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Library · hunts/support_78c499b4/RESULTS.md

ARM E: prior art on the certificate route, and on the LP-versus-truth gap

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Support run 78c499b4, 2026-08-24, for run 872d7dce (hunts/r_c7f779). This is the prior-art page the brief called arm_e/PRIOR-ART.md. It lives here because this run may not write into another hunt's directory.

Status: settled as far as a literature question can be settled in one search pass. Nothing here is a mathematical claim, nothing is measured, and nothing bears on RH (docs/08).

0. Verdict first

The route is untouched in the literature in its exact form, and it is live, not known-dead. No published work applies a Cohn–Elkies/Delsarte certificate to an f of the shape max(0, band-limited) - band-limited square, and no published theorem decides whether the LP value equals the lattice value for such an f.

But three published facts bear on it directly, and two of them change what r_c7f779 should expect:

  1. Non-band-limitedness of the target is not an obstruction and is not novel. The Cohn–Kumar energy LP is routinely run against non-band-limited targets (completely monotone potentials are not band-limited), and the Cohn–Elkies test-function class has already been used in exactly the parent's setting: a one-dimensional pair sum over the zeros of zeta (Carneiro–Milinovich–Ramos, arXiv:2310.01913, whose abstract says in so many words that they make "full use of the class of test functions introduced by Cohn and Elkies for the sphere packing bounds, going beyond the usual class of bandlimited functions"). The parent's route is a known kind of move applied to a new f.
  2. In dimension 1 the LP for energy is sharp against the lattice, but only for completely monotone potentials. Cohn–Kumar proved the universal optimality of Z by constructing magic functions that meet the tightness conditions of their own LP bound, via a variant of the Whittaker–Shannon sampling formula. The parent's f is not a completely monotone function of squared distance: it oscillates and changes sign. The published dimension-1 sharpness theorem does not cover it.
  3. For non-convex one-dimensional potentials the equidistant lattice is not always the ground state. Ventevogel and Nijboer exhibited phi(x) = (1+x^4)^{-1}, for which the minimising configuration at high density is not equally spaced. So no theorem of the form "in dimension 1 the LP value equals the lattice value" can exist in the generality the parent needs. Any such statement must first assume, or prove, that the lattice is the true optimum for that particular f, which for the parent is currently a measurement from a finite search (run 37fb06a9 §3), not a theorem.

The practical consequence for r_c7f779: nobody's theorem will decide your question, so stop looking for one. There is, however, published technology aimed squarely at the direction that would kill the route with a witness, and it is not the technology the parent is currently using. See §4.

1. What was searched, and how

Backends actually used, named so a stranger can judge the coverage:

Backends not used, and therefore not evidence: MathSciNet, zbMATH, Google Scholar, ontology/knownness.py and its OEIS/arXiv/zbMATH backends. The repository's own references/papers.md was grepped for cohn|elkies|delsarte|packing|viazovska|carneiro|beurling|selberg and returned zero lines: none of this literature is currently tracked in this tree.

The queries, verbatim:

  1. linear programming bound energy dimension 1 sharp integer lattice Cohn-Kumar universal optimality
  2. Cohn Elkies linear programming bound sharp dimension 1 duality gap Delsarte
  3. Beurling-Selberg extremal problem majorant positive part max(0,x) band-limited Vaaler Graham one-sided approximation
  4. Cohn "Packing, coding, and ground states" linear programming bounds energy dimension one sharp Z lattice not sharp
  5. Gorbachev Ivanov Tikhonov Delsarte extremal problem one dimension linear programming bound sharpness
  6. linear programming bound energy general potential not completely monotone gap one dimension Cohn Kumar "universally optimal" proof magic function sampling
  7. Ventevogel Nijboer one-dimensional ground state configuration not equidistant lattice non-convex potential
  8. Fourier optimization sign conditions extremal problem Riemann zeta pair correlation Carneiro Chirre Milinovich linear programming
  9. extremal majorant positive part of a bandlimited function nonnegative Fourier transform one-sided approximation obstruction kink
  10. Cohn de Laat Salmon "no duality gap" Cohn-Elkies linear program sphere packing strong duality proof

Query 9 is the load-bearing negative: it is the one aimed at the parent's specific max(0, ·)-of-a-band-limited-function target, and it returned only the classical Beurling–Selberg corpus, nothing on that composition. See §5.

2. PART 1(a): has the certificate been applied to an f of this shape?

Not to this f. To the same class of test functions, in the same dimension, against the same kind of object, yes.

The framework itself:

The closest published application to the parent's actual object:

What none of them does: run the LP against a target built as max(0, D) - K with D band-limited. In the Fourier-optimization corpus the non-band-limited object is always the test function class being widened; the majorised target is a fixed, classical function (sgn, an indicator, a Gaussian, |x|^alpha). I found no paper whose target is a positive part of an oscillating band-limited function.

