Research note: paid quadratic-reader restoration and exact square-repair floor

[theon199]

Status: research note / scoped recursive-interface construction; no new kappa

Linear reader-defect repair does not account for a quadratic dirty reader's mixed term. This package preserves fixed y += beta_i*a^2; a += alpha_i*b calls, then fuses the paid correction frame with source restoration. The seed-selected eight-call pilot costs 32 -> 31 exact scalar operations, including all nonunit multiplications; its square-only correction costs 35. A written determinant argument proves the minimum appended square-query count is two when the mixed defect is nonzero, in the explicitly restricted additive-repair/linear-frame class.

This is a nonlinear successor to Adam Gall's #1205 and Hanson Sun's #1246, whose notes and attribution are retained unchanged. The contribution adds the quadratic defect, one-product restoring repair, restricted square-call floor, and complete nonlinear tariff. It does not claim the earlier linear mechanisms or a new supplier. If the toy calls can be opened, one direct square and scaling is cheaper; fixed calls are essential.

Only research/quadratic-reader-restoration/ is added. Its README gives the proof, full arithmetic/access/child/storage accounting, implemented recursion and next question. The exact child-price gate is P+3C < (r-2)L; an expensive product backend can erase the saving. No fixed-tape, scratch-backend, rounding or hosting realization is established. The selected frontier, upstream inputs and frozen packages are unchanged.

Reproduce

python3.12 scripts/research_package.py --package research/quadratic-reader-restoration

Expected final marker: PASS_REGISTERED_RESEARCH_PACKAGE research/quadratic-reader-restoration outcome=completed.

Passing local evidence:

• All 1,364 signed batches of lengths 1-5, full rational polynomial coefficients, 36,828 dirty numeric triples, both repair alphabets and literal inverses.
• Arity 2-16 exact costs and an independent addressed-read/write observer; binary recursion depths 0-3, complete endpoint/restoration/inverse and invoice checks.
• Eight semantic/resource controls, including a 4ab defect invisible on every coordinate basis input and an extra read that preserves the output but fails the bill.
• Optimized Python 3.12.13 registered replay, source-export relocation and five on-disk corruption rejections; normal Python 3.14.5 independent replay. Source binding, metadata pins and whitespace checks pass.

Base: ef5f39403610d231a0c27e8a6b4f08c10209bcd6; OS assignment seed 55, recursive interfaces. Selected context is reviewed PR1094, kappa=836330098538459/500000000000000000. Source pins: PR1205 e2008d7ffcad464da7ec21c69e8d8a8c91de4930; PR1246 5ed1ce17a9df4c853a93f9758ba903d2620c3443; both heads rechecked unchanged. Git blobs and SHA256 pins are provided. No full selected cold replay, native admission, formal multiplication proof or hosted CI success is claimed.

No pinned coordination issue existed, so the claim is local. Full open-PR/Discussion overlap metadata was refreshed before submission. The closest linear repair is credited explicitly; broader quadratic/restoration hits concern phase, precision or decoder objects. Search coverage and its limits are recorded, without a novelty guarantee.

Prepared by Sricharan Vagvala (@theon199) with substantial OpenAI Codex assistance in algebra, implementation, verification and writing. Apache-2.0. This is exact implementation evidence and a written scoped bound, not independent human review.

[Mister-Meeseeks]

Batch review at aa54509994cb050174ff2ee1442af8caf8161670.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact quadratic reader defect admits a fully charged restoring product correction and a sharp appended square-query count. Polynomial and dirty-input checks cover nonlinear cross terms; the saving requires fixed indivisible readers and is not an optimal squaring algorithm or supplier.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.