Reply to `hunts/higher_xi/CROSS-ARM-TRANSFER.md`

Verdict: the discrepancy resolves, your measurements reproduce exactly, and the transfer does not survive — for a reason that is not in your data, because it is outside your scan window. You asked to be distrusted first and named the discrepancy to chase. Chasing it is what found this, so the instruction did its job.

1. The four-order discrepancy — resolved, both numbers correct

Your 2-pair shallow budget reproduces here to six figures: 9.417199e-05 at (d = 6.640 grid, y1 = y2 = 0.01) against your 0.0000942. The two numbers are not comparable, and the reason is sharper than "different regimes":

At shallow depth the cap is exactly zero. Measured at d = 6.640:

So the shallow budget being four orders smaller costs nothing: the damage it has to pay for is identically zero there, and the relative margin is 1. My recorded worst (0.2907) is a deep budget, where the cap actually bites. Your number measures the erosion of the budget; the verdict consumes budget - cap. Both are right; only the second one binds.

This also confirms your reading of the depth-uniformity argument: the shallow end is safe, and depth_uniform.py closes it by homogeneity rather than by cells for exactly this reason (the relative margin there is +0.559, the best on the whole depth cover; the binding cells are the three deepest, at +0.173).

2. Your k-monotonicity is real — and it reverses outside your window

At your spacing it is exactly as you report. Relative margin at y = 0.49, d = 6.640 grid = 1.0568 mean gaps:

k=1: 0.3825 k=2: 0.5031 k=3: 0.6538 k=4: 0.7809 k=6: 0.9062

Rising, and k = 1 is the worst — stronger than your k = 2 claim. Adding pairs helps, as you said.

Now the same measurement across spacings (y = 0.49, relative margin):

The k-dependence changes sign with spacing. Below about 1.2 mean gaps it rises (your regime, and there k = 1 binds). At and beyond 2 mean gaps it falls monotonically, and keeps falling — 0.2935 at k = 6 and still decreasing.

3. Why your data could not have shown this

Your scan was d in [5.5, 7.5] grid units, which is [0.875, 1.194] mean gaps. The binding family sits at 2.002 mean gaps = 12.579 grid units — a factor of ~1.7 outside the top of your window. Your grid was 25x25x201 and dense; it simply did not extend far enough in d to reach the region where adding pairs hurts.

That region is not incidental. It is where this arm's own instruments have independently landed all day: cluster_universal.py's rho map has its argmax at (2.002 mean gaps, y -> 1/2) with sup rho = 0.9286, and truncation_bridge.py's finite-size ladder degrades with m at exactly that spacing. Three separate instruments, same address.

4. What this does and does not kill

Killed: the reduction to a compact 3-parameter problem in (d, y1, y2). The margin is not minimised at k = 2, so the configuration space cannot be truncated at two pairs.

Not killed, and worth keeping:

• Your shallow-limit ratio sum_slack/|pair_term| = 4.16, flat in y, is a genuine number on the homogeneity argument's limit, and it agrees with the independent constant slack/y^2 -> 8 L2/A = 0.6199944 that depth_uniform.py derives. Two routes, one limit.
• Your observation that only the nearest-neighbour gap contributes negatively, with every longer gap net positive, is correct and is the right way to see why the budget rises monotonically in k.
• The three-arm convergence you name — comparison against a fixed nonnegative autocorrelation kernel — holds. This arm's obligation reduced to exactly that shape (E[F_on + F_p] >= theta E[F_on] + (1-theta) n + 4k against c2 = phi^2 * phi^2), and the pair half of it is now kernel-checked.

A correction you will want: this arm's PairEnergy.lean result, which I described earlier today as research-grade, is prior art - a corollary of the source paper's Lemma 3.1 and Lemma 3.3, with the exact numerical specialisation printed in its section 7.5(a). If the higher-xi arm is leaning on a similar Gram-positivity bound, check the same place before claiming novelty. See NOVELTY-CHECK.md.

5. Suggested next probe, if you want one

The question your method is well placed to answer, and this arm has not: does the falling branch at d >~ 2 mean gaps have a positive limit in k, or does it cross zero? adversary_evolution.py fits it as converging to +0.0212 (with a log term, residual 9.5e-05) with no crossing, but that is a fit, not a bound. Your subset-local charging machinery may bound the tail directly. If it does, this arm's last quantifier closes.