Route-MTSC v1: does the MT survivor of leg 301's screen survive on the real target? (leg 320)

Status: PARKED, escalated to the user. This leg answers the scoping gate the re-posed ℓ¹-Fourier / radii-polynomial ban's lift condition names. A YES here does not lift the ban, only the user's ruling does. Nothing beyond the artifacts below is built.

1. The gate, quoted from the Decision Maker's spec before the work began

Gate. Does the scoping establish, for each of the three death mechanisms, a structural argument (backed by the float measurement for the unmeasured nonlinearity) that it cannot recur in MT, AND produce the transform build cost? yes → ESCALATE to the user with the full packet: the lift condition is satisfied ON PAPER, the lift ruling is the user's alone. Build nothing. no → bank at full strength: MT joins the dead list with the killing mechanism named.

Answer: YES, on the evidence below, with one clause (S3's top test point) explicitly window-limited, reported as such rather than oversold.

2. What "on the real target" means, and why it changes the question

Leg 301 screened structurally, on the bare closed-form MT differentiation matrix (Iserles–Webb eq. 3.3) alone. That establishes l_min = 1, growth +1 for the differentiation operator in isolation. It does not establish anything about what happens once that operator is composed with the real target's coupling: multiplication by S(X) = U + c_l X + c_r (itself built from Uop, a dense antiderivative-of-Hilbert-transform operator, not a simple multiplier), and the V-equation's own H(Ω) coupling. This leg builds that composed operator and measures it.

Method. Solve HL_S2_nonsymmetric for real (solver/bordered_hl.py, unmodified) at n = 801: Newton converges to ‖F‖_∞ = 2.276e-14 in 16 iterations (c_l = 1.5765, c_omega = -0.6205, c_r = 0.4377). Take the converged Jacobian's top-left 2n × 2n block J_sub (drops the 3 border/gauge columns: a finite-rank correction, irrelevant to the asymptotic-in-truncation questions S1/S2 ask). Transport it into MT coefficient space by a quadrature-Galerkin change of basis: L_MT = Ā · J_sub · Φᵀ, with Φ[j,i] = φ_{n_j}(X_i) the MT closed form evaluated on the SAME grid, and Ā = Φ̄ · diag(w) the trapezoid-quadrature analysis operator. This is a discretisation of the true continuous operator's Galerkin matrix, not a hand-derived approximation: it takes the operator this repository already built and verified, and re-expresses it.

A quadrature-resolution failure was caught before it was reported as a finding, and this is itself part of the leg's evidence base. A first pass at grid n = 201 gave σ_min collapsing to 1e-16–1e-17 for truncations M ≥ 32, which would have read as a clean M1 recurrence. A CONTROL (lesson 90: a control that cannot come out differently is not a control) (the analysis/synthesis Gram matrix G = Ā·Φᵀ of the MT basis against the grid's own quadrature, with no operator in it at all) showed the identical collapse, proving the grid, not the operator, was the cause (lesson 86: a bound dominated by its own evaluation error is a statement about the code). A convergence sweep (n = 201…3201) found the requirement is roughly n ≥ 6×(2M+1); n = 801 keeps σ_min(G) ≥ 0.945 through M = 64, so all numbers below are reported only where this control passed (gram_control_trustworthy = True at every row of M_ladder in the JSON).

3. (a): does M1 (zero-diagonal / block coupling, leg 54/62) recur?

No, measured over an 8× truncation range (M = 8, 16, 32, 64).

l_min is exactly flat at 0.47732, the coupling to Uop/H does not erode it, and the growth exponent stays near +1 (0.89–1.00), matching Cadiot's A1_GROWTH. Most decisively: σ_min does not collapse toward zero as M grows (0.0314 → 0.0255 → 0.0292 → 0.0314), leg 301's own pre-committed S1 kill condition (σ_min → 0 with N, "leg 127's verdict recurring") is not triggered. δ stays in [1.22, 1.45] median (≤ 6.77 max) (above BDL's < 1/2, exactly the wall leg 301 flagged) but leg 62's test 14 already refuted δ as the coordinate that decides invertibility; the coordinate that matters, the diagonal's nonzeroness and growth, holds throughout. M1 does not recur, named realization: the real target's 2n×2n linear block, MT-Galerkin-transported at n = 801, M = 8…64.