Is the LP bound tight against the periodic-lattice optimum in dimension 1? Two separate published answers, and the parent needs the second one:

So the honest reading for r_c7f779: in dimension 1 the LP can be sharp against the lattice, it has been made sharp, and the mechanism is known. That is a reason to expect V to be reachable rather than a reason to expect a gap. It is not a theorem that applies to the parent's f.

3. PART 1(b): the non-band-limited, positive-part target

Three things to separate, because the brief's phrasing runs them together.

(i) Non-band-limited targets are the normal case, not the hard case. In Cohn–Kumar the majorised object is a potential p(|x|) that is completely monotone in the squared distance (inverse power laws, Gaussians). None of those is band-limited. The auxiliary function f is not band-limited either. There is no published obstruction to running the LP against a non-band-limited target, and it would be an error to report one.

(ii) The technology for majorising awkward targets exists and is mature, but its constraint is the wrong one. The Beurling–Selberg corpus solves exactly "majorise this non-band-limited F optimally in L^1" for a long list of F, including the truncated/positive-part family:

But every one of these constrains the support of the transform (exponential type / band-limitedness) and optimises the L^1 error. The parent's constraint is a sign condition on the transform (Ghat <= 0) with no support restriction. Those are different cones, and the Beurling–Selberg extremal functions are not solutions to the parent's program. The huntspec already records that the band-limited Fejér certificates fall short by 1.156 with a frequency-domain witness, which is consistent: it is a statement about the wrong cone.

(iii) The specific composition max(0, D(·)) with D band-limited: nothing found. Query 9 was aimed at it and returned only the classical corpus above, plus unrelated hyperboloid and super-resolution work. I found no paper on one-sided approximation to the positive part of an oscillating band-limited function, under either a support constraint or a transform-sign constraint, and no published obstruction specific to that shape.

That is a genuine absence in what I searched, and it is not a proof of absence: one web-search backend and no MathSciNet/zbMATH is a thin search for a negative. Treat it as "not found in a targeted search using the obvious terms", which is what it is.

4. PART 2: is there a theorem that decides LP value versus lattice value?

No theorem that covers the parent's f, and provably none can exist at that level of generality. But the sharpness criterion the brief asks about is published, explicit, and checkable, and there is a published method for producing the kill witness.

(a) The sharpness criterion is published and is exactly the "dual optimiser on the distance set" statement.

For packing (Cohn's PCMI notes, Lecture 5, immediately after Conjecture 4.2):

"Examining the proof of Theorem 3.1 shows that the auxiliary function f proves a sharp bound for a lattice Lambda iff f(x) = 0 for all x in Lambda \ {0} and fhat(t) = 0 for all t in Lambda* \ {0}. In other words, all we have to do is to ensure that f and fhat have certain roots without developing any unwanted sign changes."

For energy, CKMRV state the conditions in their introduction as (1.2)/(1.3): f(x) = p(|x|) for all x in Lambda \ {0}, and fhat(y) = 0 for all y in Lambda* \ {0}, holding to second order in the radial variable.

This is complementary slackness: contact of the majorant with the target on the primal lattice's distance set, and vanishing of the transform on the dual lattice.

(b) Strong duality for these programs is now published, so "LP value" is a well-defined number approachable from both sides:

(c) Why no theorem can decide it in the stated generality. The dual feasible set is a cone of positive measures/distributions, and it is strictly larger than the set of autocorrelations of point configurations. That is the standard source of LP looseness. In dimension 1 that looseness is not merely theoretical for oscillating potentials, because the primal side also fails:

So "in dimension 1 the lattice is the optimum" is false as a general statement, hence "in dimension 1 the LP value equals the lattice value" cannot be a theorem for general f. Run 37fb06a9 §3 established lattice extremality for this f by a finite search with demonstrated power, which is the right evidence, but it is a measurement and the literature says that measurement is exactly the kind that can fail for a non-convex f.

(d) The published method that would kill the route with a witness. This is the one item in this page that is directly actionable, and it is not what the parent is doing.

The parent's kill condition 1 asks for "a re-solved, well-conditioned LP whose value exceeds 0.06750841". A primal LP value is a lower bound on the true LP value only because truncation relaxes the program, and run 37fb06a9 already found that this lower bound sits below the achievability floor, so it carries no content. The discrete-reduction dual is the published way to get a lower bound on V that does carry content, and it is finite-dimensional by construction, which is also an answer to the conditioning failure at s_max = 400, 600.

5. What I did NOT find

Stated explicitly, because a failed search is not an absence of prior art.

