[petalas]

Summary

Every dyadic rank-one common-chart shear admits one borrowed source as its sole readout hub, with source-only preparation and exact restoration. This is a scoped constructive proof extending David Ganz / Dugongue's #4502 sufficient single-pivot adapters, using #4494's integer factorization setting.

M = a b, b primitive integral
prepare X_0 = bX using integer source ADDs
Y_j += a_j X_0 once per nonzero target row
reverse preparation and restore every source

Only research/dyadic-rankone-borrowing/ is added. The theorem permits every finite matrix size; the standalone compiler and verifier have explicit finite input limits. No extra field, clean entry, CLEAR, SCALE or odd-denominator native coefficient is required. Classical Bezout, Euclidean reduction and uncomputation retain their existing credit. Produced by Nick Petalas.

Evidence

• Before: Borrowed rank-one adapters: 674 → 666 scalar allowance with no extra fields #4502's sufficient source and target pivot methods both exclude its explicit [[9,15],[15,25]] fixture because the needed ratios are not dyadic.
• After: the new compiler realizes the fixture exactly, including arbitrary dirty inputs and restoration, with one readout hub. Complete bill: 32 ADDs, allowance 270. Direct entrywise evaluation costs four ADDs and allowance 76; lower scalar cost is not claimed.
• Exact symbolic replay passes 3,939 exhaustive two-by-two parameter cases, 768 wider fixtures, two named fixtures and three coefficient/role/Euclidean boundary cases. Nine invalid inputs reject, and five mathematical omission controls detect missing preparation/restoration/readout, a wrong hub and a wrong inverse order.
• Both registered commands pass locally with warning-strict CPython 3.12, each in about 0.77 seconds; the complete registered launcher took 1.83 seconds. These are check durations, not performance comparisons. AST parsing, function-size inspection and whitespace checks pass. No separate Python linter or type checker is installed locally.
• Official committed export integrity and read-only Git-free normal/optimized replays pass.
• Committed-byte and declared source-history preflight pass at 391355b8e537b82cedabd45af509177496e7a7a6. Receipts are generated by the verifier and bound to all six package files.

Reproduce from the repository root:

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

Any CPython 3.12 executable can replace the interpreter path. Expected mathematical marker: PASS_DYADIC_RANKONE_BORROWING; registered marker: PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations

The new result is universal admissibility of this one-hub topology. Direct entrywise ADDs already give a zero-scratch implementation of arbitrary dyadic shears. No circuit optimality, favorable actual native window, coefficient-growth bound, rounded precision, physical caller, stock/tape bill, supplier admission or new kappa is supplied. Independent human and formal proof-kernel review are unclaimed; no Lean build or selected cold replay was run.

The original card's historical ceiling example raises KeyError: 'complex_profile' against the changed selected schema. It was attempted, left unchanged and not used as evidence. Current selected metadata orientation passes separately. The selected frontier remains unchanged.

Related open work

No open PR studying this common-chart rank-one borrowed-hub admissibility question was found in the publication refresh. #4502 and #4505 are merged predecessors: the former supplies sufficient pivot adapters, and the latter installs its four specific native windows. Their results are cited, not audited or replayed.

Open next questions

1. Can a short unimodular preparation improve the complete bill on an actual no-pivot native window? First identify one such window and compare direct evaluation, both pivot methods and the full preparation/readout/restoration invoice.
2. What coefficient-growth and rounded-error bounds admit the method in a real caller? First bound all intermediate row norms for one concrete preparation and its inverse. No numerical precision prescription is supplied here.

CI notes

CI pending for 391355b8e537b82cedabd45af509177496e7a7a6. Both registered normal/optimized commands and publication/source-history checks pass locally. No hosted check is claimed yet.

Merge danger

Two-way door; blast radius is the isolated research package. No shared code, frozen source, predecessor credit or selected-frontier file changes.