TECHNICAL, P2 Route-TMS v1: validated integration and Taylor models, scoped

Leg 315. Branch leg/315-tms-v1. Runner experiments/p2_route_tms_v1_scoping.py (runtime 1.9 s). Curated data writeup/data/p2_route_tms_v1.json. Figure fig79_route_tms_v1_reach.png. Novelty pass writeup/novelty/leg_315.md (committed first). Journal, with the integration notes, experiments/journal/leg_315.md.

SCOPING ONLY. NOTHING IS BUILT. No solver module was written or edited. No ban was lifted, narrowed, or argued against. requirements.txt was not touched. Naming a reachable object is not reaching it: Walls 1 and 2 stand, no link of the L1→L4 chain moved, and Clay odds remain ~0.05%, unchanged.

1. The gate, in its pre-committed and immutable wording

Does the scoping name at least one concrete object currently unreachable by the radii-polynomial/NK apparatus that Taylor-model arithmetic reaches, with the reaching mechanism stated and its build cost classed?

Answer: YES. Two objects are named, O1 and O2, each with a named reaching mechanism and an evidence-backed cost class. A third, O3, is named as reached by neither, so that the YES is not read as larger than it is.

2. What the thesis got right, and the part of it that is false

The dispatched thesis says requirements.txt's "no adaptive ODE integrators are used" is a policy, not a finding. That is confirmed: the line is a project decision, and it is recorded nowhere as the output of a measurement.

But the naive form of the surrounding argument (radii-polynomial/NK cannot reach time-dependent objects, Taylor models can) is false, and this document does not make it. The novelty pass located van den Berg, Breden & Sheombarsing, arXiv:2305.08221, Validated integration of semilinear parabolic PDEs (2023): rigorous time integration by Fourier-space variation-of-constants, Chebyshev in time, domain decomposition, and a Newton–Kantorovich argument, applied to Fisher, Swift–Hohenberg, Ohta–Kawasaki and Kuramoto–Sivashinsky. NK does time. That result is carried in the reach matrix as control C2 precisely so the matrix can, and does, report against its own author's thesis.

So the reach argument must turn on something sharper. It does, and §4 states what.

3. The target, and the verifier's finding it rests on

The object is leg 251's Phase-1 candidate: BCG (Buckmaster, Cao-Labora, Gómez-Serrano, Smooth imploding solutions for 3D compressible fluids, arXiv:2208.09445, Forum of Mathematics Pi 13 (2025) e6) at the diatomic value γ = 7/5.

The leg-251 verifier's finding, in writeup/novelty/verify_251.md §C, is the hinge:

At BCG's scaling there is no self-similar profile system of the dissipative equation. […] the self-similar reduction (the autonomous (W, Z) ODE system that Thms 1.1/1.2 solve) is the Euler system. […] Dissipation […] enters the dynamically rescaled system as a forcing term, F_dis […] non-autonomous and exponentially decaying in self-similar time: the bounds […] all carry an explicit e^{−δ_dis s₀} prefactor. […] The gap […] is in the stability step: the r-restriction that lets the profile dominate F_dis.

and its re-posing:

Before Phase 1 is authorized, obligation 1 should be re-posed as: a rigorous enclosure of the linear/nonlinear stability step with the dissipative forcing retained, at a similarity exponent outside BCG's dominance regime.

The gap is therefore NOT a profile enclosure. Every object below is stated against that re-posed obligation, not against the retracted one.

3a. The measured numbers, carried forward and not re-derived

Repo-internal, from leg 266, corrected by leg 300 (experiments/journal/leg_300.md:40-51; leg 266's 6.855 was a slipped digit) and propagated repo-wide by leg 319:

This leg re-derives none of these. It uses them only to say where the named object aims: at r beyond 1.1909830.

4. O1: the named object, and its named reaching mechanism

O1, the sonic-crossing r-tube. A rigorous enclosure of BCG's autonomous (W, Z) self-similar Euler ODE flow through the sonic point, carried as a Taylor model polynomial in the similarity exponent r over an r-interval at γ = 7/5, whose output is the profile-vs-F_dis domination margin as a certified function of r extending across and beyond the dominance-window upper endpoint r = 1.1909830.

4a. Why the NK apparatus does not reach it, with the realization named (lesson 91)

"Measured dead" with no realization named is not an admissible negative. The negative here holds in three named realizations, all of them this repository's own:

1. the ℓ¹_w coefficient basis, leg 54: best Z₁ improvement 1.167× where >8× was needed;
2. the sup-norm collocation basis, leg 56: (H,D) consistency defect over budget by 1.85e7× (derivative) and 2.04e11× (Hilbert);
3. origin-H² capped at a = 0, with no transfer to the real target, legs 163/176.

