150-item roadmap status

The 150-item feature proposal this file tracks is a roadmap, not a claim that every research system can safely be delivered in one pre-release. This file is the authoritative scope boundary: it records what is implemented, what is partial and precisely what is missing from it, and what is deferred. No deferred item is exposed as a placeholder, and nothing is replaced by Python or JavaScript mathematics.

Every implemented operation is performed by an existing library routine (PARI/GP, FLINT/Arb, YAFU, Msieve, CADO-NFS, GMP-ECM, primesieve, primecount) or, where no library provides it, by an optimized C program in numerisect/native/. The governing policy is stated in CONTRIBUTING.md, and tests/test_native_computation_policy.py enforces it.

Implemented

Factorization. 2, 3, 6, 9, 11–13, 15. Item 4 is complete for the algorithms this YAFU build provides plus SQUFOF; item 5 is the resumable GMP-ECM campaign manager; items 7 and 8 cover special-form recognition, SNFS suitability, and algebraic and Aurifeuillean factors verified by division. Item 14 is implemented for 64-bit interval enumeration and standard k-tuplets. Item 20 supplies headless access to every route.

Primality and special primes. 21–30, 31–36, 38–43. Deterministic Miller–Rabin witness sets, Pocklington and Pratt certificates with independent verification, the probable-prime taxonomy, the Carmichael analyzer, Sierpiński and Riesel covering sets, bi-twin chains, and the provable constrained-prime generator.

Modular arithmetic and structure. 44–57, 58–60. Reciprocity traces, congruences over composite moduli by Hensel lifting and CRT, four selectable discrete-logarithm algorithms, and finite-field arithmetic over F_p and F_{p^m}.

Arithmetic functions and integer structure. 61–74. Divisor lattices, smoothness and roughness, highly composite and colossally abundant families, a proven weird-number subset check, amicable and sociable cycles, and Cornacchia with a step trace cross-checked against qfbcornacchia.

Algebraic primes. 109–115. Quadratic rings with units and class numbers, general number fields with prime-ideal decomposition, and Chebotarev density experiments.

Visualization and education. 117–126, 128, 130, with 116, 127 and 129 as noted below. Ulam, Sacks, and polar spirals, the Eisenstein lattice, arbitrary-base modular wheels, residue heatmaps, gap timelines, the prime-race animation, four sieve animations, the complexity dashboard, and the educational proof viewer.

Analytic prime distribution. 75–93. Approximation-error and prime-number-theorem convergence charts, nth-prime bounds cited with their validity ranges, prime races and Chebyshev bias, progressions with expected-versus-observed counts, Hardy–Littlewood singular series with a rigorous tail bound, constellation predictions, Bateman–Horn estimates, maximal-gap search with merit and the Cramér, Granville and Firoozbakht comparisons verified against the published table, Maier-matrix experiments, and a two-parameter density surface.

Zeta and L-functions. 94–108, and 80. Explicit-formula prime counting from certified zeros, Chebyshev psi reconstruction, Riemann–Siegel remainder analysis, Euler-product comparison, zero-spacing histograms, pair correlation against the GUE prediction, Gram blocks and Gram's-law exceptions, the Backlund S(T) remainder, Dirichlet characters and L-functions with independent Hurwitz cross-checks, L-function zeros for GRH experiments, and Dedekind zeta functions through PARI.

Application and reproducibility. 131–142, 144–150. Item 18 adds distributed CADO-NFS sieving; read docs/DISTRIBUTED.md before enabling it, because CADO authenticates clients by IP address only. Complete CLI/API parity, six export formats, batch import, workspaces, searchable history, revision-keyed result caching, job priorities and reordering, per-job resource limits, pause and resume, desktop notifications, declarative engine adapters, performance history, client examples in five languages, and notebook integration.

Partially implemented

• 1 and 116 — the factor view shows the input, every factor with its exponent, primality status and discovering engine, and one total elapsed time. It renders as a flat root-plus-leaves list, not a hierarchy, and per-factor discovery time is not captured. Composite cofactors can be continued as linked child jobs, so the parent and child relationship exists in the data but is not drawn as a tree.
• 4 — every algorithm the installed YAFU build exposes is selectable (rho, p−1, p+1, ECM, SIQS, NFS, SNFS, Fermat, trial division) with expert parameter panels, and SQUFOF is supplied by numerisect-squfof because no installed library provides it. SQUFOF is limited to inputs below 2^62 by its 64-bit cycle; larger inputs are rejected explicitly rather than answered.
• 10 — cross-verification compares YAFU and Msieve factor multisets and rejects a disagreement. Comparing cofactors, per-engine primality conclusions, and per-engine timings side by side is not yet done.
• 13 — certificates are generated and independently verified for every prime factor of a completed job. Certifying a factor that is only a probable prime still depends on PARI proving it first.
• 14 — prime ranges and standard k-tuplets of sizes 2, 4 and 6 use primesieve. Sizes 3 and 5 stay on PARI: primesieve emits both admissible shapes together, and separating them would require offset arithmetic outside the engines.
• 17 — CADO-NFS parameters are exposed and stage progress is parsed from its log. Running an individual stage in isolation is not implemented.
• 29, 30, 37 — the taxonomy, Korselt analysis, and covering-set verification are complete within documented finite bounds; results beyond those bounds are inconclusive rather than negative.
• 69, 71 — record-number families and sociable cycles are enumerated within bounded searches; exceeding a bound is reported as inconclusive.
• 112–113 — splitting, inertia, ramification, and prime-ideal decomposition work for general number fields of bounded degree, with class-group work gated by a time budget.
• 127 — bounded educational step traces exist for Pollard rho, p−1, and ECM, computed by PARI/GP. Stage traces for the quadratic sieve and NFS are read from engine logs rather than instrumented.
• 129 — the engine decision tree is computed by PARI/GP and returned as an explicit decision path. It is rendered as a list, not yet as a diagram.
• 128 — the complexity table is cited literature and the timings are local measurements from this machine's job history, not a benchmark of the engines.
• 135 — job and report search covers text, status, engine, and date. There is no saved-query or faceted-search interface.