4. (b): does M2 ((H,D)-consistency defect, leg 56) recur?

No, structurally, and re-verified on this leg's own transcription. D (Iserles–Webb) and the Hilbert diagonal (∓i) are closed-form rational entries with no interpolation step: leg 56's mechanism (a spline-interpolated H and a full-basis D disagreeing at the endpoints) has nothing to attach to. Self-tested here, not merely quoted from 301: D is skew-Hermitian to 0.0 (exact, machine-zero). The Hilbert-diagonal claim was checked against the code's own line_hilbert_matrix (the object every certificate constant in solver/bordered_hl.py actually uses) applied to sampled MT modes: relative disagreement mean 3.63%, max 3.63%, this is the discretised Hilbert transform's own approximation error against the exact continuous eigenvalue, not a defect of the MT construction; the continuous claim is exact by Hardy-space theory (Cayley transform to the disk), independent of any grid. M2 does not recur.

5. (c): does M3 (a=0 exactness / non-transfer, legs 163/176/182) recur, and what does the nonlinearity measure?

M3: does not recur, and not merely by argument: this leg never touched a = 0. Every number above is computed on the real HL_S2_nonsymmetric Newton solution (c_l/c_omega ≠ any CLM value, non-symmetric, converged from generic data). The MT completeness/Hardy-splitting argument was never asked to transfer from a toy profile because no toy profile was used.

The nonlinearity, measured in float on the target (the clause 301 flagged as entirely unmeasured):

Method: v = ε·Re(φ_n) (a real perturbation built from one MT mode), Q(v,v) computed exactly (solver/bordered_hl.py::quadratic, no remainder, F is degree-2), projected onto 64 MT modes via the same quadrature-verified analysis operator (gram σ_min = 0.945).

‖Q(v,v)‖_∞/‖v‖² stays bounded, O(0.34–1.37), across two orders of magnitude of input frequency, no blow-up. The output-bandwidth ratio shrinks from 32× at n = 1 to 3.06× at n = 16, inside the verified window: evidence of increasing, not decreasing, locality at higher input frequency, the opposite of what would kill the algebra property.

The n = 64 row is reported and immediately discounted, not silently kept: its output bandwidth reads exactly 64, coinciding with the projection window's own cap (M_PROJECT_NL = 64, chosen because that is where the Gram control was independently verified trustworthy, not because it matches this test point). Lesson 84 (a known-answer probe has a WINDOW): this row cannot distinguish "genuinely fills the whole window" from "the window is too narrow to see the true edge," and it is not used as evidence for either reading. The n = 16 row, resolved with 49 < 64 well inside the window, is the best-resolved data point and it shows shrinking, bounded spread.

M3 does not recur; the nonlinearity, measured (not argued) on the target, shows bounded amplitude across the resolved range and no sign of the algebra property failing: but only up to M = 64, and the top test point is explicitly unresolved, not a clean pass.

6. (d) (the validated MT transform's build cost

No validated (interval/CAP) MT transform exists anywhere) reconfirmed by this leg's own novelty pass (writeup/novelty/leg_320.md), same conclusion as leg 301's counterweight (b), for a different, load-bearing reason: arXiv:1904.10755 (Shindin–Parumasur–Aluko) supplies a classical (non-interval) convergence/stability theory for MTC collocation on the Benjamin equation, a quadratic nonlinearity coupled to a non-smooth-symbol Hilbert-type multiplier, structurally the same pairing HL_S2_nonsymmetric presents. That paper reduces the approximation-theory tier of the build; it does not supply any rigorous/interval machinery.

