Phase-2 P2 (Route D v1: a rigorous interval-arithmetic core and the a=0 Newton–Kantorovich framing

Status: Level-1 tooling + a Level-2 scoping/framing result) NOT a certificate. This is the project's first brick on the rigor ladder (the Level-1 → Level-2 jump): the step from a numerically-found approximate profile toward a computer-assisted certification of an actual solution. It delivers (a) a hand-rolled, dependency-free rigorous interval-arithmetic core, and (b) the deterministic a=0 measurements + operator framing that decide how a Newton–Kantorovich (NK) certification of the gCLM two-scale traveling wave must be set up. It makes no certificate claim, and does not move overall Clay odds (still ~0.05%). Its value is turning the banked Tier-1 "guess" (the GA two-scale profile of TECHNICAL_P2_KLADDER.md) into a certifiable question, and answering, with evidence, whether that question can be framed.

Rebuild the figure from committed data (no re-derivation): python writeup/4_p2_lottery/p2_route_d_evidence.py → writeup/figures/fig19_p2_route_d.png (reads writeup/data/p2_route_d_probe.json; regenerate the data (deterministic, ~10 s) with python experiments/p2_route_d_probe.py).

Code: solver/interval.py (the interval core) + test_interval.py (5/5, gated with fractions as an exact oracle); the probe experiments/p2_route_d_probe.py reuses the banked residual solver/gclm_family.py::GCLMResidual.residual_two_scale and Hilbert operator solver/line_hilbert.py. Full test suite (8 files) green.

Update (Route-D v2, 2026-07-28). The dress rehearsal proposed in §8 has been run: see TECHNICAL_P2_ROUTED_DRESS.md / fig20. The ball does not close, at any truncation, gauge or weight. Three claims below are superseded and should be read with that in mind: (i) §6's "Z₀ + Z₁ < 1 is plausible" (it is not; Z₁ ≥ N+1 from the truncation coupling alone; (ii) §7's ordering of the open risks) G (gauge) is exonerated by a three-gauge ladder, and R (the θ = ±π far field) is the blocker, confirmed by ablation; (iii) §5's "on the decaying subspace" caveat on H(cos kθ) = sin kθ, the identity is in fact unconditional (a Hardy-space argument, v2 §1). A k = 0 coefficient error in this leg's closed-form band was also found and fixed (v2 §1). Everything else here stands.

0. Why this brick, and what "Level-2" means here

Everything banked before this leg is Level-1: a novel numerical map (the GA finds an approximate self-similar profile and measures a residual floor). A GA proves nothing: its output is a guess. Level-2 is the first genuinely "new maths" rung: a rigorous, computer-assisted statement, an interval / Newton–Kantorovich argument that a true solution exists in an explicit neighbourhood of the guess, with every inequality machine-checked. This is a real genre (Chen–Hou, Gómez-Serrano, van den Berg–Lessard) and, honestly, an incremental one: even full success is a toy-model certification, not a Clay solve.

The honest first question is not "certify it" but "can a certifiable fixed-point statement even be set up?": can we bound the linearized-operator inverse, the defect, and the Lipschitz constant well enough for an NK ball to close, gated against the a=0 exact anchor where the answer is known? This leg answers the prerequisites to that question and frames the attempt. It deliberately stops short of the attempt itself (the next brick, §7).

1. The object and the anchor

The two-scale (traveling-wave) rescaled residual of the gCLM a-family (HQW25 = arXiv:2401.14615), at a=0 first:

F(Ω, c) := Ω H(Ω) − c Ω_X = 0,        (TW)

with H the line Hilbert transform, Ω even and decaying, and c the traveling-wave speed. The exact anchor is Ω₂ = −1/(1+X²), c = 1/2 (HQW25's a=b=c=1 normalization; H(Ω₂) = −X/(1+X²)), at which the discretized residual is ~10⁻⁹. The general a≠0 case adds −a U Ω_X, U = ∫₀ˣ H(Ω); we frame a=0 first and treat a>0 as a perturbation (§7).

2. The interval-arithmetic core (solver/interval.py)

A Level-2 statement is only as trustworthy as its arithmetic, so we do not use floating point for the bounds. solver/interval.py is a self-contained, numpy-backed Interval type (scalar or whole-vector), no scipy / no mpmath, with guaranteed outward-rounded + − × ÷, reciprocal (zero-guarded), and the three rigorous reductions the residual needs: isum, dot, and matvec (point-matrix × interval-vector, for the Hilbert and derivative operators).

