Route-C-PILOT v0: the certificate-weight fitness, measured on an object whose answer is known

Phase 2, leg 49. Stage C-PILOT of the plan of record: evolve the Lyapunov weight, on an object with a KNOWN answer, with the six-property viability gate re-run on the new fitness before any GA compute. Code: solver/weight_search.py, test_weight_search.py (8/8), experiments/p2_route_c_pilot_v0.py → writeup/data/p2_route_c_pilot_v0.json → fig44. Deterministic, 70 s.

VERDICT: the gate says FAIL, 4 of 6, and the GA was not run. The two failures are specific and priced. On the way to them the leg reproduced the premise the stage was built on, found where that premise actually lives, and measured a second wall on the search space that nobody had looked for.

0. What was pre-committed, and what it cost

plan_of_record.py carried the gate before the leg started:

GATE: does the new fitness pass the six-property viability gate? YES → proceed to stage B. NO → STOP. Do not run the GA.

plus a standing ban, any GA compute on an unvalidated fitness, whose lifting condition is this stage. The six thresholds are frozen constants at the top of solver/weight_search.py, written before the gate was run. The gate returned FAIL, so C0-5's search is a deterministic grid (295,245 evaluations, 11 s) and not the GA. That is the ban working, not a shortcut: the grid was needed for property 6 anyway, and it makes the search reproducible to the last digit.

1. The novelty check came first, and it narrows the claim

Leg 48 spent one leg on a novelty check and correctly closed an entire stage. The same habit here, fourteen arXiv queries, ledger in solver/weight_search.PRECEDENTS, verdict computed by novelty_verdict() rather than remembered:

Verdict PROCEED_NARROW. Nothing found searches the norm of a radii-polynomial certificate. But the idea of searching a certificate is not new, and this leg claims no originality for it, only for its object. Chen–Hou §5.3.3 is worth quoting because it is the human version of this stage, written by the people who do it best:

"We adjust the parameters in φ₁ so that we have a good damping factor d₁(x) from the local term for ω₁. Then we can estimate the nonlocal terms and the constants … we choose the exponents of different powers in φ₂ and adjust the parameters so that we have better stability factors."

2. The substrate: the plan's named object cannot supply the fitness

The plan named Chen–Hou's certified 2D Boussinesq profile as the known-answer substrate. It cannot be one, and this project's own record says why: the 2D relaxation limit-cycles and its residual grows under refinement (Route-K, §32), so there is no fixed profile to take a defect of, and port_certification.radii_polynomial_status returns BLOCKED_AT_STEP_ONE by design. A fitness whose Y₀ does not exist cannot be validated at all.

Substituted: the a = 0 CLM profile, which carries four known answers where the 2D object would have carried one.

K1: the exact profile is closed form. Ω₀(X) = −4X/(1+4X²), H Ω₀ = 2/(1+4X²) (CLM 1985; HQW25 arXiv:2401.14615). It nulls the continuum residual identically, so every defect measured here is ours.

K2: two knobs, two ladders, and both have the right sign.

Refining the spacing converges the profile; refining the reach converges the recovered constant, as 1/X_max (7.4× of reach buys 7.4× of accuracy, twice over). These are different knobs and they fix different things: the truncated Hilbert transform is what holds c_ω back, and no amount of n touches it. c_l comes out at 1.0000018 (n = 401) without ever being told.

K3: an analytic wall. Ω₀ ~ −1/X, so a weight ν ~ |X|^(p+q) gives the true profile an infinite weighted sup norm as soon as p+q > 1. p* = 1 exactly, derived, not fitted. Gated: past the wall the norm grows with reach as X_max^(p−1), measured ×7.39 against a predicted ×7.39 over a 54.6× reach.

K4: an exact invariance of the fitness itself. Scale every weight by one constant s: Y₀ → sY₀, Z₁ unchanged, Z₂ → Z₂/s, so budget → s·budget and the ratio is invariant. Verified to 4.4e−16. The overall scale of the norm is a gauge, not a search direction: a searcher cannot win by making the norm big, and w_ω = 1 is fixed rather than searched because of it.

3. The premise reproduces, and the gauge is what carries it

Leg 46 reported 5186× in Y₀/budget between a naive weight (w_l = X_max) and a hand-tuned one (w_l = 0.01 X_max) on the HL object, and called it "closure is a property of the space, not of the object". That single table is the empirical case for this whole stage, so it was pre-committed as a claim to be tested rather than assumed.

On the known-answer object, the same one-constant change is worth 5604×: naive +1.268 (does not close), hand-tuned −2.480 (closes). The premise is not a one-object accident.

But the ablation says where it lives, and it is not the weight family. Run the identical weights on the same equation with the gauge changed (c_l pinned directly by a border row instead of coming out implicitly) and the gain collapses to 0.56×: the "tuned" weight is now slightly worse.

The mechanism is visible in the constants. Pinning c_l makes that row of A = DF⁻¹ a unit vector, so the weighted operator norm never sees it; leaving c_l implicit puts ‖A‖_w = 1.69e+08 at the naive weight against 1.69e+06 at the tuned one, a clean factor of 100, and Z₂ follows it. The weight is not tuning a function space here; it is preconditioning the border rows. That is worth knowing before spending a stage evolving weight families, and it is exactly the kind of thing a known-answer object with a switchable gauge can tell you and a certified black box cannot.

4. The gate

Frozen predicate, n = 201 against n = 401, 40-weight roster (2 controls, 6 designed degeneracies, 32 random inside the box).

