Route-D v13, the turning point: correcting v12, and locating the wall

Status: Level-1 tooling + a correction to the previous leg + an attributed negative. NOT a certificate, NOT rigorous, NOT a Clay result. Plain float64.

Figure: fig31 · Data: writeup/data/p2_route_d_v13_turning.json Code: solver/turning_point.py (+ test_turning_point.py, 6/6), experiments/p2_route_d_v13_turning.py, p2_route_d_v13_evidence.py

0. What this leg is for

v12 found that the a > 0 two-scale profile ends at a finite radius X_c where the effective transport coefficient E = c + aU crosses zero, with an algebraic zero of order 1/a; and it measured that the gauged inverse's graded norm diverges with J at that profile (J^+2.86 at a = 0.2) while it is flat at the a = 0 anchor (J^−0.003). Both stand.

It then attributed the divergence to a homogeneous mode h ~ (X_c − X)^{−1/a}. That attribution is wrong. This leg corrects it, finds where the obstruction actually is, attributes the measured divergence to it by measurement rather than by argument, and disqualifies the cheap repair v12 recommended.

1. The correction

Beyond and around X_c the profile vanishes, so the linearization is

L h = H(Ω) h − E h_X ,    E(X) ≈ a h_c (X − X_c) ,   h_c = H(Ω)(X_c)

with H(Ω) and E fixed functions of the profile. Put s = X_c − X, so h_X = −h_s and E = −a h_c s. The homogeneous equation is

