# Finite-field factorization: defer the new backend, test exact audit integration

9 October 2026. The first product recommendation remains the [Facility Plan Auditor](https://mathideas.stera.ventures/facility/). This review narrows opportunity 016 to a low-confidence specialist integration and adds a conventional bounded checker. No source theorem or competitive backend advantage was established.

## What the selected source actually requires

Family 142 takes a prime p in binary and a nonzero dense polynomial of degree n over Fp. It claims a deterministic complete factorization with monic irreducible factors and positive multiplicities in O(((n+1)L)^(10^12)) bit operations, L=ceil(log2 p). Constants return their leading coefficient and an empty factor list. This is polynomial factorization over a field, not integer factorization.

The driver fixes B=20+(n+1)(L+1). At p<=B^200000 it uses an older Berlekamp-algebra scalar search. The new auxiliary-prime/geometric branch only applies beyond that inequality. [Exact guard arithmetic](source-guard-analysis.json) finds the first possible upper-endpoint crossing for n=2 at L=4,753,080: 2^4,753,079 is still at most its corresponding B^200000, while 2^4,753,080 exceeds its corresponding bound. Because 2^L is an even endpoint, this is a necessary eligibility boundary, not construction of an eligible prime. Larger degree increases B. Every prime of at most 4,096 bits remains in the old branch for n>=2.

For each prime q<=n the new branch needs a prime ell not in {2,3,p,q}, with ell=1 mod 12q and p^((ell-1)/q) unequal to 1 mod ell. The algebraic reduction is polynomial in the **numeric** largest auxiliary prime E, not merely its bit length. The paper gives O(B^500000+B^100 E^20), and an auxiliary bound ell<=c0 B^20000000 with an unselected absolute constant c0. Substitution yields exponent 400000100 before the looser headline bound. These upper bounds do not establish a runtime lower bound, nor show that every run consumes them. They supply no demonstrated practical advantage.

The table is assembled in increasing q: factoring Phi_q only requires degrees q-1 and earlier entries. This is the paper's claimed way to avoid a circular factorization oracle. Large divisor coefficients are stored in binary and principal functions in circuits; their numerical magnitudes must not automatically be treated as loop counts. The norm/divisor construction and its geometric correctness were not fully reviewed or implemented.

## The analytic dependency is stronger than infinitude of primitive roots

The exact input is companion family 029, Theorem 1.2 of *Primitive roots for every admissible integer base*. It claims a zero-free strip Re(s)>1-10^-6 for every finite-order Hecke character over every cyclotomic field containing mu_12, including principal characters with their pole at one permitted. The width has no conductor or height cutoff. The introductory discussion distinguishes field-dependent constants/starting points from the shared numerical saving. Neither an all-base primitive-root infinitude statement nor an ordinary shrinking zero-free region supplies this exact input.

The factorization paper applies the strip to K=Q(mu_12q) and K(p^(1/q)), bounds discriminant/degree terms by B^3, and chooses x=T B^20000000. The relative error has the form O(T^-delta B^-17), delta=10^-6. A sufficiently large fixed T produces the asserted auxiliary-prime bound; no numerical T was selected here. The primitive-root paper in turn cites a Quasi-Riemann companion as context for mechanisms and develops its own extension. We inspected selected statement/setup sections only. The source presents its claim as unconditional; our acceptance of that dependency and the full proof remains open. Assuming the appropriate GRH would supply the required strip condition, but no conditional or unconditional source backend was executed.

## A practical, conventional capability

[finite_field_factorization_audit.py](../../tools/finite_field_factorization_audit.py) verifies submitted factor lists using exact modular arithmetic, deterministic prime checking, multiplication with multiplicities, and the classical Rabin irreducibility criterion. For a degree-d factor it checks x^(p^d)=x modulo the factor and gcd(x^(p^(d/q))-x,f)=1 for each prime q dividing d. Exponentiation uses repeated Frobenius steps and binary modular powering.

