--- slug: viazovska-magic passed: 2026-10-18 by: the assay line (research/assay-line/claude-1018b/), Claude fixer (claude-opus-5-5) directed by claude-assaying-boyle-1014; source record by a Claude agent (claude-sonnet-5-5), checked by a Claude agent (claude-opus-5-5) checked: 154 confirmed: 111 wrong: 11 unverifiable: 0 observed: 32 stale: 0 --- This was the assay line's source-first re-read of The Function That Left No Room (viazovska-magic), batch claude-1018b, tier B. A Claude agent (claude-sonnet-5-5) wrote the source record (research/assay-line/claude-1018b/viazovska-magic/N1-sources.json); a Claude agent (claude-opus-5-5) checked every entry against ar5iv 1603.04246 and math/0110009, the arXiv abstract pages, Crossref and the layer's own code (N2-claude.md, read at e7fb271c26); the Claude fixer decided each of the 11 findings (triage.json) and fixed the one WRONG on this layer: the check panel described its tolerance as "every digit the HTML prints" when the code allows 0.75 to 7.5 units of the last printed digit. The other WRONG, the placard's reach line failing for a reader on the edge-injected beacon, is a shared class held under board note 81816d. The nine MINOR lines are counted under wrong and listed, as tier B does. No shipped number changed. ## Claims - CONFIRMED | [ce-2003-ceiling] In 2003 Henry Cohn and Noam Elkies showed that a single well-chosen function puts a ceiling on how densely you can pack spheres. | https://ar5iv.labs.arxiv.org/html/math/0110009 | Theorem 0.3 bounds the center density by f(0)/(2^n f-hat(0)) for one admissible f; Annals 157 (2003). Viazovska Sec. 2 also writes "In 2003 Cohn and Elkies [4] developed linear programming bounds" - CONFIRMED | [ce-2003-year] In 2003 Henry Cohn and Noam Elkies showed | https://arxiv.org/abs/math/0110009 | journal reference "Annals of Mathematics 157 (2003), 689-714" (preprint v1 1 Oct 2001; the page gives both dates later) - CONFIRMED | [numeric-gap-dek] In 2003 the best numerical function anyone could find in dimension eight left a gap of about one part in a million above the E8 lattice. | https://ar5iv.labs.arxiv.org/html/math/0110009 | Introduction: "too high by factors of 1.000001 and 1.0007071 in dimensions 8 and 24"; 1.000001 is one part in a million - CONFIRMED | [numeric-gap-improvable] In 2003 the best numerical function anyone could find in dimension eight | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 2: "their upper bound for Delta_8 was only 1.000001 times greater than the lower bound"; the next sentence says more computation could improve it, which does not contradict "best ... anyone could find" in 2003 - CONFIRMED | [mar2016-function] In March 2016 Maryna Viazovska wrote down a function that left no gap at all | https://arxiv.org/abs/1603.04246 | "[v1] Mon, 14 Mar 2016 13:00:35 UTC" - CONFIRMED | [laplace-modular] built as a Laplace transform of a modular form. | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 2: "it is defined as an integral transform (Laplace transform) of a modular form of a certain kind" - CONFIRMED | [thirteen-years-later] Thirteen years later Maryna Viazovska wrote the function down. | https://arxiv.org/abs/1603.04246 | 2003 (the dek's own start year, Annals 157) to 14 Mar 2016 is 13 years; from the 2001 preprint it would be 14 - CONFIRMED | [it-left-no-room] It left no room at all. | https://ar5iv.labs.arxiv.org/html/1603.04246 | Theorem 3 gives g(0) = g-hat(0) = 1 and "Theorem 2 applied to the optimal function f(x)=g(sqrt2 x) immediately implies Theorem 1" - CONFIRMED | [ce-slot-conditions] The Cohn–Elkies theorem