Deferred, with no placeholder implementation

None. Every proposal item is now implemented, partial with its gap named, or declined with the reasoning recorded below.

Declined, by decision rather than omission

These were considered and deliberately not built. They are recorded here so the choice is visible and can be revisited.

• 16, autotuning half — delivered in a narrower form. POST /api/factor-lab/tune runs YAFU's own tune, which measures this machine's SIQS/NFS crossover, and reports the result as a suggested NUMERISECT_CADO_THRESHOLD. Numerisect never rewrites its own configuration. Using the engine's own tuning routine is what the native-computation policy requires; writing a separate tuner would have reimplemented it.
• 16, benchmarking half — declined. Timing YAFU against Msieve, GMP-ECM and CADO on a fixed synthetic input set measures one machine on one day with one set of engine builds. It reads like a general fact about the engines and is not one, and it goes stale the moment an engine is rebuilt. The durable version already exists: GET /api/history/performance aggregates real job runtimes by engine and digit bucket, so the numbers come from work actually done.
• 19, FactorDB integration — declined. The code would be cheap: the permissioned network path, the two-key opt-in and native verification of claimed factors already exist for OEIS and local catalogues. The objection is to what it does to the application's meaning. Numerisect's claim is that it computes locally and proves what it asserts; FactorDB is a database of unverified community claims, and consulting it turns "your machine factored this" into "someone once said this factors that way, and we checked the division." The submit direction is worse, because it publishes your work to a third party, and any such switch will eventually be flipped without thought. It also buys little: numbers FactorDB knows are usually ones these engines factor quickly anyway, and for numbers it does not know it says nothing. Local catalogue files under STATE_DIR/catalogues remain the supported way to use known-factor tables, entirely offline.
• 143, side-by-side engine comparison — declined. The correctness half is already covered: cross_verify runs YAFU and Msieve independently and rejects a factor multiset disagreement. The timing half would present a single sample as a comparison, when the result is dominated by thread count and ECM luck. Engine choice is better served by the strategy adviser, which decides from digit count, algebraic form, small factors and completed ECM depth.

Beyond the proposal

An audit compared what the installed engines expose against what the application calls, and the following were added because they belong to the subject, not because the proposal listed them.

Exposing more of the installed libraries. A prime-counting algorithm comparison across Legendre, Meissel, Lehmer, Lagarias–Miller–Odlyzko, Deléglise–Rivat and Gourdon; Legendre's phi(x, a) and the nth-prime inverse approximations; the integer-structure predicates isprimepower, ispowerful, istotient, isfundamental and ispolygonal; Lenstra's divisors in a residue class, with the hypotheses PARI does not itself check enforced before the call; factorint's strategy flags as a factoring-method comparison; and a binary quadratic forms and continued fractions workbench covering reduction, composition, class groups, reduced-form enumeration, representation, exact expansions of quadratic irrationals, and Pell equations.

Things no single engine can do. Cross-engine verification computes the same quantity by every independent method available and reports whether they agree: pi(x) from primecount's six algorithms, primesieve's sieve and PARI's primepi; primality from PARI's proof, PARI's Baillie–PSW and GMP's independent test. On disagreement it reports every value and refuses to choose, because a majority of implementations sharing a bug is what a vote would conceal. An engine self-test asks each installed engine questions whose answers are published constants, each carrying a citation, so a miscompiled build is caught before its output is trusted.

A gap no library fills. numerisect_bigsieve.c enumerates primes in an interval of any magnitude. primesieve refuses inputs at or above 2^64 outright, and PARI's forprime is single-threaded and roughly 130 times slower per window at 10^30. Results at or above 2^64 are labelled probable primes from Baillie–PSW, never proofs.

Notes on scope and honesty

A search that reaches a documented bound, a test that times out, or a catalogue that has no entry is reported as inconclusive. None of these is presented as a negative result, and none is presented as a proof of primality.

Results are labelled by their strength. PARI isprime gives a proof; ispseudoprime gives a probable prime and is labelled as such. FLINT/Arb ball enclosures, certified zero intervals, and Turing-method counts are certified; plot samples use enclosure midpoints and are exploratory. Externally supplied values are unverified until an engine re-checks them.