The advection scope of the Route-D bound programme: the velocity is logarithmically divergent

Phase-2 P2, Route-D scoping note. Code: solver/advection_scope.py + test_advection_scope.py (6/6). No figure and no committed JSON (every number here is reproduced by running the gates, and a plot of two log-slopes would add nothing a table does not.

Rigor level: 0–1) a scoping measurement, in plain float64. Nothing is interval-enclosed, nothing is rigorous, and this is a statement about which space the later rigorous work has to be done in, not a statement about blow-up.

0. The question nobody had asked

Eleven Route-D legs built a function space, priced it, and bounded constants in it. Every one of them worked at the exact a = 0 anchor, because that is where the known-answer gate lives.

And a = 0 is precisely the value at which the advection term −a U Ω_X is absent.

So the space was chosen, tuned and optimised on the one member of the family where the term it would have to carry does not exist. Whether it can carry that term at all was never checked. It cannot.

1. The velocity is logarithmically divergent

The rescaled velocity is U(X) = ∫₀ˣ H(Ω) dX'. Route-D v3 (§S6) already established the far-field law H(Ω)(X) → (∫Ω)/(πX), so integrating it,

U(X) → (M/π) log X ,      M := ∫ Ω dX .                              (U)

There is no cancellation available: M ≠ 0 for every profile in this family, the a = 0 anchor itself has ∫ −1/(1+X²) = −π, giving M/π → −1.

Measured against an independently integrated mass:

A subtlety that had to be got right. dU/dX = H(Ω) exactly, so dU/d log X = X·H(Ω)(X) is an identity; the content of (U) is that it tends to M/π. The right predictor is therefore the mass over the fit window, not over the whole truncated domain, on the sinh grid the outermost decade is coarse (the same far-field under-resolution v11 found), and at a = 0.5 that moves the full-domain mass by 15% while the windowed mass and the fitted slope agree to 0.3%. Both are returned by velocity_log_rate so the discrepancy stays visible.

2. What that does to the operator

The linearization carries the advection term as two pieces,

dR₂/dΩ · h  ⊃  −a [ (V H h)·Ω_X  +  U·h_X ] ,

and in the decay-graded codomain ‖g‖_Y = sup (1+X²)^{(α+1)/2}|g| that v3–v11 use, they behave completely differently:

• (V H h)·Ω_X ~ (log X)·X⁻³ → weighted, X^{α−2} log X, which decays for α < 2 (the whole working range). Harmless.
• U·h_X ~ (log X)·X^{−α−1} → weighted, (a|M|α/π)·log X, which diverges. For every a ≠ 0.

Measured at α = 1.4 (the operating point of v8–v10), the transport piece's log rate is +0.317 against a predicted +0.318 at a = 0.3, and +0.480 at a = 0.5. At a = 0 both pieces are identically zero.

So DF does not map the domain class into the codomain class for any a ≠ 0, and neither does the residual: Y₀ is infinite in that norm too. The eleven-leg bound programme is a = 0-only. That had never been stated.

3. Which half of this measurement is real

The stretch piece carries Ω_X, and for a ≠ 0 the profile has collapsed to the far-field discretization noise floor by X ~ 10. A log-rate fit out there measures amplified noise, not the operator.

The discriminator is reproducibility, not magnitude:

The transport piece is built from U (fixed by the profile's core mass) and an analytic test function, so it repeats under refinement. The stretch piece does not. Only the reproducible half is quoted as a result: the gates enforce that, and a first version of this note that quoted both would have been reporting noise at a = 0.5.

4. The fix is a grading, and the project already has it

The two-scale (traveling-wave) residual balances Ω H(Ω) against c Ω_X, whose far field is X^{−α−1}; that is why its codomain carries α+1. The one-scale (self-similar) residual balances against c_l X Ω_X ~ X^{−α} instead, so its natural codomain grading is α, one power weaker. One power is exactly what the log needs: X^α · (log X) X^{−α−1} = (log X)/X → 0.

Measured on the same profile, same h, same grid:

So the constructive reading is not "a ≠ 0 is out of reach" but "a ≠ 0 needs the one-scale formulation", which this project already has a validated solver for.

5. A crossing found on the way, and what it turned out to be

Because U is log-divergent and negative, the effective speed c_eff = c + aU(X) changes sign at a finite, grid-independent radius: X* = 7.16 at a = 0.3 and 3.10 at a = 0.5, each stable to ~1% across three grids, with no crossing at a = 0.

This note originally read that as a stagnation point: the far field of an a ≠ 0 "traveling wave" moving opposite to its core. A parallel line of work (solver/finite_support.py, Route-D v12/v13) reads the same crossing as something sharper and better: it is the EDGE OF SUPPORT X_c, beyond which the profile is identically zero with an algebraic zero of order 1/a. That reading is the one to carry forward, and it retro-explains §3's noise floor: the Newton profile collapses to ~10⁻⁹ by X ~ 10 while X* = 7.16. There is no tail out there. There is numerical dust past the end of the profile.

The gates here depend only on the reproducible half, so they stand under either reading.

6. Where this sits relative to Clay

It moves no link of the L1→L4 chain. It narrows the scope of an L1 sub-programme: the bound machinery eleven legs built is now known to be a = 0-only, and the repair is named. If anything it is a cost finding, work that looked general was specific, and its value is that it was found before more estimate legs were built on the assumption.

7. Reproduce

.venv/bin/python test_advection_scope.py     # 6/6, ~4 min

Gates: the log law against an independently integrated windowed mass; the transport/stretch split with an a = 0 control where both vanish identically; the one-scale fix; the reproducibility discriminator; the grid-independence of X*; and an input-sanity gate that the d/dX matrix is not double-counted, which it was, once, costing exactly one power of X and hiding the entire effect.