is a machine with a very narrow slot. Feed it a function f on R^d that is (i) admissible, (ii) non-positive outside radius 1, and (iii) whose Fourier transform is non-negative everywhere. Out comes a pro... | https://ar5iv.labs.arxiv.org/html/1603.04246 | Theorem 2 (Cohn, Elkies): admissible, not identically zero, f(x) <= 0 for norm >= 1, f-hat >= 0, bound f(0)/f-hat(0) Vol B_d(0,1/2). The page omits "not identically zero", harmless - CONFIRMED | [density-decimal] the number Viazovska's paper states as | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 1: "Delta_8 = pi^4/384 ~ 0.25367" - CONFIRMED | [density-decimal-value] For the magic function they meet at | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 32-35) | pi^4/384 = 0.253669507901048 by hand; shipped 0.2536695079 is correctly rounded - CONFIRMED | [lower-bar-formula] The lower bar is not the same computation as the upper bar. The upper bar is 2⁴ · g(0)/ĝ(0) · vol B₈(0, ½), read off the witness. The lower bar is vol B₈ · (√min²/2)⁸ / √det, read | https://ar5iv.labs.arxiv.org/html/math/0110009 | Sec. 2: lattice packing density pi^(n/2)/(n/2)! (r/2)^n / covolume; with r = sqrt(min^2), covolume = sqrt(det) this is the page's formula - CONFIRMED | [upper-bar-formula] The upper bar is 2⁴ · g(0)/ĝ(0) · vol B₈(0, ½), | https://ar5iv.labs.arxiv.org/html/1603.04246 | f(x) = g(sqrt2 x) gives f(0)/f-hat(0) = 2^4 g(0)/g-hat(0), and Sec. 2 says optimal means f(0)/f-hat(0) = 2^4; redone by hand - CONFIRMED | [volB8] Both bars carry the same factor vol B₈ · 2⁻⁴ = π⁴/384 | https://en.wikipedia.org/wiki/Volume_of_an_n-ball | n = 8 volume pi^4/24 R^8; pi^4/24 x 2^-4 = pi^4/384 (and 2^4 x vol B8(1/2) = the same) - CONFIRMED | [bars-meet-pi4-384] every digit they share is a digit of that one constant. | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 34) | algebra: upper = V8 2^-4 g(0)/g-hat(0), lower = V8 2^-4 (min^2/2)^4/sqrt(det); shared factor is pi^4/384 identically - CONFIRMED | [dimensionless-ones] Two ones, reached from unrelated data. | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 18, 19, 29, 30) | g(0) = g-hat(0) = 1 is Theorem 3 (5) of the paper; (2/2)^4/sqrt(1) = 1 - CONFIRMED | [e8-coordinates] Λ₈ = {x ∈ Z⁸ ∪ (Z+½)⁸ : Σxᵢ even} | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 1 definition, sum of x_i congruent 0 mod 2 ("even") - OBSERVED | [e8-enumerated-counts] is enumerated out to squared norm 8; that is where the minimum non-zero squared norm | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 30 "E8 minimum non-zero squared norm = 2", PASS 31 shells 240, 2160, 6720, 17520 from enumeration - CONFIRMED | [e8-shell-counts] the enumerated shell counts are checked against 240σ₃(k), the coefficients of the theta series of the | https://oeis.org/A004009 | "Theta series of E_8 lattice = 1 + 240 * Sum sigma_3(m) q^(2m)" - CONFIRMED | [e8-shell-values] 2160, 6720 and 17520 come from. | https://oeis.org/A004009 | "1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8"; also 240 sigma_3(k) for k = 1..4 by hand - CONFIRMED | [e8-min-norm] Every non-zero E8 vector has length at least | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 2: "norm of l >= sqrt2 for all l in Lambda_8 minus 0" - CONFIRMED | [e8-cartan-det] and Bareiss elimination over the integers gives its determinant | https://en.wikipedia.org/wiki/E8_(mathematics) | "The determinant of this matrix is equal to 1."; verifier PASS 29 det = 1 - CONFIRMED | [e8-gram-is-cartan] The Gram presentation is the E8 Cartan matrix | https://en.wikipedia.org/wiki/E8_lattice | excerpt says simple roots form a