The accepted JSON contract uses canonical integer residues in ascending degree order, nonzero input polynomial, distinct monic positive-degree factors, positive multiplicities and the original leading coefficient. Constants use an empty factor list. Duplicate factors must be combined. The deliberately bounded implementation accepts primes at most 2^31-1, degree at most 64, at most one MiB of input, and two million charged scalar operations. These are reference limits, not customer-scale performance promises. Exhaustion returns unknown. Operation charges do not constitute a wall-clock or memory guarantee. The output hashes input bytes and checker code; hashes do not authenticate a user or prove the program correct.

[A concrete false-confidence case](multiply-back-false-confidence-report.json) submits x^2+3x+2 as its own supposed irreducible factor over F5. Multiplication matches perfectly; irreducibility fails. [A valid nonmonic repeated-factor example](repeated-valid-report.json) accepts 2(x+1)^2 over F5. Characteristic-p repeated inputs and constants are covered by tests.

The second mode illustrates source driver Step 5 with a **supplied** nonscalar Frobenius-fixed separator, after checking those prerequisites and square-freeness. For (x-(p-2))(x-(p-1)) and b=x, the first successful scalar is p-2. Recorded p=101 and p=1009 cases use 100 and 1,008 trials. At p=65537, a 256-trial cap returns unknown. The source does not require every implementation to choose b=x; this is one lawful choice illustrating a scan's possible cost, not a lower bound for all algorithms. A verified divisor is not reported as a complete irreducible factorization.

## Existing software and measured scope

[SymPy's official documentation](https://docs.sympy.org/latest/modules/polys/internals.html) already describes finite-field factorization, square-free decomposition, Berlekamp, Cantor-Zassenhaus, Shoup methods and irreducibility checks. The installed SymPy 1.13.1 factored all three scan examples, with each of three recorded runs per case taking approximately 0.11-0.28 milliseconds; each output passed our separate checker. These tiny local measurements are not a general benchmark or an independent implementation of the source's new method. [Saved comparisons](scalar-scan-comparisons.json).

[FLINT's official univariate modular-polynomial documentation](https://flintlib.org/doc/nmod_poly.html) documents an existing algebra surface and notes that many field operations assume, without checking, a prime modulus. Explicit input validation and portable evidence can be useful integration requirements, but this does not establish a missing commercial product. The separate factorization-document URL returned 403 during review, so it supplied no additional evidence.

## Validation, reading extent and business decision

[456 passing controls](../../snapshots/2026-10-08-baseline/finite-field-factorization-validation-2026-10-09-v1.json) include all 401 monic polynomials of degrees 1-6 over F2, 1-4 over F3 and 1-3 over F5. A separately implemented exhaustive monic-divisor oracle checks irreducibility and SymPy supplies complete factorizations that the new checker verifies. Additional controls cover multiplicities, constants, false factors, duplicate/noncanonical inputs, malformed JSON, resource/scan exhaustion, exact branch arithmetic and fresh-output preservation. Finite testing is not formal correctness proof.

Reading: the complete tracked factorization introduction, auxiliary table, driver, complexity and analytic-implication TeX sections; visual PDF pages 1,38,41,45. Companion reading: introduction lines 1-180, zero-detection section lines 1-160, visual PDF page 3, and both bibliographic READMEs. No full-paper review, Lean/kernel execution or source algorithm run occurred. [The source-location supplement](source-location-supplement.json) records tracked build/ TeX omitted by the preserved initial metadata. [Source/file hashes](../finite-field-factorization-artifact-manifest-2026-10-09-v1.json) and [ledger](../../checkpoints/manuscript-ledger-2026-10-09-v16.json) preserve exact extents.

Defer a direct port of the new backend. A specialist reproducibility adapter could preserve a CAS result and rerun a separate verifier, but existing libraries already cover the mathematics. The earlier AUD 5,000-20,000 pilot range was speculative and has no supporting buyer evidence. Do not build a standalone subscription until a real team shows recurring failures, valuable evidence requirements and willingness to pay. This module can share the wider evidence platform; it is not an additional forecast customer or revenue stream.