The structural reason, stated so it can be checked rather than believed: a radii-polynomial certificate is a zero-finder. It requires a fixed function space, a numerical candidate, and an approximate inverse of the linearization in that space. The (W, Z) trajectory must pass through the sonic point, a degeneracy of the vector field, where a global-basis linearization is not boundedly invertible. And this repository's standing, re-posed ban forecloses proposing a fourth space/basis without its own scoping leg. So the NK route here is closed by this repository's own law, not merely by difficulty.

This is lesson 87 in its own terms: a certification method has a SHAPE, and the shape is a property of the operator. NK's shape is a zero; what is wanted is a flow map.

4b. The reaching mechanism, named in three parts

SONIC-POINT-DESINGULARIZED TAYLOR-MODEL STEPPING IN THE SIMILARITY PARAMETER.

1. The object is a finite-dimensional autonomous ODE: the verifier's own words, "the autonomous (W, Z) ODE system that Thms 1.1/1.2 solve". A validated IVP integrator encloses the flow map, and therefore requires no function space at all. That is the first load-bearing half of the reach.
2. The sonic degeneracy is crossed by quasi-homogeneous desingularization plus local analytic continuation. This technique is already recorded in this repository, at writeup/data/p2_route_w2l_v1_lit.json:180, verbatim: "quasi-homogeneous compactification plus rigorous integration in interval arithmetic". That is the second load-bearing half.
3. Wrapping control by Lohner-QR, mean-value form and shrink-wrapping (Berz–Makino). Without it, naive interval stepping cannot cross an interval of useful length. This is the part that is Taylor-model arithmetic proper rather than merely interval arithmetic, and it is exactly what solver/interval.py cannot supply.

Carrying r as a symbolic Taylor variable then certifies an r-interval in one integration rather than one r per run.

4c. One claim explicitly NOT made

The parameter-interval property in 4b is a convenience, not a load-bearing distinction. Radii-polynomial certificates are routinely run with interval parameters for branch continuation, so "Taylor models sweep r and NK does not" would be false, and is not claimed. The reach rests on 4b(1) and 4b(2) only.

5. Cost classing: what each estimate rests on

The useful part of this leg is the distinction between an existing library, a hand-roll of the kind leg 256 was forced into, and a genuine research problem.

5a. O1: class C, a genuine research problem, with a class-B implementable core

Not class A. No maintained, pip-installable, Python Taylor-model ODE integrator was located. python-flint 0.9.0 (PyPI, released 2026-07-03, SPDX MIT AND LGPL-3.0-or-later, FLINT/Arb themselves LGPL v2.1+, requires only Python ≥ 3.10, does not require scipy) supplies arb/acb rigorous balls and arb_series, but no Taylor-model IVP integrator. The field's implementations are out-of-language: CAPD (C++, arXiv:2010.07097), VNODE-LP (C++, unmaintained), Flow* (C++), COSY Infinity (restricted licence), TaylorModels.jl (Julia). TERA (arXiv:2607.01189, 2026) is Python, but is control-systems reachability, brand-new, and unvalidated for a fluid profile.

The class-B core. A Taylor-model type over solver/interval.py's existing outward-rounded Interval, plus a Lohner-QR wrapping layer, is a hand-roll of exactly the kind leg 256 was forced into. Leg 256's own words (experiments/journal/leg_256.md:69-77): "with no scipy there is no Gauss–Laguerre rule to call, so nodes come from Sturm bisection on the Jacobi matrix […] followed by Newton on the scaled recurrence." That was one leg for a static quadrature rule; a validated IVP integrator with wrapping control is strictly larger, so ≥ 1 leg and plausibly several.

Why C and not B: the research obstacle is not the integrator. O1 is only the first rung. The verifier's re-posed obligation is the stability step with F_dis retained over s ∈ [s₀, ∞): a non-autonomous PDE trapping argument, not an ODE. The standard bridge from a validated ODE integrator to a validated PDE enclosure is Zgliczyński's self-consistent a-priori bounds (math/0005247, Zgliczyński–Mischaikow, 2000), whose hypotheses are dissipative/parabolic. BCG's rescaled system is quasilinear hyperbolic, with dissipation entering as an exponentially decaying forcing (e^{−δ_dis s₀}) rather than as a smoothing principal part. That mismatch is the research problem, and naming it is this leg's most useful output.

5b. O2: class A / B split

O2. Rigorous enclosure of the degree-4503 Gauss–Laguerre nodes and weights, and of the resulting quadrature, that leg 256 had to hand-roll in float64, i.e. closing leg 256's own recorded ceiling.

capabilities.py:496-498 records that ceiling verbatim: "CEILING: float64, NOT interval arithmetic, this reproduces their CONSTANTS, not their proof." Breden–Chu prove their Theorem 42 in interval arithmetic over 16384-bit BigFloat; leg 256 could not, and said so.