What this leg reused at zero build cost: Iserles–Webb's closed-form D; the exact-diagonal Hilbert claim (Hardy-space theory, no computation needed); solver/bordered_hl.py's exact quadratic remainder (F degree-2) and Jacobian, both reused unmodified.

What does not exist and would have to be built, in order, each item blocking the next:

1. A validated (interval) MT forward/inverse transform, nothing published, nothing here. Cost floor: comparable to what realization 1's ℓ¹-Fourier machinery took to reach scoping strength (legs 51–54, four legs) plus the extra step of an interval implementation of a rational, not polynomial, basis (no off-the-shelf interval package does this), ≈ 2–3 leg-equivalents.
2. A rigorous bilinear-form / Z_2 bound in MT coefficients, i.e. an MT-basis analogue of BDL/Cadiot's dominance machinery, informed by but not identical to Shindin–Parumasur–Aluko's float convergence theory (their bound is not rigorous and does not cover this leg's specific bordered system): ≈ 2–3 leg-equivalents, contingent on (1).
3. Porting solver/bordered_hl.py's bordered system (3 gauge unknowns) into MT coordinates and re-deriving Y_0/Z_1/Z_2 in the new norm, analogous to what leg 46/47 (Route-PORT) took for the sup-norm collocation realization, ≈ 1–2 leg-equivalents.
4. A tail-decay proof for the target's own MT coefficients (301 flagged this as target-dependent (geometric for rationally-decaying targets, only algebraic |n|^{-5/4..-9/4} for others) and unmeasured for HL_S2_nonsymmetric specifically) ≈ 1 leg-equivalent, and the item most likely to force a restart of (2)/(3) if it comes back slow.

Floor estimate: 6–9 further legs (~40–65 leg-hours at this repository's own per-leg cadence), with item (1)'s interval-rational-basis implementation as the largest unpriced risk, nothing in the literature bounds how long a first validated rational-basis transform takes, because nobody has built one.

7. Counterweights re-tested, not inherited

• δ ≈ 1 is confirmed, not = 1 exactly, on the real operator (1.22–1.45 median, up to 6.77 max): worse than the bare differentiation matrix's exact 1.0000, because Uop/H add genuine off-diagonal mass. Scored not-a-death for the same reason leg 301 gave (leg 62's test 14): the diagonal stays nonzero and growing, and δ is not the coordinate.
• The nonlinearity is no longer unmeasured: it is measured here, in float, on the real target, and reads bounded, with the caveat in §5 stated plainly rather than smoothed over.
• σ_min was the sharpest test available and it passed (did not collapse) once the quadrature artifact was caught and removed: this is the single measurement most capable of having produced a clean NO, and it did not.

8. What this leg did NOT do

Did not build a certificate. Did not compute Y_0, Z_1, Z_2 for any operator in MT coordinates. Did not lift the ban, only the lift condition's own wording, and the user's ruling on it, can do that. Did not resolve the n = 64 nonlinearity test point (window-limited, reported as such). Did not move a link of the L1 → L4 chain: Clay odds stay ~0.05%. Territory: solver/bordered_hl.py, solver/line_hilbert.py, solver/hl_rescaled.py read and imported unmodified; nothing in solver/ was edited.

9. Provenance

Iserles & Webb, DAMTP NA2019/03 eq. (3.3); Cadiot arXiv:2505.03091 (Assumption 1, as quoted and machine-verified by leg 304); Shindin, Parumasur & Aluko, arXiv:1904.10755. Internal: legs 54, 56, 62, 127, 163, 176, 182, 301, 304, and solver/bordered_hl.py (legs 46/47). Every real-target number in this file is read from writeup/data/p2_route_mtsc_v1.json, produced by experiments/p2_route_mtsc_v1.py; none is re-derived by hand here.