[petalas]

Summary

Scoped algebraic obstruction with finite corroboration: rational rank one does not bound the number of eligible observations needed for a dyadic decoder. For every n>=1, the exact incremental acquisition count can be n. The prime-cover construction preserves every feasible selection and every positive weighted optimum of an arbitrary finite Set Cover instance.

mandatory observation: (p_1 ... p_n) x
eligible observation i: (product_{j != i} p_j) x
allowed decoder coefficients: Z[1/2]
minimum additional acquisitions: n
rational observation rank throughout: 1

The three-entry fixture is -105x + 2*35x + 21x + 15x = x; every two-entry selection leaves an odd-prime obstruction. The proof is self-contained. Ali Romman #5789 supplies the rational-versus-dyadic question, cited rather than audited or replayed. Bezout, prime factorization, localization and weighted Set Cover are classical ingredients. No new complexity-theory classification or kappa is claimed.

Evidence

• Before acquisition, the mandatory observation has a rational inverse but no dyadic inverse. After the complete singleton menu is acquired, constructed Bezout coefficients decode exactly. Each proper selection has a missing-prime obstruction.
• Warning-strict CPython 3.12.15 normal and optimized direct checks pass locally. Each registered command takes less than one second locally, below its 120-second limit. Registered replay and metadata checks pass.
• Complete finite checks cover 689 ordered menus, 5054 menu selections, 16609 weighted instances, 4530 literal scalar decodings and 8190 selections in the sharp family through n=12. All 15 rejection, four boundary and three semantic controls pass. Counts and SHA256 receipts are generated on stdout.
• Exact committed bytes and declared source-history preflight pass at 26309a3936985b3c7780856f708b0526508483bf. The only Git-history source dependency is reachable base de5ec2e7a6074068452f45b1a191b5c72832df4a. The pending predecessor is contextual, not an executable or mathematical dependency.

Reproduce from the repository root:

PY=/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12
"$PY" -W error -B research/sparse-dyadic-normalization/verify.py
"$PY" -W error -B -O research/sparse-dyadic-normalization/verify.py
"$PY" -B scripts/research_package.py --package research/sparse-dyadic-normalization
"$PY" -B scripts/check_research_commit.py --package research/sparse-dyadic-normalization --check-source-history

Any CPython 3.12 interpreter can replace the local path. Expected markers are PASS_SPARSE_DYADIC_NORMALIZATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. No third-party dependencies are used. The arbitrary-size theorem has a written proof; finite execution corroborates only its declared domain.

Limitations

The menu is synthetic and its eligible observations are a declared interface. No current physical source is claimed to realize it. Observation production, coefficient preparation and growth, legal delivery, dirty restoration, native execution and the complete supplier bill remain unpriced. Adding an arbitrary observation such as x itself changes the problem. This is not a lower bound for unrestricted circuits, nonlinear decoding or a ring that inverts odd primes. The five framework assumptions and selected frontier are unchanged. No selected cold replay, Lean, Docker, rounded multiplier or independent proof-kernel validation was run.

Open next questions

• Derive one nontrivial eligible prime-loss menu from a named physical producer/consumer cut, with complete acquisition and restoration charges.
• Prove a physical menu restriction, such as nested prime-loss sets or bounded incidence, that permits a bounded or tractable acquisition rule. Rational rank alone supplies no such restriction.

The package records first steps and blockers for both questions.

Related open work

• Export exact rational rank and complete kernel certificates #5789 provides rational rank and kernel certificates and excludes dyadic completion and physical delivery. This adds a sparse eligible-menu obstruction without changing or replaying that tool.
• Prove sharp tensor composition of dyadic helper capacity (no new κ) #5882 studies dyadic unit-channel capacity under tensor composition. This fixes scalar rational rank one and minimizes acquired menu entries, a different objective and interface.
• Prove all-precision dyadic affine feasibility (no new κ) #5889 studies dyadic feasibility of one supplied affine system with Smith charts and strict inequalities. This selects the observation system itself under acquisition charges and does not recompute its precision values.
• Construct minimum-cost binary query bases with exchange certificates #5791 and Prove the sharp codimension bound for partial query acquisition #5878 study binary query-basis selection and partial acquisition dimensions. The odd-prime obstruction here uses Z[1/2] and asserts no agreement or disagreement with their binary values.
• Prove exact constant-weight phase COPY error (no new κ) #5903 computes constant-weight phase COPY approximation coefficients and leaves their Gaussian-dyadic realization open. This characterizes exact scalar acquisition over Z[1/2]; no approximation-error value is shared or compared.

The completion refresh inspected this account's eight other open PRs created since the CLAIM: #5897, #5898, #5899, #5900, #5901, #5902, #5903 and #5905. None computes a common scalar-menu acquisition value. No extra computations were run to validate their results.

CI notes

At exact head 26309a3936985b3c7780856f708b0526508483bf, run 40501296498 finished with plan failed; replay skipped. The only failed step is Onboarding helper tests. Direct inspection found ten failure/error entries, all in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The assertion failures concern frozen-card/current-command routing, missing historical references and sparse omission behavior.

This matches the shared onboarding failure reported by the orchestrator for recent PRs based on main de5ec2e7a. The same failed step, test file, summary and all ten failure/error headers were directly confirmed on PR #5903 run 40501261851. Maintainers will repair it. No shared implementation, tests or package files were changed in response.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific and publication checks reported above have run. No hosted scientific success is claimed.

Merge Danger

Door: two-way. Blast radius: package. Only six files under research/sparse-dyadic-normalization/ are added.

This contribution was produced by Nick Petalas. Advance claim.