Route-PORT v2: reach makes the truncation gap worse, so the tail lemma is forced

Phase 2 / P2, leg 47. Code: experiments/p2_route_port_v2_reach.py → writeup/data/p2_route_port_v2_reach.json. Working notes: PHASE2_P2_NOTES.md §36. 4.4 s, 3/3 pre-committed clauses.

Leg 46 left one number unmeasured and the whole road depended on it. Measured: extending the domain does not close the truncation gap; it opens it, by half an order of magnitude per unit of reach. A rigorous analytic far-field enclosure is FORCED, not optional. No chain link moved; Clay unchanged at ~0.05%.

1. The question, and why it was open

Leg 46's certificate closes around the truncated object, at 1.55e+08 × the ball radius. I initially called that "not close to affordable" and then corrected myself: the grid is log-radial, so reach costs logarithmically, and if the distance kept falling like leg 46's power law the gap might be a handful of extra grid points rather than a wall.

That correction rested on applying leg 46's X_max^−0.437 law (which was measured for the contraction ratio, not for the weighted distance) to a quantity it was never fitted to. Both readings were guesses. This leg measures the thing itself.

Pre-committed before the run, with both branches actionable so it could not be argued either way afterwards: fit log₁₀(distance / r_max) against ρ_max. Slope ≤ −0.05 → brute force closes, report the cost. Slope > −0.05 → tail lemma forced, and do not propose "just refine" as the next leg.

2. The measurement

Radial resolution held fixed at dρ = 0.02 so the ladder varies reach and nothing else.

d log10(distance) / dρ  =  −0.0196
d log10(r_max)    / dρ  =  −0.4899
d log10(ratio)    / dρ  =  +0.4703      (gate: −0.05)

Both of my guesses were wrong, in opposite directions.

The distance does not fall. It is essentially flat, slope −0.02 per unit ρ, and over the last three rungs it rises, 0.184 → 0.205 → 0.331. That is what an algebraic far field does: every unit of reach exposes more un-resolved tail, and the two truncations keep differing by about the same weighted amount. The X_max^−0.437 law I extrapolated belonged to the contraction ratio and does not transfer.

And the ball shrinks fast. r_max falls −0.49 per unit ρ, a factor of ~70 across the ladder. That is not mysterious: the tuned weight is w_l = 0.01·X_max by construction, so the norm the ball is measured in changes as the domain grows, and Z₂ grows with it.

Net: the ratio rises +0.47 per unit ρ. Every unit of reach costs a factor of ~3 in the wrong direction. The gap at ρ = 10 is 63× worse than at ρ = 6 (4.374e+09 / 6.944e+07 = 62.99; a leg-60 reproduction audit caught this cell quoting the ρ = 8 → 10 factor, 28.16×, instead: the wrong baseline, understating the effect by 2.25×).

3. The verdict

Reach cannot close this at any size. Not expensively, at all. The trend has the wrong sign, so there is no X_max, however large, at which the float ball contains the true object. The pre-committed gate fires on the tail-lemma branch.

What that means concretely. Certifying HL_S2_nonsymmetric on the whole line requires an analytic far-field enclosure: for |X| > X_max, a rigorous bound on the solution from its asymptotic expansion, with the error controlled and folded into the budget, so that the finite-dimensional certificate plus the tail estimate covers ℝ. This is standard apparatus in validated numerics on unbounded domains. It is also real mathematics rather than more compute, and this leg is what promotes it from "one of the things we'd need" to "the thing that decides whether L1 is reachable."

And it re-prices the L1 road, which is the actionable output. Before this leg the two gaps between here and a certified L1 were interval arithmetic (engineering) and truncation (unknown). Now: interval arithmetic (engineering, and solver/interval.py exists as an arithmetic layer that has never been wired to a certificate), and a tail lemma (mathematics, forced, and nobody here has written one).

4. What this does not say

• It does not say the object cannot be certified: Chen–Huang–Li's profile is presumably as certifiable as any, and groups doing validated numerics write tail lemmas routinely.
• It does not invalidate leg 46. The certificate still closes in float at every rung; what changes is that "extend the domain" is now known not to be the way to make it mean something.
• It says nothing about Clay. L1 is a 1D toy, and L2/L3 are occupied by Chen–Hou while L4 is out of reach of interval arithmetic entirely.

5. Reproduce

.venv/bin/python -u experiments/p2_route_port_v2_reach.py   # 4.4 s

Q1 every rung converges or is refused, Q2 distance falls with reach (passes only on the first-to-last comparison: see §2, it is not monotone and the writeup says so), Q3 the verdict is decided. 3/3.