Rigor model. We do not portably own the FPU rounding mode from numpy, so we use the standard round-outward-by-one-ulp discipline: evaluate the real interval-extension formula in IEEE-754 double (each elementary op correctly rounded to ≤ 0.5 ulp), then push the lower endpoint down and the upper endpoint up one ulp with np.nextafter. One ulp conservatively covers the ≤ 0.5 ulp elementary rounding, so every result is a guaranteed enclosure. For the reductions, the classic running-error bound |fl(Σ) − Σ| ≤ γ_m Σ|tᵢ|, γ_m = m·u/(1−m·u), u = 2⁻⁵³, is added before the outward push: giving a rigorous, fully-vectorized enclosure of an m-term accumulation (here m ~ 1200) with no Python-level sequential loop.

Validation (test_interval.py, 5/5). Each op is gated against fractions (exact rational arithmetic) as the oracle: + − × ÷ of point intervals enclose the exact rational result; 1/3 straddles the true value at ulp-scale width; a fresh point interval is width 0 while every arithmetic result is a ≥ 1-ulp box that still encloses the exact value; inclusion-monotonicity holds; the reciprocal zero-guard fires; and isum/dot/matvec enclose the exact answer, with the box-matvec covering 64 random samples from the input box.

3. Q1, arithmetic precision at the anchor (fig19-A)

Enclosing F(Ω₂) rigorously at the exact anchor gives (n = 2001 grid):

The enclosure width is ~10% of the defect it encloses, the interval core is comfortably precise enough to carry a Newton–Kantorovich defect bound Y₀; the enclosure sup|F| ≤ 9.8 × 10⁻¹⁰ is dominated by the true defect, not by rounding. Over a genome box of radius r around the anchor parameters (A,B) = (−1,1), sup|F| grows linearly (r = 10⁻⁶ → 1.9×10⁻⁴, slope 1), i.e. dominated by genuine residual sensitivity, an honest Lipschitz slope of ~190, not by wrapping. The defect term is enclosable.

4. Q2 (the degeneracy, counted (fig19-B)

The a=0 zero set is not an isolated point: every single Lorentzian A/(1+BX²) is an exact traveling wave (speed −A/2√B), so the solutions form a 2-parameter scaling valley generated by two continuous symmetries) amplitude (Ω,c) ↦ (λΩ, λc) and dilation (Ω,c) ↦ (Ω(·/μ), μc). A naive interval-Newton bound on ‖DF⁻¹‖ is therefore infinite; the gauge quotient is not optional. Singular values of the finite genome-map Jacobian count the kernel:

Both directions vanish when c is slaved (the full 2-D valley); fixing the speed removes exactly one. So two scalar gauge conditions (speed c + one normalization) isolate a nondegenerate zero. Concretely: fix c = 1/2 and impose Ω(0) = −1; the residual symmetry that preserves c is the fiber λμ = 1, i.e. Ω ↦ (1/μ)Ω(·/μ), which sends Ω(0) ↦ (1/μ)Ω(0), so Ω(0) = −1 forces μ = 1 → isolated. (The precise Fredholm-index bookkeeping (whether c floats with the 1-D cokernel bordered, or is fixed) is open sub-task G, §7.)

5. Q3: the diagonalization that makes it tractable (fig19-C)

Compactify the line to the circle by θ = 2 arctan X, X = tan(θ/2). Then the line Hilbert transform equals the circular conjugate-function operator (diagonal in the Fourier basis, cos kθ ↦ sin kθ) verified numerically to ~10⁻⁷ (discretization-limited) for k = 1…6 on the decaying subspace (endpoint- vanishing at θ = ±π ⇔ X = ±∞):

modes [1,3]: 8.3e-8   [2,4]: 1.4e-7   [1,5]: 2.7e-7   [0,2]: 8.3e-8   [2,6]: 3.3e-7
anchor (k=1): 2.1e-8

In this basis the anchor is a 2-term Fourier object: Ω₂ = −(1+cosθ)/2 (so a₀ = a₁ = −1/2, all higher zero) and H(Ω₂) = −(1/2) sinθ. This is the structural gift the whole framing rests on.