VERDICT FAIL. The GA was not run.

4a. P3: the failure is partly the probe's, and saying so is the point

The probe pushes the converged state off the solution by ε in a fixed direction. Since F(z*) ≈ 0, A F(z*+εd) = εd + O(ε²), so Y₀ is linear in ε and the fitness must fall with slope exactly 1 per decade: a known answer, not merely a known direction.

It does not, at ε = 10⁻²…10⁻⁶. The diagnosis is ‖A‖:

The linear regime does not begin until ε ≲ 1/‖A‖ = 5.9e−07. Inside the corrected window ε ∈ [10⁻⁹, 10⁻⁶] the known answer comes back: 0 monotonicity violations, median |slope−1| = 0.0018, worst 0.092 over 20 finite weights.

So P3 splits: monotonicity passes, the slope-1 known answer passes at the median and fails at the worst weight against the frozen 5% threshold. The frozen predicate is reported as it stands, FAIL, because retuning a threshold after seeing it miss is the move this project keeps a plan-of-record to prevent. What the corrected window buys is not a pass; it is a number for how well the fitness tracks a defect: to 0.2% typically and 9% at worst, which is the resolution at which two weights can honestly be compared.

4b. P2: the censoring is real, and it has a mechanism

Nine of 40 roster weights return no fitness. Every one of them has Z₁ ≥ 1: A stops being an approximate inverse in that norm, so there is no budget at any residual. All nine have far-field power p+q ≤ −2.41; the converse does not hold (a finite weight exists at −4.04), because the two scales L, l decide where the decay begins. And no censored point lies within 10% of the box near the optimum (0 of 400 sampled).

The censoring is therefore informative rather than pathological, but the frozen threshold was 0.90 and the measurement is 0.775, and a fitness that is undefined on 22% of its own box is not one to hand an optimizer without carrying that boundary explicitly.

4c. What P4 passes, and what it does not cover

P4 as written asks whether the ranking survives refinement. It does, comfortably: Spearman 0.995 between n = 201 and n = 401. The level does not, and the gate never asked it to: see §5, which is the leg's most consequential number.

4d. The substantive diagnostic

Gate 4's lesson was that a 6/6 can still be a false pass and that what catches it is a correlation nobody gated. Run here: the best single-gene predictor of the roster fitness explains R² = 0.163 (log₁₀ w_l/X_max), and the far-field power p+q explains 0.211. No single coordinate organizes this landscape, which is the opposite of the ν_crit failure, where max|ω₀| explained 75%. This landscape is genuinely five-dimensional.

5. Two walls, and only one of them is mathematics

The analytic wall p+q ≤ 1 was known before the run. Whether anything else bounds the search space was a measurement, bisect the far-field power on the single-power slice ν = (1+X²)^(p/2) and find where Z₁ crosses 1:

The admissible band is (p₋, 1], and it shuts. At n = 3201 there is no weight at all: Z₁ > 1 everywhere on the slice, and the Newton solve itself has degraded to a residual of 1.3e−10. The lower wall is not a property of the equation: it is the float rehearsal's conditioning, κ(DF)·ε_mach, and it climbs until it meets the analytic wall from below.

The certificate on this object closes only at the coarsest grid. That is the honest reading of row 1 against rows 2–5, and it re-prices two things at once. It says the 5604× of §3 is measured where Z₁ is 1e−09 and therefore where the weight is fighting Y₀ and Z₂ alone. And it puts a number on why stage L1 needs interval arithmetic rather than more care: the float stand-in for Z₁ fails at a measurable resolution, and on this object that resolution is ≈ 3.2e+03.

6. The search, and what it actually bought

Deterministic grid, five genes, 295,245 evaluations. ν(X) = (1+(X/L)²)^(p/2) (1+(X/l)²)^(q/2), the discrete cousin of Chen–Hou's "different powers".

θ* = (p, log₁₀L, q, log₁₀l, log₁₀(w_l/X_max)) = (−1.788, −0.750, 2.711, −0.274, −2.878): 48,270× over the naive weight and 8.61× over the hand-tuned one, interior in every gene (margin 0.161), and with far-field power 0.923, inside the analytic wall. Removing the wall entirely moves the optimum by 1.5e−04 decades: the wall never binds on this object, which is worth recording because it was the one constraint the plan told this stage to respect.

Where the 8.61× comes from is not where a reader would guess. The searched weight's budget is barely better than the hand-tuned one's (1.19e−08 against 1.03e−08); B is 4.3× worse. The entire win is Y₀, 7.45× smaller, bought by shrinking ‖A‖_w a further 5× with w_l ≈ 0.99: a factor 750 below X_max, where the hand had stopped at 100. The hand was searching the right direction and stopped early.

7. What this does and does not mean

Does: the fitness is one number, it is gauge-invariant, it tracks a known defect to 0.2%, its landscape is genuinely five-dimensional, and a deterministic search beats the best hand-picked weight this project has by 8.6× on an object whose answer is known.

Does not: pass its own viability gate. Two of six properties fail on the frozen predicate, and per the plan of record the GA does not run and stage B does not start. The repairs are named and neither is research: carry the measured lower wall in the box as the analytic one already is (P2), and state the fitness's defect-tracking accuracy as a resolution rather than assuming it is exact (P3).

And the ceiling. No link of the L1→L4 chain moved. The object is CLM, in closed form since 1985; nothing here is certified, and no novelty is claimed for the idea of searching a certificate. What the leg produced is a measurement about a method, taken where the answer was checkable, which is what a pilot is for, including when it says no.