R-6F0F63: the ceiling of the Delsarte LP for kissing numbers

Everything below is float grade (rung 1, "measured"). No interval arithmetic is used anywhere in probe.py, so no statement here climbs past measured except where it reproduces a value that is a theorem elsewhere. Nothing here is evidence for or against RH (docs/08).

Reproduce with .venv/bin/python hunts/r_6f0f63/probe.py (7 s, numpy + scipy HiGHS only). Raw output: results.json.

Labels: VERIFIED = checked here against an independent oracle; MEASURED = computed here, no external check; INFERRED = read off the measurements, not directly computed.

0. The planted-fault ladder, which fires first

A verifier that cannot fail is not a verifier. Five items, all as expected (results.json → planted_faults):

Fault 4 exists because the interval scan is not the whole verifier: a certificate can be inadmissible with f(t) ≤ 0 everywhere, and only the coefficient sign test sees it.

1. Reproduction, field for field

The two exactly tight cases were rebuilt from their contact structure, not from printed coefficients: the optimal polynomial vanishes to order 2 at each inner product realised by the configuration and to order 1 at the two endpoints of [-1, 1/2].

• E8 contacts {-1, -1/2, 0, 1/2} → f(t) = (t+1)(t+1/2)² t² (t-1/2), degree 6. Gegenbauer coefficients, normalised to f_0 = 1: [1, 8, 25, 52, 66.5, 60, 27.5], all non-negative, sum 240 exactly. VERIFIED (sum matches τ₈ = 240 to 4e-14 relative).
• Leech contacts {-1, -1/2, -1/4, 0, 1/4, 1/2} → f(t) = (t+1)(t+1/2)²(t+1/4)² t² (t-1/4)²(t-1/2), degree 10, all coefficients non-negative, sum 196560 exactly. VERIFIED (1.7e-11 relative).

That both come out with non-negative Gegenbauer coefficients without being asked to is the reproduction: the structure, not the arithmetic, is what is being checked.

2. The soundness read: the acceptance step is not sound as usually run

This is the load-bearing finding.

The standard way to compute this bound replaces the semi-infinite condition (P3) f ≤ 0 on [-1, 1/2] with f ≤ 0 on a finite node set. That is a relaxation: it enlarges the feasible set, so the LP optimum on a node set can sit below the true LP value, and a number below the true LP value is not a bound at all.

Measured, dimension 24, degree 10 (node_sweep_n24_d10). τ₂₄ = 196560:

MEASURED: at every node count tested, up to 20000 nodes in dimension 24 and 40000 in dimension 8, the node-discretised LP optimum is strictly below the true kissing number. At m = 600, a node count nobody would call coarse, the LP reports 196505.76 for a quantity that is exactly 196560. Published as a bound that number is false by 54, and it is false in the direction that looks like an improvement. The same happens in dimension 8: m = 600 reports 239.9930 for a quantity that is exactly 240.

INFERRED: the discretisation error is one-sided and decays like m⁻². The sup of f over the interval falls by a factor of ≈ 4 for every doubling of m (1.5e-3 → 3.8e-4 → 9.4e-5 → 2.2e-5 at m = 600, 1200, 2400, 5000). One-sided, because the LP always exploits the gaps between nodes.

The constant that encodes the target rather than the mathematics is the node set. Move the target and acceptance moves with it: the interval [-1, 1/2] comes from the 60° angle, and the node placement on it is a free choice made by whoever ran the optimisation. Nothing in the LP output records it.

The repair. A failed verification is not fatal, it is a worse number. If M = sup f > 0, then f - M satisfies (P3) exactly, leaves every f_k for k ≥ 1 untouched, and gives the valid value (f(1) - M)/(f_0 - M) whenever f_0 > M. Every "repaired value" above is that number, and it converges to 196560 from above as m grows, which is what a sound acceptance step must do. The true LP value is therefore bracketed: grid optimum ≤ truth ≤ repaired value.