6. Q4: the linearized operator is banded + rank-1 (fig19-D)

F maps an even profile to an odd residual (Ω even ⇒ Ω H(Ω) and Ω_X odd), so DF is a map from cosine coefficients {aₖ} to sine coefficients {bₘ}. With Ω₂ degree-1, each term of

DF[h] = h·H(Ω₂) + Ω₂·H(h) − c h_X,     h_X = (1+cosθ) h_θ

couples cos kθ only to sin (k−1)θ, sin kθ, sin (k+1)θ. Built in closed form, DF is tridiagonal (bandwidth 1) plus a rank-1 column ∂/∂c = −Ω₂,ₓ = −(1/2)sinθ − (1/4)sin2θ. A banded operator whose tail is dominated by the c·(ik) diagonal has an explicit O(1/(cN)) tail-inverse bound: this is the concrete reason the finite-section NK bounds Z₀ + Z₁ < 1 are plausible, not merely hoped. The closed-form band matches the dense grid operator to 3.9×10⁻²; the residual is the known θ = ±π (Cayley) endpoint correction: flagged sub-task R (§7), not swept under.

7. The a-posteriori Newton–Kantorovich framing and its open risks

We instantiate the standard radii-polynomial theorem (van den Berg–Lessard). For an approximate zero x̄ = (Ω̄, c̄), an approximate derivative A† ≈ DF(x̄), and an injective approximate inverse A ≈ DF(x̄)⁻¹, with the Newton-like operator T(x) = x − A F(x), require bounds

Y₀ ≥ ‖A F(x̄)‖,   Z₀ ≥ ‖I − A A†‖,   Z₁ ≥ ‖A(A† − DF(x̄))‖,   Z₂ ≥ ‖A · D²F‖.

Then p(r) = Z₂ r² − (1 − Z₀ − Z₁) r + Y₀; if Z₀ + Z₁ < 1 and (1 − Z₀ − Z₁)² ≥ 4 Y₀ Z₂, T is a contraction on B(x̄, r₀) and F has a unique zero there, the certificate. Two simplifications are real: F is quadratic, so D²F is constant and Z₂ is r-independent (no third-order term); and the anchor being a finite trig polynomial means its convolution tail is zero beyond mode 1. The space is a weighted ℓ¹_ν of cosine coefficients (a Banach algebra under convolution) ⊕ ℝ for c.

Honest open risks (what could keep it from closing). - G: gauge / Fredholm index. The exact well-posed square system (fixed-c + one normalization, vs c floating with the 1-D cokernel bordered). The crux; get it wrong and Z₀ is meaningless. - R: endpoint rank-1 correction. The diagonalization holds on the decaying subspace; the θ = ±π Cayley correction (the 3.9×10⁻² of §6) must be carried explicitly or shown negligible. - T: truncation tail. Make the O(1/(cN)) tridiagonal tail-inverse bound rigorous (the "eventually diagonally dominant" argument), enclosed. - a ≠ 0. −a U Ω_X becomes a bounded but non-banded operator under the map, and, decisively, off a=0 there is no exact anchor (residual floor ~10⁻²), so Y₀ jumps from ~10⁻⁹ to ~10⁻² and the ball very likely will not close. The probable honest outcome is "certifies at a=0, not yet at the a≈0.5 boundary."

8. The next brick, and the honest ceiling

Next: a float dress rehearsal, build DF as a finite (N+1)-mode matrix in plain floating point, invert the finite section, and compute Y₀, Z₀, Z₁, Z₂ and the radii polynomial across a small N-ladder with the §4 gauge. This answers the only question that gates the interval build (does Z₀ + Z₁ < 1 and does the ball close at the anchor?) at near-zero cost, and either green-lights the verified-interval enclosure or surfaces exactly which sub-task (G/R/T) blocks it. A legitimate, publishable result either way (including the negative "why the naive NK doesn't close here yet"). Only after it closes in float do we harden with the interval core.

Ceiling, said plainly. This leg is validated tooling and a scoping/framing result. It does not climb the rigor ladder by itself: it makes the Route-D attempt concrete and grounded. Even the eventual success it aims at is a computer-assisted toy-model certification, not a Clay solve. What is genuinely new here is only that the certification question is now framed in a basis where its central operator is banded, and that we can say so with evidence rather than hope.