basis; Gram = Cartan follows because E8 is simply laced with roots of norm 2 (the excerpt does not state that step itself) - CONFIRMED | [gauss-fourier] whose eight-dimensional Fourier transform is exactly | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 28) | by hand: FT of r^2 e^(-pi r^2) in R^8 is (4/pi - s^2) e^(-pi s^2), so FT of (1 - a r^2)e^(-pi r^2) is (1 - 4a/pi + a s^2)e^(-pi s^2) - CONFIRMED | [gauss-a-range] Both conditions hold for a < π/4, so it proves a theorem. | research/viazovska-magic/verify-viazovska-magic.mjs | h-hat >= 0 iff 1 - 4a/pi >= 0; at a = pi/4 h-hat(0) = 0 and the bound is infinite, so a < pi/4 - CONFIRMED | [gauss-best-bound] The best it can do, at a = π/5, is | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 100) | bound = (pi^4/6144)/(a^4(1 - 4a/pi)), minimised at a = pi/5, value 3125/6144 = 0.50862630208; ratio 3125/(16 pi^4) = 2.005074659 - CONFIRMED | [gauss-half-bound] Push a to ½ so its cutoff sits exactly at √2, the E8 minimum distance, and it still only reaches | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 100) | (pi^4/384)/(1 - 2/pi) = 0.69808285815; cutoff 1/sqrt(a) = sqrt2 at a = 1/2 - CONFIRMED | [poisson-unimodular] E8 is unimodular and self-dual, so Poisson summation over it reads | https://en.wikipedia.org/wiki/E8_lattice | "It is unimodular ... Equivalently, Gamma_8 is self-dual" - CONFIRMED | [poisson-eq6] Σ f(ℓ) = Σ f̂(ℓ) with no extra | https://ar5iv.labs.arxiv.org/html/1603.04246 | (6) "sum g(l) = sum g-hat(l)" over Lambda_8 - CONFIRMED | [poisson-chain] Now suppose a witness f has cutoff at most √2. Every non-zero E8 vector has length at least √2, so every non-origin term on the left is ≤ 0, and every term on the right is ≥ 0. Therefore f(0) ≥ Σ f(ℓ) = Σ f̂(ℓ) ≥ f̂(0) | https://ar5iv.labs.arxiv.org/html/1603.04246 | (7)-(8); the page's chain is the same argument for a general witness with cutoff at most sqrt2 - CONFIRMED | [poisson-zeros-forced] That is where the zeros come from. | https://ar5iv.labs.arxiv.org/html/1603.04246 | "Therefore, we deduce that g(l) = g-hat(l) = 0 for all l in Lambda_8 minus 0" - CONFIRMED | [ledger-sums] For the Gaussian at a = ½ the two sides are both | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 36) | recomputed by hand-script: 1 + sum 240 sigma3(k)(1 - k)e^(-2 pi k) = 0.99237917172; both sides agree to 2.22e-16 - CONFIRMED | [ledger-deficit-split] So the parked Fourier mass is | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 100) | recomputed: 0.6289989441 / (2/pi) = 0.98802923093 - CONFIRMED | [ledger-ifnowaste] Zeroing it alone would leave the bound at | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 100) | 1/0.99237917172 = 1.0076793513 - CONFIRMED | [ramanujan-de4] Ramanujan's DE₄ = (E₂E₄ − E₆)/3 | https://en.wikipedia.org/wiki/Eisenstein_series | "q dM/dq = (LM - N)/3" with L = E2, M = E4, N = E6 - CONFIRMED | [e2-at-i] together with E₂(i) = 3/π and E₆(i) = 0 | https://arxiv.org/html/2405.06497 | "E_2(i) = 3/pi" (secondary source; standard, follows from the E2 transformation law at tau = i) - CONFIRMED | [e6-at-i] together with E₂(i) = 3/π and E₆(i) = 0 | https://arxiv.org/html/2405.06497 | "E_6(i) = 0" - CONFIRMED | [poisson-equals-de4] Poisson summation for E8 against (1 − a r²)e^(−πr²) is equivalent to the classical identity DE₄(i) = E₄(i)/π, which follows from | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 37, 38) | by hand: Poisson against (1 - a r^2)e^(-pi r^2) reads E4(i) - 2a DE4(i) = (1 - 4a/pi)E4(i) + 2a DE4(i), i.e. DE4(i) = E4(i)/pi for a not 0 - CONFIRMED | [ledger-e4i-values] The page