Physical operation frames with deferred readouts: conditional κ = 1.11192082577e-4

[DanieleCorso]

Summary

This draft introduces per-operation physical frame optimization and partial deferred readouts on the PR #118 complex multiplication network.

Under the inherited analytic, fixed-tape and stopped-product assumptions, the resulting finite numerical certificate proposes:

T(n) = O(n (log n)^(1−κ))

κ = 111192082577 / 10^15 = 1.11192082577 × 10⁻⁴

This is a conditional exponent improvement, not a measured runtime speedup or a standalone proof of the complete multiplication algorithm.

Improvement over previous candidates

Reference| Conditional κ| Improvement
PR #118| 1.08542805968e-4| +2.44077%
PR #125| 1.10591581647e-4| +0.54299%
This candidate| 1.11192082577e-4| —

The comparison with PR #125 reflects the open-PR frontier at the time of preparation and is subject to ongoing submissions and verification.

Technical contribution

1. Per-operation binary frames: Assign independent nondegenerate frames to physical occurrences of scalar operations, allowing different copies of a logical value to use different frames.

2. Backward construction and local optimization: Construct legal frames backward from the output constraints and optimize the resulting frame chains while preserving predecessor/successor inclusions.

3. Partial deferred readouts: Select additional deferred readouts while explicitly accounting for exterior costs, first transitions and target-front charges.

4. Preserved underlying construction: Retain PR Deferred readouts on PR117’s complex producer: conditional κ = 1.085428e-4 #118's scalar DAG, carrier matching, rational-center decoder, source injections, endpoint charges and the existing round-seven bit supplier.

Complex network comparison

Metric| PR #118| New candidate
Scalar additions| 91,770| 91,770
Physical roles| 28,705| 28,705
Carrier links| 71,185| 71,185
Deferred roles| 3,020| 4,107
Complex saving| 1.08566486e-4| 1.11216930e-4
Final κ| 1.08542805968e-4| 1.11192082577e-4

No global optimality is claimed.

Verification and current blocker

Checks passed:

• Five focused mathematical/control tests
• Exact moment and next-grid checks
• All 47 strict numerical constraints and seven assembly margins
• Full compiler reconstruction, arbitrary-scratch replay and binary frame-chain checks

Outstanding correctness issue:

The GitHub Actions verification currently fails on Python 3.11, 3.13 and 3.14 during the independent reflection audit.

"research/physical-frame-kappa/reflection_audit.py" rejects an expanded readout coefficient exceeding unit magnitude ("maxcoef > 42" with denominator 42).

The complete physical readout transfer and associated accounting must be corrected and revalidated before the conditional witness can be considered fully verified. Full repository verification has not yet passed.

Reproduction:

make verify-physical-frame

Attribution and dependencies

This contribution builds upon Rohan Arun's PR #118, eumemic's PR #117, Avi Eisenberg's physical-role compilation, Swapnil Jain's round-seven bit construction, and the stopped-product and nondegenerate-frame interfaces developed by icekylinx and other upstream contributors.

The source references, acknowledgements and original licensing notices are preserved.

Status: Draft — mathematical and implementation review requested.