Research construction: sharp restoration cost for fixed readers

[thakrarsagar]

Change and mathematical scope

Research construction and sharp boundary cost; no new kappa. Fixed children
C_i(D,W)=(D+R_i W,-W) can keep their readers unchanged while the parent
negates the smaller output between calls. The exact uniform boundary minimum
is (k mod 2)N + 2 floor(k/2) min(N,d) in the declared complete-register
negation alphabet. A full two-pass dirty echo makes the returned output depend
only on the source and restores all incoming scratch.

The seed-selected five-child, width-eight, one-output example uses 200
scalar instructions versus 256 eager, saving 56 negations. The package
includes the written proof, recursive ADD/NEG and storage recurrences, source
copies, both child directions, data accesses, descriptor reads and preparation
charges. All ten added files are in research/output-sign-restoration-seed51/.

The child boundary is a real restriction. A compiler allowed to rewrite
inside the explicit matrix leaves can beat both words. No optimal matrix
algorithm, physical supplier, runtime gain, tape realization, rounding theorem,
or complete multiplication proof is claimed. Intermediate observers also need
an additional priced interface because the physical output can be negated.

Proof and scope ·
Exact certificate ·
Source pins

Task and overlap

The user's recursive-interface assignment was refined to this exact immutable
reader contract. OS-random assignment seed 51; local claim made within the
first eight minutes. No scope change after the claim and no pivot.

Investigation base: 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe. The final fetch
found upstream 49b3157d1d89adcef16df8be98445451672be5ba, with unchanged selected
PR1094 evidence. Its new submission guidance was inspected. This standalone
archive format retains its own exact replay and source pins, rather than
inventing a catalogue assignment for the user's explicit track.

Nearest predecessors are #1155 at 9821c6e9388f3fc097e94b72be9fa7b41a849527,
#1199 at 85ac2880ea5bda44f28632b71ffa2b4e35ec6b7d, #1203 at
38459854fd5bb93099871a84324413bd0e338d8e, and #1205 at
e2008d7ffcad464da7ec21c69e8d8a8c91de4930. Compatible nonzero readers and
observer distinctions are already their results. This package adds paid
output sign changes, the sharp uniform boundary count, and a recursively
executable source-only echo. It does not claim priority for dirty echo
cancellation, which also appears in #889.

Immediately before submission, checked 834 open PRs through #1225. The broad
keyword match #784 optimizes leaf-call counts for ordered echo trees, a
different object. #1211 prices balanced factor-tape travel; #1216 excludes
cheap restored CK/JP8 donors. No matching wrapper was found. All five current
discussions were checked; no pinned sprint issue existed, so the claim is local.
This is bounded overlap evidence, not proof of novelty.

Validation

Standard library only. From the repository root:

python3 -I -O -B research/output-sign-restoration-seed51/verify.py
python3 -I -O -B research/output-sign-restoration-seed51/test_package.py

Expected: PASS_OUTPUT_SIGN_RESTORATION ... saved_NEG=56 source_pins=3 kappa=unchanged
and PASS_OUTPUT_SIGN_PACKAGE relocation=1 corruptions=4.

Passed with CPython 3.14.8 in normal and optimized mode and CPython 3.11.17
in optimized mode. Evidence includes all 21,840 gap schedules through six
children, 384 exact price comparisons, 384 recursive cases on all 3,168 basis
columns, literal inverses, rational dirty probes, depth-two/three dense cases,
seven mathematical controls and seven invalid-input controls. The standalone
package test rejects four real on-disk corruptions. Certificate regeneration
is byte-identical; all retained source hashes and base-context hashes match.

These are finite implementation checks alongside the written general proof,
not formal verification of the full multiplication theorem. The selected cold
replay, Linux execution and human peer review were not run. Hosted CI status
must be assessed separately. No selected construction or upstream file changed.

Open next questions

The next question
is whether a supported supplier has an indivisible sign-drifting child with
fewer outputs than workspace coordinates. First trace one candidate's complete
source-return map and retained boundary, then price routing and precision. No
such supplier has been identified here. If its terminal negation can simply be
removed, reject that candidate immediately.

Credit

Prepared by Sagar Thakrar with substantial OpenAI Codex assistance in algebra,
implementation, verification and writing. Credits Aaron Wang / navinor for
#1155, Sagar Thakrar for #1199, mightyOil for #1203, Adam Gall for #1205, and
CubeJerry for #889. Three predecessor notes are retained byte-for-byte with
their notices; no predecessor implementation is imported. New material is
Apache-2.0 under the repository license. The selected Rohan Arun / Jeremy Kothe
construction and its earlier contributor chain remain unchanged.

[Mister-Meeseeks]

100-PR batch review at fea44a4d0d78570021d854881d15d9efe62e1713.

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

Reviewed scope: Fixed-reader sign-frame interface: uniform sharp boundary cost (k mod2)N+2floor(k/2)min(N,d), with complete dirty restoration. Children are indivisible; exposed or cancellable internal drift is outside the optimization model.

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.