evaluates both sides from the shell counts: | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 37-39) | E4(i) = 3 Gamma(1/4)^8/(2 pi)^6 = 1.4557628922687; /pi = 0.46338372054864 - CONFIRMED | [kernel-laplace-g] g(r) = (π/2160) · sin²(πr²/2) · ∫₀^∞ A(t) e^(−πr²t) dt | https://ar5iv.labs.arxiv.org/html/1603.04246 | (62), stated for r > sqrt2 - CONFIRMED | [kernel-laplace-ghat] ĝ(r) = (π/2160) · sin²(πr²/2) · ∫₀^∞ B(t) e^(−πr²t) dt | https://ar5iv.labs.arxiv.org/html/1603.04246 | (70) - CONFIRMED | [kernel-A] A(t) = −t²φ₀(i/t) − (36/π²)ψᵢ(it) | https://ar5iv.labs.arxiv.org/html/1603.04246 | "A(t) = -t^2 phi_0(i/t) - (36/pi^2) psi_I(it)" - CONFIRMED | [kernel-B] B(t) = −t²φ₀(i/t) + (36/π²)ψᵢ(it) | https://ar5iv.labs.arxiv.org/html/1603.04246 | "B(t) = -t^2 phi_0(i/t) + (36/pi^2) psi_I(it)" - CONFIRMED | [kernel-signs] A(t) < 0 for all t > 0 gives g ≤ 0 past the cutoff; | https://ar5iv.labs.arxiv.org/html/1603.04246 | "we see that A(t)<0 for t in (0,infinity). Then identity (62) implies (3)." - CONFIRMED | [kernel-signs-B] B(t) > 0 for all t > 0 gives | https://ar5iv.labs.arxiv.org/html/1603.04246 | "Our aim is to prove that B(t)>0"; "Now identity (70) implies (4)." - CONFIRMED | [sin2-in-formula] The sin² factor is where the shell zeros live | https://ar5iv.labs.arxiv.org/html/1603.04246 | (37), (53) carry sin(pi r^2/2)^2; Props. 2 and 6 give the double zeroes - CONFIRMED | [two-representations-A] They agree only because | https://ar5iv.labs.arxiv.org/html/1603.04246 | "from identities (29) and (45) we obtain the following two presentations for A(t)" - CONFIRMED | [transform-phi0] φ₀(−1/z) = φ₀(z) − (12i/π)z⁻¹φ₋₂(z) − (36/π²)z⁻²φ₋₄(z) | https://ar5iv.labs.arxiv.org/html/1603.04246 | (29) verbatim - CONFIRMED | [psiI-S] and ψᵢ/₋₂S = ψₛ are true. | https://ar5iv.labs.arxiv.org/html/1603.04246 | "psi_I/_-2 S = psi_S" (definition (45) and proof of Prop. 5) - MINOR | [paper-sections-3-5] Every formula on this page (the definitions of φ and ψ, the kernel identities, the asymptotic expansions and the coefficient bounds) is from its Sections 3 to 5. | https://ar5iv.labs.arxiv.org/html/1603.04246 | the listed items (phi, psi definitions, kernels, expansions, coefficient bounds) are in Sections 3 to 5, but "Every formula on this page" is broader than that: the Cohn-Elkies bound, the Poisson step (6)-(8) and Lambda_8 are in Sections 1-2, and Ramanujan's DE4 identity, E2(i) = 3/pi, the Gaussian transform and the Hankel formula are not from the paper [declined] fixer: tier B MINOR: the sources footer says every formula on the page is from Sections 3 to 5; the listed items are, but the Cohn-Elkies bound, Lambda_8 and the Gaussian/Hankel formulas are not. Loose scope wording, not a false fact about the source; not in a sentence being fixed. - OBSERVED | [cert-q-sub] turns the whole interval into x ∈ [0, e^(−π)] and the | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | x = e^(-pi/t) maps (0,1] to (0, e^(-pi)]; PASS 40 "no bisection needed" - OBSERVED | [cert-small] gives Q_A ≤ −37351.041 | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 40 enclosure -37351.041137910 (= -368640/pi^2, the paper's leading A_0 coefficient); printed outward - OBSERVED | [cert-mid] A is certified in | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 41 "3 + 4 sub-intervals, worst margins -456.4200 and 430.6758" - OBSERVED | [cert-tail-bound] Everything except two terms is bounded by a | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 42 "tail bound 6.527e-2" - OBSERVED | [cert-tail-A] For A the surviving | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 