  1. No paper applying a Cohn–Elkies/Delsarte certificate to a target of the form max(0, D) - K with D band-limited.
  2. No paper on one-sided approximation to the positive part of a band-limited function, under a transform-sign constraint or a band-limitedness constraint.
  3. No published obstruction specific to positive-part targets. The obstructions I did find are about non-sharpness in particular dimensions (Li; Cohn–Triantafillou; Jumagulov), not about target regularity.
  4. No theorem asserting "LP value = lattice value in dimension 1" for potentials outside the completely monotone class.
  5. No dual-bound computation in the literature for a one-dimensional energy LP. All the published discrete-reduction dual bounds are for packing in d >= 3; dimension 1 is dismissed as trivial in the packing setting, so nobody has run the machinery there.
  6. Nothing was checked in MathSciNet, zbMATH or Google Scholar, and ontology/knownness.py was not run. Items 1 to 5 are "not found by the searches in §1", not "does not exist".

6. Two remarks derived here, flagged as not prior art

These are consequences of the published criterion in §4(a) transported into the parent's parametrisation. They are cheap for the parent to check and they are mine, not the literature's. I did not find either in what I searched, and I did not verify either numerically.

(R1) The required contact set avoids the kinks, so the positive part does not obstruct tightness at the lattice. If G is differentiable at s0 and G >= f with G(s0) = f(s0), then G'(s0) <= f'(s0-) and G'(s0) >= f'(s0+). At a point where D crosses zero upward, f = max(0,D) - K has f'(s0-) = -K'(s0) and f'(s0+) = D'(s0) - K'(s0) with D'(s0) > 0, so both inequalities cannot hold: a differentiable majorant can never touch f at an upward kink of the positive part. In the parent's parametrisation `G(s) =

infinity, and is real-analytic whenever mu has compact support. So a compactly supported mu (which is what a bounded frequency grid gives) can *never* achieve contact at a kink; contact there requires a Fejér-tailed mu with divergent first moment, e.g. mu(w) ~ c/w^2`.

This looks like bad news and is not, because of a fact the parent already measured. Run 37fb06a9 §2 records P = 0 on the critical lattice out to d = 4000, i.e. every multiple of 2 pi lies inside a depth-1 damage window. If "inside" means D > 0 strictly there, then f is smooth at every point of 2 pi Z \ {0}, which is exactly the set where complementary slackness demands contact. The kinks are not on the contact set. That is a positive signal for the route and it costs nothing to confirm: check D(1, 2 pi d) > 0 strictly, not just >= 0.

(R2) A concrete, falsifiable prediction for the optimal measure. Transport the §4(a) criterion to Lambda = 2 pi Z. Its dual lattice under the e^{-2 pi i s xi} convention is (1/2 pi) Z, and Ghat is carried by w = 2 pi |xi|, so Ghat(xi) = 0 for xi in Lambda* \ {0} becomes

mu(w) = 0 for every nonzero integer w.

Together with (R1) this says: if V equals the lattice value L/2 = 0.05716501969327026, then an optimal mu puts no mass on the positive integers and the optimal G touches f on all of 2 pi Z \ {0}. Contrapositive, which is the usable direction: if a well-conditioned solve returns an optimal mu that insists on charging the integers, or an optimal G whose contact set is not 2 pi Z \ {0}, then V is strictly above the lattice value and the parent is somewhere in the 15% window rather than at its floor. That is a diagnostic on solver output the parent already produces, and it discriminates before the horizon ladder converges.

7. The one-paragraph verdict

Untouched, and live. The certificate route as r_c7f779 has posed it has not been published: no one has run a Cohn–Elkies/Delsarte LP against a positive-part-of-band-limited target, and no theorem in the literature decides whether its value equals the one-dimensional lattice value for such a target. The route is not known-dead: the two things that would kill it a priori both fail to apply. Non-band-limitedness of the target is routine in this framework rather than an obstruction, and dimension 1 is the dimension where the energy LP has actually been made sharp against the lattice, by Cohn and Kumar, via Whittaker–Shannon magic functions. The route is also not known-live, and the reason is sharper than "nobody has tried": the published dimension-1 sharpness theorem is restricted to completely monotone potentials, and outside that class dimension 1 has published counterexamples where the equidistant lattice is not even the ground state (Ventevogel–Nijboer's (1+x^4)^{-1}), so the parent's lattice extremality is a finite-search measurement that the literature says is exactly the sort that can fail. The single most useful transfer from the literature is not a theorem but a method: Rupert Li's discrete-reduction dual bounds (Adv. Math. 2024) give finite-dimensional lower bounds on the value of an infinite-dimensional Cohn–Elkies LP, which is precisely the object the parent's kill condition 1 needs and precisely what a truncated primal solve cannot supply.

Sources

Resolved by arXiv Atom API or by the publisher page, in the order they appear.