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Deterministic finite field factorization library
Edition: 8 October 2026. Initial decision: Conditional research. The product interpretation and pricing are hypotheses; independent proof and buyer validation remain open.
Research finding
Family 142 claims complete deterministic polynomial bit-time factorization of dense polynomials over prime fields, with the prime supplied in binary. Its proof depends on a companion uniform Hecke zero-free result. The family has no listed scope document in the baseline.
Problem and buyer
A computer algebra or coding-theory software team may prefer reproducible finite-field factorization without randomized execution or conjectural assumptions, especially for difficult large-prime cases.
What the finding could enable
An exact backend could support deterministic symbolic workloads and reproducible build or verification pipelines. It would strengthen an existing algebra capability if its dependency chain and practical cost survive review.
Technical and commercial limits
This is polynomial factorization over a prime field, not integer factorization or a demonstrated attack on deployed cryptography. Dense coefficient input differs from sparse succinct input. Strong dependent analytic claims need independent assessment.
Minimal architecture
Dense input and prime validation -> deterministic backend -> multiply-back verification -> irreducibility checks -> benchmark suite. Pin all companion dependencies and preserve a fallback reference implementation.
Existing alternatives and differentiation
SymPy and other algebra systems already factor polynomials. Compare speed, determinism requirements and coefficient regimes before building a separate service.
Monetization hypothesis
Hypothesis: a supported backend, contract implementation or reproducibility integration rather than a mass-market application. AUD 5,000-20,000 pilot assumptions depend on a specialist customer with a real bottleneck.
Validation experiment
Use repeated factors, irreducible inputs, small and large primes and increasing degrees. Verify factor multiplication and multiplicities independently and measure actual coefficient and memory costs.
Conditions to reject or defer
Defer if the analytic dependency cannot be validated or if existing methods are faster and randomization is acceptable to customers.
Next concrete action
Map the exact companion theorem dependency and read the complete construction before starting an algorithm port.
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 142
Subject: Deterministic polynomial factorization over prime fields.
Family 029
Subject: Primitive roots for every admissible integer base.
- Primitive roots for every admissible integer base
- Simultaneous primitive roots: a conditional lower bound for prime bases
Family 003
Subject: The quasi-Riemann hypothesis.
- The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $
\Re s\gt 7/8$ - The Quasi-Riemann Hypothesis (alternate 11/12 proof)
- Uniform exclusion of Landau–Siegel zeros
- Formal scope notes
Current alternative sources
Primary documentation reviewed on 8 October 2026. Product availability demonstrates alternatives, not demand or willingness to pay for this proposal.