42 "A dominant -5.998513e+11"; 72/pi^2 e^(8 pi) = 5.9985e11 by hand - CONFIRMED | [cert-tail-B-terms] 8640t/π − 12960/π² | https://ar5iv.labs.arxiv.org/html/1603.04246 | B_inf^(6): "-12960/pi^2 ... + t(8640/pi + ...)"; 34560/pi - 12960/pi^2 = 9687.6671265 by hand - CONFIRMED | [cert-tail-A-term] For A the surviving term is −(72/π²)e^(2πt) | https://ar5iv.labs.arxiv.org/html/1603.04246 | A_inf^(6) begins "-72/pi^2 e^(2 pi t) - 23328/pi^2" - OBSERVED | [cert-truncation] First, the truncation: the page keeps the Fourier expansions out to | public/strata/viazovska-magic/index.html | KMAX = 40 in the page (line 927) and the verifier - CONFIRMED | [paper-coefficient-bound] and bounds the remainder with the paper's own coefficient estimate /c(n)/ ≤ 2e^(4π√n) | https://ar5iv.labs.arxiv.org/html/1603.04246 | (65)-(69): 2e^(4 pi sqrt n) for phi_0 and psi_S, e^(4 pi sqrt n) for the rest, so 2e^(4 pi sqrt n) bounds all - OBSERVED | [remainder-bound] that remainder never exceeds | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | cert.remainder recomputed 1.0922802e-30, printed 1.0923e-30 as an upper bound - CONFIRMED | [paper-six-terms] Viazovska's published certificate keeps only | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 5 uses A_0^(6), A_inf^(6), B^(6) and R^(6): six terms - CONFIRMED | [paper-interval-arith] has to work much harder. | https://ar5iv.labs.arxiv.org/html/1603.04246 | "Using interval arithmetic we verify that"; "has to work much harder" is a judgement, not checkable - OBSERVED | [not-verified-library] Second, the arithmetic: this is double precision with outward padding, not a formally verified interval library. | research/viazovska-magic/README.md | "IEEE double precision with outward padding of a few tens of ulps" - MINOR | [A-one-growing-term] it contains exactly one growing term | https://ar5iv.labs.arxiv.org/html/1603.04246 | A_inf^(6) has two growing terms, -(72/pi^2)e^(2 pi t) and the bare t-linear (8640/pi)t ("+t(8640/pi + 2436480/pi e^(-2 pi t) ...)"); the page itself names the (8640/pi)t term in both kernels two paragraphs later. Read "exactly one exponentially growing term". The record's where-note ("t, t^2 times decaying exponentials") misreads the source [declined] fixer: tier B MINOR: index.html line 568 "exactly one growing term"; the intended sense is the one exponentially growing term (the (8640/pi)t term grows only linearly and the page names it two paragraphs later). Loose wording, decided from the checker note; not in a sentence being fixed. - CONFIRMED | [gprime-closed] g′(√2) = −√2/60 | https://ar5iv.labs.arxiv.org/html/1603.04246 | (41) a'(sqrt2) = 72 sqrt2 i/pi, (61) b'(sqrt2) = 2 sqrt2 pi i, g = (pi i/8640)a + (i/(240 pi))b: g'(sqrt2) = -sqrt2/120 - sqrt2/120 = -sqrt2/60 by hand - CONFIRMED | [gprime-b] g′(√2) = −√2/60 | https://ar5iv.labs.arxiv.org/html/1603.04246 | (61) "b(0)=0 b(sqrt2)=0 b'(sqrt2)=2 sqrt2 pi i" - CONFIRMED | [gdef] Two Fourier coefficients decide the entire zero structure | https://ar5iv.labs.arxiv.org/html/1603.04246 | Theorem 4 definition of g; the claim is the page's own reading (the only zero order not fixed by sin^2 is the one at sqrt2, which the q^-1 coefficients set) - CONFIRMED | [ghat-double-zero] ĝ has a double zero at | https://ar5iv.labs.arxiv.org/html/1603.04246 | g-hat'(sqrt2) = (pi i/8640)a'(sqrt2) - (i/(240 pi))b'(sqrt2) = -sqrt2/120 + sqrt2/120 = 0 by hand - CONFIRMED | [b-e2pt-coeff-zero] Its e^(2πt) coefficient is | https://ar5iv.labs.arxiv.org/html/1603.04246 | (49) psi_I = q^-1 + 144 - ..., (32) phi_-4 = q^-1 + 504 + ...; second presentation of B