer transporter of one 8-dimensional joint basis #7071 constructs a transporter for those different eight-dimensional joint spaces.
• Split the c=1 sign characters that g and t move #7009 splits moved c=1 sign characters.
• Record the characteristic polynomial of one 48-dimensional eigenspace of s #7036 computes polynomials on moved-character eigenspaces.
• Record geometric multiplicity 10 for eigenvalue 0 on the pinned 48-by-48 matrix #7057 computes a zero multiplicity on a different 48-dimensional matrix.
• Record the characteristic polynomial over Q(i) of one eigenvalue-i 48-space #7060 computes a polynomial on an eigenvalue-i space over Q(i).
• Cut one moved 48-eigenspace by a 12-dimensional A_1 subspace #7082 constructs a 12-dimensional subspace of a moved 48-space.
• Record the characteristic polynomial of one exhibited 12-dimensional … #7231 computes the polynomial on that separate exhibited 12-space.
• Prove degree-881 payload involutions split a sector-3 joint space #7020 studies payload-involution traces in sector c=3, rather than this c=1 component.
• Record the characteristic polynomial of the 84-dimensional X_3=+1 subspace #7068 computes a polynomial on an 84-dimensional subspace of that other sector.
• Prove odd signs on u keep the X_3 trace -48 #7069 studies odd-sign changes of a trace in that other sector.
• Record the characteristic polynomial of one weight-4 c=3 summand. #7021 computes polynomials in a different c=3 character.
• Prove doubled centre Gram stabilization by width ten (no κ) #7172 bounds annihilators and stabilization across tensor widths; this does not extend its open questions or compute another width.
• Prove least denominator 12,480 for the A1 zero spectral projector (no κ) #7463 constructs the zero spectral projector and its denominator on the full 216-space; this constructs a multiplicity operator on W, a different arithmetic quantity.

Live main/merged searches and local claims were checked again before committing. The broad helper reports complete remote pagination and incomplete sparse local-note coverage, supplemented by Git-tree searches. Unpublished work and arbitrary historical diffs remain outside coverage.

Closeout refreshed this account's open PRs created after the CLAIM at 2026-11-12T18:22:45Z: #7461 through #7468. The only other result on this same historical A_1 operator is #7463, already listed above; its zero spectral projector is a different arithmetic quantity. The other packages concern phase-preserving widths, a selected binary address map, chronological compiler width, helper recharts, robust coordinate pairing and tensor-prefix integrality. No new computation was run to compare their results, and no shared numerical disagreement was identified.

CI notes

CI did not certify this package. At unchanged head bf5e56f3d62d13f632f6e9934112367831539600, run 43280441536 failed its plan job 131400005353 in Onboarding helper tests. I read the job log: exactly one failure among 760 tests, with 24 skipped. The failing test is

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

At tests/test_research_current_status.py:58, its AssertionError says that 'make frontier7000-verify' was not found in the current CI verification section. That logged section still describes #6030 and make frontier6030-verify. I read both base files at e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030. Neither file is changed by this six-file package addition. This is a repository-level mismatch on main.

The dependent package replay job 131402870660 was skipped because plan failed. The local normal/optimized scientific replay, metadata/pin validation and committed-byte/source-history checks remain the execution evidence; no hosted package pass is claimed.

I also read the logs of all five comparison runs below. Each has the same sole failing test, the same missing-command assertion and Ran 760 tests / FAILED (failures=1, skipped=24):

• 42905094314.
• 42903313790.
• 43279626746.
• 43279933171.
• 43280276547.

No repository-level fix, retrigger commit, push or workflow rerun was made during closeout.

Merge danger

Door: two-way. Blast radius: one isolated six-file research package. No shared implementation, frozen predecessor source or selected record is changed.