lastsolve 1.1.0

The last few solves you'll ever pay for: a certified accelerator and identification layer over any black-box parametric solver, built on resona's effective-rank dial.

lastsolve

The last few solves you'll ever pay for.

lastsolve is a certified accelerator and identification layer over any black-box parametric solver. You bring an expensive function f(k) → field — a PDE solver, a legacy code behind a subprocess, a simulator you can't even import. lastsolve spends a handful of calls on Chebyshev nodes and hands back the whole price list:

In plain terms. You have a simulation that takes 2 minutes, and your workflow — a parameter sweep, a calibration loop, an MCMC — needs to call it 10,000 times. lastsolve runs it ~10 times, learns how the answer depends on the parameter, and serves the other 9,990 calls in microseconds — with a validated error bar on every answer, and a typed exception instead of a guess whenever it can't keep that promise.

The contract: f(k) → 1-D numpy array (anything np.asarray-able; a scalar result is a length-1 array). Plain CPU NumPy in, plain NumPy out — no PyTorch/JAX tensors needed, no GPU, no training loops. The whole library is a few files of numpy + scipy + resona you can read in an evening.

Learn by doing: the COOKBOOK — ten paste-and-run recipes, each ending with the verbatim output it printed on this machine.

from lastsolve import accelerate, learn

@accelerate(warmup=5)        # watch 5 real calls, learn the range they live in,
def solve(k):                # then serve everything from the surrogate
    ...  # your expensive solver

solve(0.021)                 # microseconds, with a validated error and the Φ₁ dial
solve.stats                  # every real call counted

s = learn(solve_fn, (0.014, 0.026))          # one knob  → Surrogate
s = learn(solve_fn, [(a1,b1), (a2,b2)])      # several   → SurrogateND, same verbs
k_hat, crb = s.invert(y_obs)                 # rich Estimate, unpacks as a tuple
s.certify(n_cal=8)                           # Certificate(err ≤ 3.9e-15, ≥88.9%)
s.query(0.05)                                # OutOfRangeError — lastsolve refuses
                                             # to extrapolate; strict=False to
                                             # accept an uncertified answer knowingly

Built on resona's matrix-free effective-rank dial — the measuring instrument of the Spectra Without Matrices series. The philosophy in one line: measure the structure first, then pay accordingly.

What you get for ~10 solves

API	What it does	The honest part
@accelerate	transparent surrogate cache: in-range calls served in µs at near machine precision	out-of-range calls fall through to the real solver — never extrapolates; .stats counts every real call
Surrogate(f, krange)	the core: query(k), deriv(k) (Fisher info), adaptive node ladder, transform='auto' discovers coordinates like 1/√k	.val_err from held-out solves; .phi1 — resona's dial: ~1 healthy, ≫1 a wall, ~0 dead parameter
.certify(n_cal, alpha)	split-conformal error band	distribution-free finite-sample guarantee (and it tells you 8 calibration points buy 88.9%, not 90%)
identify(f, data, krange)	maximum-likelihood k̂ ± Cramér–Rao bar from one observation	verdict includes "the data do not contain this parameter" when they don't
audit(f, x0, sigma, prior)	field-level identifiability: how many independent numbers about N unknowns your dataset holds (matrix-free probes + resona)	reports the blind subspace no method can recover, before anyone reconstructs anything
detect_break(f, krange)	bifurcation alarm + blind localization via validation-error bisection	refuses to fit across broken physics instead of interpolating a lie
classify_wall(f, krange)	after the alarm: is the wall removable (a coordinate heals it) or genuine (Shor-class)?	resona's lift-rank saturation test — the Journey-I instrument, pointed at your family
SurrogateND(f, box)	several parameters, committee-disagreement adaptive design	45 solves matched a 125-solve tensor grid on a 3-parameter family; sloppiness (Φ₁ ≈ 1–2 at p=3) is measured, not assumed
CommandSolver + accelerate_command	accelerate a solver you cannot import — any CLI/binary	every subprocess invocation counted
TimePropagator(snapshots)	learn the propagator from one trajectory (DMD/Koopman via resona.lift)	ships |λ|max stability verdict + held-out-tail validation; says when long prediction is dishonest
identify_spectral(family, k0, λs)	recover a parameter from a measured spectral fingerprint (resona.wkernel + rayleigh_polish)	uses several eigenvalues — one alone can be ambiguous, and the docstring says so
normality_warning(A)	non-normality check before trusting any spectral dial	"the spectrum lies about where this operator acts" is a measurable condition

Why believe any of this

Because none of it is asserted — it was all run first. The methods were battle-tested on a zoo of 35 nonlinear PDE families (Burgers, KdV, Kuramoto–Sivashinsky chaos, NLS solitons, Camassa–Holm, fractional heat…), included here as lastsolve.zoo and exercised by the test suite:

• forward surrogates at ~5·10⁻¹⁵ over ±28% parameter ranges, ~570× faster queries;
• inversion saturating the Cramér–Rao bound (median error ≈ 0.7× the bound — the theoretical optimum is 0.674);
• conformal coverage measured at 88.4% against the exact 88.9% finite-sample guarantee;
• a pitchfork bifurcation localized blind to 3% of the range, zero false alarms on healthy physics.

The research trail with every number: The Price of an Answer (Journey II) and Never Quantum at All (Journey I).

Install & test

pip install numpy scipy resona   # the FULL dependency list — no ML frameworks,
pip install -e .                 # no CUDA, nothing to train
pytest tests/ -q                 # 26 tests, ~17 s, real PDEs inside

Honest limits (v1.1)

• Scalar-parameter surrogates are certified; multi-parameter (SurrogateND) is adaptive least-squares — excellent in the sloppy regime (which is most of physics), but its validation is empirical, not conformal yet.
• The surrogate is built per initial condition / configuration — it is a cache around a question family, not a general solver replacement.
• Smoothness is measured, not assumed: when the parametric manifold genuinely resists (a breathing soliton — Kolmogorov n-width), Φ₁ flags it and no node budget will help. The dial exists precisely so the library can refuse honestly.
• resona.effective_rank is a stochastic (Hutchinson) estimate; lastsolve uses 128 probes, good to a few percent on the dial.

Design principles

1. Never silently wrong — every answer is validated, certified, or refused.
2. The dial before the fit — Φ₁ says whether a cheap surrogate exists at all.
3. Everything is counted — the solver calls we spend are the price we quote.

---

MIT · Dmytro Sierikov · part of the Spectra Without Matrices series