gives -(36/pi^2)(1) + (36/pi^2)(1) = 0 - CONFIRMED | [simple-vs-double] A function that must stay | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 2 "double zeroes"; the parity argument is standard real analysis - CONFIRMED | [B-no-pole] No pole, so nothing cancels the double zero: | https://ar5iv.labs.arxiv.org/html/1603.04246 | checked by hand from (38) and (58): pole terms 36/(pi^3(r^2-2)) in a and 1/(pi(r^2-2)) in b cancel in g-hat and add in g - OBSERVED | [measured-derivs] it differentiates the g this page actually built, by a Richardson-extrapolated central difference at step | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 20 g'(sqrt2) -0.0235702260395 vs -sqrt2/60; DERIV_H = 1e-4 Richardson - OBSERVED | [measured-contrast] run it on g and it returns −0.0236; run it on ĝ and it returns a number below | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 21 "measured 3.25e-16, against -2.36e-2" - CONFIRMED | [phi-m2-720] 720, the constant Fourier coefficient of | https://ar5iv.labs.arxiv.org/html/1603.04246 | (33) "phi_-2(z) = 720 + 203040 q + ..." - CONFIRMED | [norm-g0] Both kernels share the term | https://ar5iv.labs.arxiv.org/html/1603.04246 | (41) a(0) = -8640 i/pi, (61) b(0) = 0, so g(0) = 1; (pi/2160)(8640/(4 pi)) = 1 by hand - CONFIRMED | [hankel-formula] (2πs⁻³∫ g(r) J₃(2πrs) r⁴ dr | https://en.wikipedia.org/wiki/Hankel_transform | angular convention with k = 2 pi s gives 2 pi s^-3 integral g(r) J_3(2 pi r s) r^4 dr for d = 8 - OBSERVED | [hankel-agree] So here is the eight-dimensional radial Fourier transform of g, done the stupid way | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 27 "worst relative difference 6.74e-12 at s = 1.2" - CONFIRMED | [hankel-eigen-theorem] The identity itself is a theorem in Viazovska's paper, proved by contour | https://ar5iv.labs.arxiv.org/html/1603.04246 | Props. 1 and 5 prove a-hat = a, b-hat = -b by exchanging contour integration with the Fourier transform; the Laplace forms behind g-hat use the contour deformations of Props. 2 and 6 - OBSERVED | [hankel-three-points] The offline verifier runs the same comparison at | research/viazovska-magic/verify-viazovska-magic.mjs | line 450 "for (const s of [0.4, 0.75, 1.2])" - OBSERVED | [check-panel-sentence] The one lamp on this page that a control | public/strata/viazovska-magic/index.html | globalFourier() (lines 1309-1321) turns the trace lamp red when 1 - 4a/pi < 0; the a slider (line 1945) drives the trace witness - OBSERVED | [check-panel-run] Run it yourself: | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | from the repository; see check-placard-two-files for the reader's run - OBSERVED | [verifier-claim] That script reads this HTML file, extracts every shipped number above, and re-derives each one | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | reads 45 data-check values and recomputes each; imports nothing from the page (but see the (added) tolerance line) - OBSERVED | [free-choices-quadrature] The quadrature on | public/strata/viazovska-magic/index.html | NPAN = 800 (line 928); tail integrated term by term - OBSERVED | [refine-3200] refining it to | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 17 "at most 2.2523e-11 relative on the page's 41-point grid" - OBSERVED | [quad-401] (The offline verifier repeats the sweep on a | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 17 "4.4133e-11 on a 401-point grid"; assertion quadWorst401 < 1e-9 (line 375) - OBSERVED | [boot-cost] the page uses | research/viazovska-magic/README.md | "measured: ~54 ms of a ~65 ms boot on desktop V8, against ~5 ms