[petalas]

Summary

Proves the sharp information tradeoff for compression of the coupled center output in Cinna Davis's #4694/#4698 interface. This is a scoped algebraic proof with finite corroboration, with arbitrary dirty inputs and arbitrary decoding of fixed scalar linear observations.

all 632 old output fields + center record of rank j
    -> exactly 624+j recoverable independent linear functions of the old endpoint
    -> full recovery requires rank 8

For any invertible block interface y=Ax+Bz, w=Rx+z, the minimum rank of a record of w recovering a target Tx is exactly rank(T(A-BR)^-1 B). The proof constructs an attaining linear decoder and equal-observation collision witnesses against every decoder. The 624+j corollary uses the predecessor's invertible A and injective B as cited premises. No priority is claimed for the standard kernel/Schur argument itself; the contribution is the sharp compression law at this changed endpoint and its target-specific demand criterion.

Only research/center-output-compression/ is added. No new kappa, supplier admission or selected-frontier change is claimed. Produced by Nick Petalas. Credit Cinna Davis #4694/#4698 and the retained #4470 lineage, Cao Yifan #4610, Chris R #4839, zxv, Jacob Alan Sussman, ywz65's observation-floor lineage, Ross Richardson #2480 and Ali Romman #4720.

Evidence

• Before: Construct a reversible 640-field center interface and exact raw dual (no new K) #4694 retains eight coupled center fields and leaves a paid consumer open; Normalize eight center chains and construct a conditional two-call consumer (no new K) #4698 normalizes the helper observations. Neither supplies this compression tradeoff.
• After: the written all-field proof gives the exact 624+j dimension and the target-specific minimum. The self-contained finite checker passes every 0..2-row center record over F2 and F3, every dirty input of the n=4,k=2 chain fixture, and every one-row target plus two multirow targets. It enumerates 112 record matrices, 67,683 input executions, 7,931 target tests, 125 nonzero zero-fiber collision witnesses and seven rejection controls. These finite-field fixtures do not reinterpret the predecessor's dyadic native word in positive characteristic.
• Both registered warning-strict normal and optimized checks pass locally, taking 0.125 and 0.126 seconds. The package metadata/pins and committed-byte/source-history checks pass at 8c76d9aebf94ba40c856fbd1437bca04e4d905a7. The five-file committed export passes integrity validation and a Git-free optimized replay, preserving every exported byte and matching scientific stdout. No handwritten receipt is used.

From the repository root, use the actual local interpreter path below, or substitute CPython 3.12 on another machine:

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

Expected scientific marker: PASS_CENTER_OUTPUT_COMPRESSION. Standard library only; the scientific verifier has no external runtime inputs. The source manifest pins the merged predecessor proof as provenance rather than importing or rechecking it.

Limitations and unrun obligations

The contract grants every old output and a fixed scalar linear center record, with no other input-dependent state. The 624+j count concerns independent linear functions, not necessarily particular coordinate entries. Additional caller state, changed targets, nonlinear measurements, intermediate access or repeated calls can change the problem. No information dimension is priced as a native call.

Physical chart conversions, reads, routing, restoration, precision and a complete profitable consumer remain open. No predecessor native replay, selected cold replay, Lean, compiler or outer theorem was run. Independent human and formal proof-kernel review are unclaimed.

The historical card's first example was attempted and raises KeyError: 'complex_profile' on the changed selected schema. It was left unchanged. The current selected source-binding control passes. Neither orientation check is a premise of this theorem.

Related open work

• Construct a reversible 640-field center interface and exact raw dual (no new K) #4694 supplies the cited coupled endpoint and injective eight-column B. This result answers a new observation-compression question without replaying or repricing its construction.
• Price an actual centre-chain terminal and certify its scoped cost loss (no new κ) #4839 prices one physical terminal policy and finds a scoped loss. This result gives an information threshold for fixed output records, and makes no new comparison of that terminal's cost.

Latest account publication check on 2026-10-22 at 21:58 UTC found no additional open PR studying this coupled center-output compression question among contributions opened since the advance claim.

Open next questions

1. Supply one literal changed consumer target T and determine whether rank(T(A-BR)^-1 B)<8. Its target matrix and complete costs are not provided here.
2. Emit and price an attaining row-basis record in the actual endpoint charts, including read and restoration work. Algebraic dimension supplies no native tariff.

Stop at this result; another construction round needs a changed, fully charged consumer.

CI notes

Hosted run 39860519577 passed both checks on exact head 8c76d9aebf94ba40c856fbd1437bca04e4d905a7: plan passed in 3m20s and the registered research/center-output-compression replay passed in 1m58s. This certifies execution of the declared bounded package checks, not independent mathematical acceptance or the unrun native/compiler/outer obligations.

Merge danger

Door: two-way. Blast radius: isolated research package.