TECHNICAL, Route-WV v2 (leg 59): the conditioning wall in both weight factors

Runner experiments/p2_weight_repairs_v2.py; evidence experiments/p2_weight_repairs_v2_evidence.py; data writeup/data/p2_weight_repairs_v2.json; figure writeup/figures/fig58_weight_repairs_v2.png (fig58); gates (10)-(12) in test_weight_search.py. Every number below is in the JSON and is re-derived from it by the evidence script (18 checks, all passing).

1. The object

solver/weight_search.py scores a weight by fitness(θ) = log10(Y_0/budget) with budget = (1-Z_1)^2/(2 Z_2), on BorderedCLM: the a=0 CLM steady system bordered with (c_l, c_ω) as implicit unknowns, whose exact profile Ω_0(X) = -4X/(1+4X^2) is closed form. The genome is

θ = (p, log10 L, q, log10 l, log10(w_l/X_max)),
ν(X) = (1+(X/L)^2)^(p/2) (1+(X/l)^2)^(q/2),  w_ω = 1 (gauge-fixed),

and the norm is the weighted sup norm on z = (Ω, c_l, c_ω), same weight on domain and codomain.

2. What was frozen and what changed

FROZEN, untouched: all six thresholds (P1_SPREAD_MIN 1.0, P2_FINITE_FRAC 0.90, P3_MONO_VIOLATIONS 0 and slope-error 0.05, P4_RANK_RHO_MIN 0.90 / P4_TOP3_OVERLAP 2, P5_BAND_MIN 2.0, P6_INTERIOR_FRAC 0.05), the roster construction (2 controls + 6 degenerates + 32 random, seed 0), the resolutions (n = 201 coarse / 401 fine), the search settings (per_gene=9, refine=4), the defect grid DEFECT_EPS_DEFAULT, DEFECT_WINDOW_C = 0.1, DEFECT_MIN_WINDOW = 3.

CHANGED: only the MODEL of the conditioning wall the box carries. six_property_gate(..., wall_model=...) defaults to "1d" and reproduces leg 50 exactly; "2d" swaps the half-plane in p+q for the level set of the weight's log-range. in_box, FitnessEngine and roster take an opt-in wall2d argument whose default None is the previous behaviour (gate (12) asserts the opt-in).

3. The mechanism, and why the boundary cannot be 1-D

A = DF^-1 in float, so M = I - A DF is roundoff and

Z_1 = max_i w_i Σ_j |M_ij|/w_j  ≲  ‖M‖_max · (max_i w_i / min_j w_j),

i.e. the conditioning wall is a statement about the weight vector's dynamic range

R(θ) = log10( max_i w_i / min_i w_i ),   w = (ν(X_1..X_n), w_l, w_ω).

R is gauge invariant, like the fitness. For the two-factor family, log10 ν is stationary in X^2 where p/(X^2+L^2) + q/(X^2+l^2) = 0, i.e. at

X_ext^2 = -(p l^2 + q L^2)/(p+q),

so R is the spread of four candidates: log10 ν(0) = 0, log10 ν(X_max), log10 ν(X_ext) when that root is real and inside the domain, and the border entries log10 w_l, log10 w_ω = 0. This is log_range_analytic; it agrees with the range the grid actually carries (weight_log_range) to 0.057 decades over 300 in-box weights, the residual being grid discreteness at the interior stationary point (gate (10) asserts < 0.15).

Two weights with the same p+q therefore have different R whenever L ≠ l (the interior extremum exists precisely when p and q have opposite signs) so the Z_1 = 1 boundary is a level set of R, a curve in the (p,q) plane, and p+q is one of its coordinates and not the boundary.

4. The wall's calibration is inherited

two_factor_wall(engine) runs lower_wall(engine) (the SAME bisection, unchanged) and reads the log-range at its crossing:

p_- = -3.7370  ->  r_crit = 11.6060 decades   (n = 201)
fine grid:              r_crit =  9.2954 decades   (n = 401)

No constant is fitted to the 2-D data (WALL_RANGE_DELTA = 0; the standoff already lives in the 1-D wall's own WALL_LOWER_DELTA = 0.05). On the measuring slice the two models agree by construction, which gate (12) checks explicitly. The fine grid's smaller r_crit is the same shrinking-band effect gate (7) already banks: refinement raises p_- and tightens the admissible range.

5. Measurement A: the growth law re-derived in the 2-D geometry (known answer)

With Ω_0 ~ -1/X, g(X) = ν(X)|Ω_0(X)| has

d log g / d log X = p r_L/(1+r_L) + q r_l/(1+r_l) + (1-4X^2)/(1+4X^2),
r_L = (X/L)^2, r_l = (X/l)^2,

and sup g is attained at a root of that explicit equation or at X = X_max (weighted_sup_analytic, a bisected root find on a closed form, not a max of sampled data). The 1-D reading, "the norm grows like X_max^(p+q-1)", is the edge branch alone.