   h_c h − a h_c s h_s = 0   ⇒   h_s/h = 1/(a s)   ⇒   h ~ s^{+1/a}
```: a mode that **vanishes** at `X_c`. v12 wrote `s^{−1/a}`, from a sign dropped in
converting `d/dX` to `d/ds`. Nor is the inhomogeneous problem singular there: with
the integrating factor `s^{−1/a}`,
`h = s^{1/a}[C − (1/(a h_c))∫ g σ^{−1/a−1} dσ] → g(X_c)/h_c` as `s → 0`.
**Nothing blows up at the turning point.**

Measured (fit of the inner mode, integrated inward from `0.8 X_c` on the profile's
own `H(Ω)` and `E`, `J = 400`):

| a | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.50 |
|---|---|---|---|---|---|---|
| fitted `p` | **+5.28** | +4.21 | +3.51 | +3.01 | +2.65 | +2.14 |
| corrected prediction `+1/a` | +5.00 | +4.00 | +3.33 | +2.86 | +2.50 | +2.00 |
| v12's claim `−1/a` | −5.00 | −4.00 | −3.33 | −2.86 | −2.50 | −2.00 |

Positive at every `a`, within 5–7% of `+1/a` (the same systematic overshoot a
leading-order fit over a finite window gives everywhere in this series).

## 2. Where the obstruction actually is

Outside `X_c` the same equation has the same exponent, and now it is a **growing**
mode. For `X ≫ X_c`, with `m = ∫Ω dX < 0`:

H(Ω) ~ m/(πX) , E = c + aU ~ (a m/π) log(X/X_c) [E(X_c) = 0] ⇒ h_X/h ~ 1/(a X log(X/X_c)) ⇒ h ~ ( log(X/X_c) )^{1/a} ```

The domain space of the entire Route-D programme is the decay class |h| ≲ X^{−α}. A growing mode is not in it, and the constant multiplying it is fixed by matching to the inner solve rather than free, so the image of the inverse generically leaves the space. That is a codimension-1 range obstruction of the continuum operator, and no amount of grid refinement touches it. It is a strictly worse situation than the local singularity v12 named, which would at least have been a resolution problem.

Measured with an instrument independent of the matrix inverse, integrate h_X = (H(Ω)/E) h outward from 1.5 X_c on the profile's exact H(Ω) and E, out to X = 1e8:

Agreement is 0.02–0.6%, with no fitted constant anywhere. The quadrature is converged (the exponent moves 1.6e-4 over a 16× refinement of the integration grid).

The row that did not fit, refined rather than dropped. At a = 0.5 the measurement returns q = 0.054 against a prediction of 2. v12's own rate table (T4) already said the collocation profile stops converging around there, so the question is whether the prediction fails or the profile is unresolved:

a = 0.4 is converged to four digits; a = 0.5 moves by a factor of 34 on the first refinement and is still drifting. The outlier is the instrument.

3. So 1/a appears three times, in three roles

One exponent, in one problem:

• the order of the profile's zero at X_c (v12 T3),
• the exponent of the vanishing inner mode of the linearization (§1),
• the power of the logarithm by which the outer mode grows (§2).

All three are forced by the same leading balance Ω H(Ω) = E Ω_X with E vanishing linearly, and none of them has a fitted constant.

4. Attributing v12's divergence

An argument that says where a divergence comes from is not a measurement. The measurement: recompute ‖A‖ with the domain sup restricted to a fixed outer radius, so that the grid's own outer radius (~4J/π, which grows with J) stops being the thing that moves.

The divergence is monotone in the outer radius and nearly gone when the far field is excluded, and the a = 0 control is flat at every cutoff, so the ladder is measuring the profile and not the discretization. The residual J^+0.3…0.5 at X ≤ 20 is real and not attributed by this leg; it is small compared to what the far field contributes, but it is not zero and should not be described as such.

Where the extremal row is sourced is the complementary measurement: the fraction of its mass coming from codomain slots within 10% of X_c is 69→86→92% (a=0.2), 87→93→97% (a=0.4) over J = 200/400/800. So the picture is coherent: the perturbation is sourced at the turning point and does damage in the far field, which is exactly what a growing homogeneous mode excited at X_c does.

5. The cheap repair, disqualified

v12 recommended bordering the system with an extra unknown to supply the missing range direction, and the obvious candidate is the speed c, which the certificate's gauge freezes. It cannot work, and the reason is already in the project's own notes: dilation Ω(X) → Ω(X/μ), c → μc is a symmetry of the zero set at every a, so restoring c supplies a kernel direction, not a range direction. A symmetry cannot discharge a solvability condition.

Measured anyway, because a prediction that is not measured is an opinion:

• the square bordered system [gauge ; DF | dF/dc] at a = 0 has cond = 4.4e18, σ_min = 4.4e-17 (singular to machine precision, which is Route-D v1 Q2's and v11 V0's finding read forward;
• the overdetermined version's norm grows with J even at the anchor (J^+1.40 at a = 0, where the plain system is flat), and at a = 0.2/0.3 it grows J^+1.57/J^+1.86) better than J^+2.86 but still divergent.

6. Ledger, gate-check, honest reading

CORRECTED: v12's stated mechanism for the ‖A‖ divergence. UNCHANGED: the profile ends at X_c with a zero of order 1/a; ‖A‖ diverges with J at the a > 0 profile and is flat at a = 0; Y₀ in the graded codomain norm is not under budget once the operator is priced at the right point. NEW: the obstruction is a codimension-1 range condition from a growing far-field mode, not a local singularity; and the cheap repair is disqualified.

GATE-CHECK (a) which link does this move? L1, and only in the sense of knowing what is wrong. It corrects a published claim and replaces an argument with an attributed measurement. Nothing here is rigorous and nothing climbs the ladder.

(b) is another L1 leg the best use of the next chunk? The repair is now sharper than it was in v12, because the diagnosis changed what it has to do. It is not "border the operator". It is remove the far field from the domain: pose the problem on [0, X_c] with X_c an unknown and perturbations supported there, so that the growing mode has nowhere to live. That is consistent: the residual Ω H(Ω) − E Ω_X vanishes identically outside the support because every term carries a factor of Ω or Ω_X, even though H(Ω) does not vanish there.

(c) a cheaper experiment that kills the route? Unchanged and still the right first move: build the finite-interval system at one a and measure ‖A‖ against J. Flat ⇒ the framing is repaired and eleven legs of far-field machinery are simply not needed. Still divergent ⇒ the framing needs replacing, and the alternative lanes (the coupled-system HL question; writing the P2 arc up) become primary.

HONEST CEILING (unchanged). Plain float64; nothing interval-enclosed; nothing rigorous; a toy model. What v13 adds is that the previous leg's headline number was right and its explanation was not, which is the sort of thing that is only cheap to find if you go looking.