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Matrix tensor construction research toolkit
Edition: 8 October 2026. Initial decision: Defer performance product. The product interpretation and pricing are hypotheses; independent proof and buyer validation remain open.
Research finding
The square matrix result bounds the complex arithmetic exponent by nine fourths, with related rectangular and every-field bounds. Scope notes explicitly distinguish arithmetic complexity from bit complexity and practical crossover sizes.
Problem and buyer
An algebraic-complexity research team needs to manipulate tensor constructions, verify parameter inequalities and explore their tradeoffs. An ML systems team instead needs fast, accurate hardware kernels at finite sizes.
What the finding could enable
A symbolic tensor-construction toolkit could improve research reproducibility and parameter search. A production matrix library would require additional finite constructions, numerical stability and hardware evidence.
Technical and commercial limits
Complex asymptotic arithmetic does not directly establish a faster GPU kernel, stable mixed-precision behavior or cheap extraction. Finite-field applicability varies across the companion bounds.
Minimal architecture
Construction parser -> exact parameter arithmetic -> tensor or restriction records -> certificate export -> small finite benchmarks. Avoid claiming a hardware advantage before comparing complete end-to-end kernels.
Existing alternatives and differentiation
Research scripts and existing linear algebra libraries serve different users. The realistic initial buyer is a research group funding tooling, with a limited market size.
Monetization hypothesis
Hypothesis: grant-funded tooling or specialist contract work. A lucrative infrastructure company requires a demonstrated production crossover or broader reuse outside tensor research.
Validation experiment
Validate a small construction symbolically, measure extraction overhead and coefficient growth, and compare stable finite kernels with established matrix multiplication at realistic dimensions.
Conditions to reject or defer
Defer if only asymptotic existence is accessible or if finite constructions exceed feasible memory and precision budgets.
Next concrete action
Read the extraction and uniformization details and list which parts can be materialized at small dimension.
Source evidence
Repository sources are pinned to revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Selected scope notes and manuscript statements have been reviewed to the extent described above; these links do not represent successful kernel checks.
Family 107
Subject: Matrix multiplication with exponent at most 9/4.