Growth of sup ν|Ω_0| over a reach of 54.598x (X_max 13.65 -> 745.24), five cases all at p+q = 1.5:

The pre-committed window is the banked 1-D answer (x7.39 predicted, x7.39 measured, capabilities.py); row 1 meets it to 6.7e-16. Worst 2-D error over the battery 1.9e-10. The last two rows are the point: at identical far-field power the sup sits at an interior peak, the edge branch is the wrong branch, and the 1-D law is off by up to 7.39x. Gate (11) asserts both halves.

6. Measurement B: what each model explains of the Z_1 >= 1 failure set

400 weights drawn inside the box (analytic upper wall only), seed 11, n = 201. 123 have Z_1 >= 1.

Ratio of misclassification 22.67x, with no new free constant. Gate (12) asserts the 2-D model both misclassifies strictly less and at least halves the failure rate among admitted weights.

7. Measurement C (the frozen gate, re-run

Verdict FAIL, 5/6 (leg 49: 4/6, leg 50: 4/6). P2's residual is a single roster weight, rand_10, still without a finite score: one of the 0.72% the 2-D wall admits and should not have.

P3's max_slope_error is 0.3421493449940881 here and 0.3421493449940881 at leg 50: identical to sixteen digits across two rosters that share only the eight non-random weights. That is the tell that P3 is not measuring any property of the weights the wall model selects.

8. Measurement D: what carries P3 (diagnosis, NOT folded into the gate)

defect_ladder bounds ε above (ε <= C/‖A‖_w, leg 50's P3 repair) and never below. But Y_0(ε) = max_i w_i |A F(z*+ε d)|_i stops tracking ε once the perturbation falls under the roundoff already in A F(z*), at

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

so each weight's probe has a window [ε_min, ε_max] and the decade grid puts points below its floor. Over the 21 resolved weights:

• Spearman(window width in decades, |slope-1|) = -0.878.
• Worst |slope-1| = 0.3421 at a 2.10-decade window.
• Windows ≈ 5.2 decades fit slope 0.999; ≈ 3.1-3.5 decades fit 0.825-0.841; ≈ 1.8-2.2 decades fit 0.658: the same 0.658 to five digits for five distinct weights.
• The pass/fail populations overlap only between 3.53 decades (narrowest window meeting P3) and 3.88 decades (widest failing it).

An independent ladder on rand_12 shows the same thing directly: local slope stays within 0.976-1.025 from ε = 1e-6 down to 1e-9, then collapses below 3e-10, while that weight's admitted window is ε <= 2.86e-9, two of its three grid points sit under the floor.

This is stated as a diagnosis and is deliberately NOT applied to the gate. Applying it would be moving a goalpost mid-leg; the gate is frozen and its answer stands.

9. Gate answer

"With a 2-D wall model, does the FROZEN six-property viability gate pass 6/6 (in particular P2 >= 0.90 and P3 max |slope-1| <= 0.05?"

NO) 5/6. P2 passes at 0.975 (floor 0.90). P3 fails at 0.342 (ceiling 0.05) with 8 monotonicity violations, unmoved by the wall model. Per the pre-committed no-branch: Stage B's fitness is dead as parameterized, and any future B proposal must change the fitness's DEFINITION (sec 8 says exactly where, the probe's missing noise floor) and not its wall model. No GA compute was run, which the ban requires on either branch.

10. Ceiling

No link of the L1->L4 chain moved. The substrate is the a=0 CLM linearisation, closed form since CLM 1985, and every constant here is float: Z_1 measures conditioning, not a truncation tail, so this is a rehearsal of a certificate and not a certificate. Nothing here is a statement about HL_S2_nonsymmetric.

Carried from the leg's brief, and load-bearing for how the result is read: 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. Passing the gate would have been worth knowing; it would not have been a route.

11. Novelty

writeup/novelty/leg_59.md, run and committed before construction. Chen–Hou arXiv:2210.07191 sec 5.3.3 uses this same two-factor weight family and picks it by hand, publishing no admissible set; arXiv:2203.02404 carries a single decay parameter; arXiv:1702.07421 states the "which weight is least conservative" problem for contraction metrics of DAEs, not for a Newton–Kantorovich norm. No source states the admissible weight-parameter set of a radii-polynomial argument as a measured region in any dimension. The claim here is correspondingly narrow: it is a MEASUREMENT of one search space's geometry on one known-answer object in float, not a theorem.