Reaching mechanism: arbitrary-precision ball arithmetic with rigorous special-function and root enclosures. This is the validated-integration half of the route name, not the Taylor-model half, and is labelled as such rather than smuggled in. Arb's exponent range makes leg 256's e^{−1.8e4} × 1e1220 product representable directly, removing the per-node log-renormalization scaffolding leg 256 was forced to build.

Class A for the arithmetic substrate, with licence and version pinned as above, and note it does not even engage the "no adaptive ODE integrators" policy line, which is about scipy and about ODE integration, neither of which this is. Class B for the quadrature layer: it is not confirmed that rigorous quadrature (acb_calc_integrate) is exposed through the Python bindings, the API page reached documents v0.3.0 and lists no .integral() method. That is recorded as under-evidenced, not assumed either way.

Territory: leg 312 (Route-APIA) already owns the arbitrary-precision interval build (solver/interval_mp.py). O2 is a consumer of APIA, not a new build leg.

5c. O3: reached by neither, and deliberately not costed

O3. Forward invariance (a trapping region) for the dynamically rescaled compressible-NS perturbation system with F_dis retained, over the semi-infinite self-similar time interval s ∈ [s₀, ∞), at r outside the dominance window.

This is the verifier's re-posed obligation in full, and no mechanism is claimed for it. Costing an object that neither apparatus reaches would be vibes, which the spec forbids. Whether any computer-assisted work has ever enclosed a non-autonomous forcing-domination / trap-region argument of this shape, in any field, is the explicit territory of leg 267 (Route-FDL). This leg does not pre-empt or predict FDL's answer, and does not claim the precedent is absent.

6. The reach matrix, and why it is not a tautology (lesson 90)

A matrix whose every row said "Taylor models win" would be a restatement of the author's intent. So the matrix carries two controls that come out the other way, and the runner asserts it in code: _check_matrix_is_not_a_tautology fails the run if no row is NK-only, or if no row is "neither". It is wired to the same field the headline reads, so it cannot pass vacuously.

C1 is this repository's own bordered certificate machinery (capabilities.py:238-243, :333-337, :359-360): Taylor models have no notion of an infinite-dimensional zero-finding problem and lose that row. C2 is scored off an external paper that directly contradicts the naive form of this leg's thesis, so it could not have come out the author's way by construction.

7. Instrument controls, including one instrument found broken

The arXiv search UI returns zero for ANY query containing two quoted phrases ANDed. "Taylor models" alone returns 84 results; "validated integration" alone returns 20; every two-phrase conjunction returns 0. Six all-zero queries were therefore treated as a broken instrument and discarded, not banked. pypi.org/search was likewise blocked by a "Client Challenge" page and discarded; individual project pages load and were used instead.

Three controls passed: the positive control (arXiv:2505.03091, Cadiot, fetched and matching the in-repo record), the negative control (a fabricated arXiv path returning HTTP 404), and a spelling-variant control, "Navier--Stokes" with the LaTeX double hyphen returned exactly this repository's own recorded negative-control citation, arXiv:2604.09949. Full log and eleven banked links in writeup/novelty/leg_315.md.

8. capabilities.py, grepped as the standing ban requires

Terms Taylor model, validated integration, rigorous integration, interval ODE, CAPD, Lohner, COSY, VNODE, flow map, wrapping: one hit, capabilities.py:98 ("exact-Taylor inner integrals" inside Xu's resolvent, an exact Taylor expansion, unrelated to Taylor-model arithmetic). Every other term returns zero. What is registered nearby is solver/interval.py, "rigorous interval arithmetic … hand-rolled outward-rounded intervals; no scipy, no mpmath": scalar arithmetic, no integrator. The absence is corroborated verbatim in-repo by leg 285's spec, experiments/p2_route_p2s_v1_spec.py:493: existing="solver/interval.py supplies the arithmetic but no validated integrator".

9. What this leg does not do

It does not build. It does not edit requirements.txt, the policy observation is routed as an integration note in experiments/journal/leg_315.md, per the spec's own instruction, and it is worth stating that on this leg's evidence the policy is not the binding constraint: the tooling that would matter (python-flint/Arb) is not scipy, and the Taylor-model IVP integrators are not Python at all. It lifts no ban; the one adjacent ban, the re-posed radii-polynomial ban, is recorded as disjoint in subject matter (a Taylor-model flow enclosure proposes no function space, so it is not the "fourth space/basis" the lift clause contemplates) and that reading is routed to the user, not decided here.

No link of the L1→L4 chain moved. Clay odds remain ~0.05%.