3. The ceiling of the parameterisation

Held fixed: the method (Delsarte LP, Gegenbauer basis, f_0 = 1). Swept: the degree d, 1…30, at m = 600 Chebyshev-clustered nodes. Reported: the smallest degree within 1e-6 relative of the best value the sweep reached, and the bracket at that degree.

All MEASURED. Dimensions 8 and 24 are additionally VERIFIED: the bracket contains 240 and 196560, the values the LP attains exactly, and the ceiling degrees the sweep finds, 6 and 10, are exactly the degrees of the two certificates rebuilt in §1, the sweep discovers the published parameterisation rather than being handed it.

The headline of §3: degree is not the binding parameter. For every dimension from 3 to 24 the LP value stops moving by degree 14 at the latest, and adding degree up to 30 buys nothing measurable. The parameterisation's ceiling in d is reached early and cheaply. What binds is the node count, §2, and the node count is the free parameter the literature does not report, because it is an implementation detail of an optimisation whose output is a single number.

So the answer to "what could this parameterisation achieve that the published run did not" is, for this method, nothing in d. That is a negative ceiling result and it is the useful kind: it says the headroom in the Delsarte LP is not in the polynomial degree, so anyone looking for headroom must change the method (more constraints, i.e. the SDP/Bachoc-Vallentin side), not tune this one.

4. What this procedure cannot decide here

Stated plainly, per item 4 of issue #110:

1. The SDP half was not touched. Cohn-Elkies sphere packing in R^n and the three-point Bachoc-Vallentin SDP for kissing numbers are the other half of the family the issue names, and neither was attempted at this budget. Nothing here bounds their ceiling.
2. No claim of novelty, and no new bound. The values in §3 are the literature's; the point of computing them was the bracket and the ceiling degree, not the number.
3. Float grade throughout. The sup in §2 comes from Chebyshev interpolation and companion-matrix roots in double precision, not from ball arithmetic. A rung-2 version of this hunt would recompute the sup with zeta/rigor.py's enclosures and rationalise the coefficients; that would turn the repaired value into a genuinely enclosure-carrying bound, and it was not done.
4. The published node counts are unknown to this run. §2 shows that a discretised LP under-reports; it does not show that any published table under-reports, because this run did not read what node sets those runs used or whether they verified (P3) globally afterwards. The soundness finding is about the procedure as usually implemented, not an accusation against a specific table.
5. Dimension 3 at degree 27 is the one row where the sweep's saturation degree is high, and this run did not establish whether that is real structure or the 1e-6 saturation tolerance chasing float noise.

Loose threads

• The m⁻² law, and the node placement that would beat it. The one-sided discretisation error decays like m⁻² with Chebyshev-clustered nodes. Placing nodes adaptively at the near-active constraints (the contact values) should do far better, and the contact values are known a priori for the lattice cases. Why it matters: the whole soundness problem in §2 is the cost of m; a node rule that reaches sup < 1e-9 at m = 100 makes the repair penalty negligible and the bracket tight. First step: re-run node_sweep(24, 10, ...) with nodes concentrated near {-1/2, -1/4, 0, 1/4} and compare the sup at equal m.

• The repair is a general instrument and this hunt built it by hand. Any semi-infinite LP solved on nodes admits the same f → f - sup f repair, which converts an invalid optimisation output into a valid weaker bound with the sup as the explicit price. Why it matters: it turns "the grid was too coarse" from a fatal objection into a quantity. First step: see core_candidates in HANDBACK.json.

• The n=3 degree-27 row. Either real, or the saturation tolerance chasing noise. Why it matters: it is the only counterexample to "degree saturates by 14", which is §3's headline. First step: re-run n=3 at m = 20000 with tolerance 1e-9 and see whether the saturation degree drops.

• The three-point SDP ceiling is the natural next instance, not this one. §3 says the two-point LP has no headroom in its degree, which is exactly the argument for measuring the ceiling of the method that does. Why it matters: it is where the issue's premise ("headroom hides in hand-chosen parameters") would actually be tested. First step: reproduce the Bachoc-Vallentin dimension-4 bound (24) and sweep its two degree parameters.