Repository navigation
Conversation
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
This scoped proof answers which cube-zero multiplication maps retain the two-role cost-three outer-product witness. The exact cost is three precisely when the quadratic relation space is zero or an irreducible quadratic line; every other relation space has an explicit cost-two word. It also proves the minimum product-image dimension and gives a five-dimensional attaining algebra when the field has an irreducible quadratic. The complete kappa calculation is unchanged.
Proof and scope, exact checks, source pins, registered replay. Base: canonical main
32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7. Contribution head:ccd6d17483349ad1995369d4776f96482a40fa60.The algebra is split commutative local with radical cube zero, and the two chosen elements are independent modulo the radical square. Paid factors are I+u*v^T with base-field right vectors; arbitrary free GL2 maps and singular factors are allowed. The all-field proof includes characteristic two. It classifies this outer-product family, not arbitrary cube-zero targets or every local algebra.
CPython 3.12.14 registered normal and optimized checks pass: all 224 quadratic-relation subspaces over F2,F3,F5,F7, 10320 projective products, 2016 complete basis-input actions and 20 controls. There are 39 exact cost-three contexts and 185 with explicit cost-two constructions. Polynomial-root and product-zero tests agree. Both modes print
PASS_CUBE_ZERO_PRODUCT_MAP_COST; the launcher printsPASS_REGISTERED_RESEARCH_PACKAGE.Python 3.12 and the standard library suffice. Committed-byte checks pass. No extension-field census, predecessor scientific replay, physical coefficient implementation, selected cold replay, Lean or Docker run is claimed. No independent human or proof-kernel acceptance is claimed. Routing, restoration, precision, stock and complete supplier costs remain unpriced. Hosted status is unconfirmed at publication.
Credit Nick Petalas (@petalas) for PR #4764, its pinned witness and product-map question, and PR #4276 with the exact two-factor criterion. The alphabet and shared-chart predecessors retain their credit in those sources. Classical binary-quadratic and finite-field facts are not claimed as new. PR #5320 treats nonzero first-order pencil lifts, a separate question. The scoped claim records the handoff and overlap.
Next: give a physical realization and complete coefficient-operation price for the five-dimensional binary quotient. Fewer algebra dimensions alone do not establish a recursive saving.
Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.