The wall we were measuring was a shadow of the wall that was there

Route-WV, leg 59. Figure: writeup/figures/fig58_weight_repairs_v2.png. Data: writeup/data/p2_weight_repairs_v2.json. Runner: experiments/p2_weight_repairs_v2.py.

The setup, in one paragraph

We have a number that scores a function space. Given a weight (the thing that decides what "small" means for the Newton–Kantorovich argument) the number is log10(Y_0/budget), and it is negative exactly when the certificate closes. Before letting a search loose on it, we froze a six-property viability gate and asked whether the number is safe to optimise at all. It answered FAIL 4/6 (leg 49). Two named repairs moved it to 4/6 with better internals (leg 50): the fraction of weights returning a finite score went 0.775 -> 0.875 against a 0.90 floor, and the worst defect-tracking slope error went 0.366 -> 0.342 against a 0.05 ceiling.

Leg 50 left a note naming the suspect for the first of those: the wall we were measuring is a 1-D slice of a 2-D boundary, so weights were being scored against the wrong geometry. This leg models the wall in both weight factors and re-runs the frozen gate. The gate does not move. Repairing the fitness is allowed; moving the goalposts is not.

What the wall actually is

The weight is a product of two algebraic factors, nu(X) = (1+(X/L)^2)^(p/2) (1+(X/l)^2)^(q/2), plus two scalar weights on the border rows. The old model watched one number: the far-field power p+q. Below a measured crossing, Z_1 >= 1 and no certificate exists at any residual, so the box excluded everything with p+q under that line.

But in this float rehearsal M = I - A·DF is roundoff, and

Z_1 = max_i w_i Σ_j |M_ij| / w_j  ~  ε · κ · (max_i w_i / min_j w_j),

which is governed by the weight's dynamic range, not by its far-field power. Two weights with identical p+q do not share a range as soon as the two factors carry different scales (the weight then dips or peaks inside the domain) and the border weight w_l enters the range too. The admissible set is a level set of the range: a curve in the (p,q) plane. The 1-D model was that curve's intersection with one line.

Panel (a) of the figure is the whole argument. Four hundred in-box weights, red where Z_1 >= 1. The horizontal green line, the 2-D wall, separates the colours. The vertical purple line, the 1-D wall, runs straight through the middle of both.

The calibration is inherited, not refitted. The 2-D wall's one number is the log-range at the same bisected crossing the 1-D wall already measured (p_- = -3.7370 becomes r_crit = 11.6060 decades). Nothing is fitted to this leg's data, so the comparison below is a comparison of geometry, not of tuning.

The known-answer window, and where the old law is simply the wrong branch

Before trusting a new geometry we made it reproduce an answer we already had. Past the analytic wall the true profile's weighted sup norm grows like X_max^(p+q-1); over a 54.6x reach that predicted x7.39 and measured x7.39. The 2-D re-derivation reproduces it to a relative 6.7e-16, and over a five-case battery its worst relative error is 1.9e-10.

The interesting part is the cases where the two laws disagree. Give the two factors opposite-sign powers and the weight develops an interior peak; the supremum sits there rather than at the domain edge, and X_max^(p+q-1) is not merely imprecise, it is the wrong branch of the maximum. At identical p+q = 1.5, the measured growth is x1.00 where the 1-D law says x7.39: off by 7.39x, with a second case off by 4.83x. The far-field power alone does not determine the growth rate. That is the same fact panel (a) shows, seen from the analytic side.

What the 2-D wall buys, and what it does not

On the same 400 weights (123 of which fail):

A 22.7x reduction in misclassification, from a model with no new free constant.

Then the frozen gate, thresholds untouched:

• P2 (finite fraction): 0.875 -> 0.975, against the 0.90 floor. Passes. Exactly one roster weight is left without a score.
• P3 (defect tracking, worst |slope-1|): 0.342 -> 0.342. Unmoved to sixteen digits, against the 0.05 ceiling. Fails.
• P1, P4, P5, P6 unchanged and passing.

Verdict: FAIL, 5/6. The gate's question was "does it pass 6/6". The answer is no.

The part worth more than the pass we did not get

P3's number did not move by a hair, across two rosters that share almost no weights. That is not a coincidence, and it is not the wall's fault. The defect probe gives each weight an upper limit on the perturbation ε (leg 50's own repair, ε <= C/||A||_w) and no lower limit. But Y_0(ε) stops tracking ε once the perturbation falls under the roundoff already sitting in A·F(z*), at

ε_min(w) = Y_0(z*, w) / max_i w_i |d_i|.

So every weight has a probe window, and the fitted slope is a function of that window's width rather than of the weight: Spearman -0.878 over the resolved weights. Five-decade windows fit slope 0.999. Three-decade windows fit 0.83. Two-decade windows fit 0.658: the same 0.658, to five digits, for five different weights. P3 has been measuring the probe's noise floor.

That is a defect in the fitness's definition, not in the box a wall model draws. Which is exactly what the gate's no-branch pre-committed to concluding: Stage B's fitness is dead as parameterized, and a future B proposal has to change what the fitness is, not where its wall sits.

No GA was run, and a pass would not have been a route

plan_of_record.py bans GA compute on an unvalidated fitness on either branch of this gate. None ran here.

And the caveat this leg is required to carry: leg 54 measured the shape of A dead on top of leg 53's split and leg 52's space, so even a 6/6 here would have unblocked a stage whose three degrees of freedom are all separately measured worthless for this operator. A 2-D wall is worth having because it is a measured geometry of a search space. It is not a route to a certificate, and nothing here says anything about Hou–Luo: the substrate is the a=0 CLM linearisation, closed form since 1985, and the arithmetic is float.