for the 41-point sweep"; raw timing not kept - OBSERVED | [zero-simple-vs-double] Every double zero on this page | recreated: node research/viazovska-magic/verify-viazovska-magic.mjs at e7fb271c26 (exit 0, 108 PASS) | PASS 25 largest double zero 1.234e-33; PASS 26 g(sqrt2) = -3.7007434154e-18 - CONFIRMED | [sqrt2-squared] The nearest double to | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 26) | Math.SQRT2*Math.SQRT2 = 2.0000000000000004 = 2 + 2^-51 = 2 + 2 eps, eps = 2^-52 - CONFIRMED | [eps-over-60] so feeding it | research/viazovska-magic/verify-viazovska-magic.mjs (PASS 26) | by hand: near r^2 = 2, g ~ -(r^2 - 2)/120, so 2 eps gives -eps/60 = -3.7007434e-18; slope -2r/120 = -sqrt2/60 at sqrt2 - OBSERVED | [free-choices-window] The window t ∈ [0.60, 1.70] in which the kernel panel calls the two representations comparable is ours too | research/viazovska-magic/README.md | "the window t in [0.60, 1.70] ... chosen as the range where both truncations are still worth twelve digits" - CONFIRMED | [theorem-quote] No packing of unit balls in Euclidean space R⁸ has density greater than that of the E8-lattice packing, | https://ar5iv.labs.arxiv.org/html/1603.04246 | Theorem 1 verbatim; the tool's NOT FOUND is ar5iv printing each formula twice (the raw entry is FOUND) - CONFIRMED | [theorem-quote-raw] No packing of unit balls in Euclidean space R⁸ has density greater than that of the E8-lattice packing, | https://ar5iv.labs.arxiv.org/html/1603.04246 | Theorem 1, as rendered - CONFIRMED | [delta8] Δ₈ = π⁴/384. | https://ar5iv.labs.arxiv.org/html/1603.04246 | "Delta_8 = pi^4/384" - CONFIRMED | [periodic-uniqueness] The paper adds that combining its proof with Section 8 of Cohn–Elkies implies E8 is the unique periodic packing of maximal density. | https://ar5iv.labs.arxiv.org/html/1603.04246 | "combined with arguments given in [4, Section 8] implies that the E8-lattice sphere packing is the unique periodic packing of maximal density" - MINOR | [uniqueness-stronger] Uniqueness among all packings is a different and stronger | https://ar5iv.labs.arxiv.org/html/math/0110009 | Sec. 8 says just before the record's excerpt: "it is trivially not unique, since for example removing a single sphere does not change the global density". So uniqueness among all packings is not an unproved stronger statement but false; the page could say so [declined] fixer: tier B MINOR: index.html line 802 says uniqueness among all packings is stronger and not proved there, which is true; the checker notes Cohn-Elkies Sec. 8 calls it trivially false (removing one sphere keeps the density). An omission, not a false statement; not in a sentence being fixed. - MINOR | [scaling-min-distance] The E8 lattice as usually written has minimum distance | https://ar5iv.labs.arxiv.org/html/1603.04246 | The page repeats the paper's Sec. 1 sentence "packing of unit balls with centers at (1/sqrt2)Lambda_8", but by the paper's own definition (unit balls, centres at least 2 apart) (1/sqrt2)Lambda_8 has minimum distance 1, so those balls overlap. Consistent forms: balls of radius 1/2 at (1/sqrt2)Lambda_8 (the Cohn-Elkies normalisation the page then uses) or unit balls at sqrt2 Lambda_8. The page's "so" derives the slip [declined] fixer: tier B MINOR at the checker level: index.html line 805 "the packing of unit balls is centred at (1/sqrt2)Lambda_8" repeats the paper's own Sec. 1 normalisation slip (those centres are distance 1 apart, so the balls have radius 1/2). A slip of normalisation precision; the page's numbers use the consistent Cohn-Elkies scaling. Not in a sentence being fixed. - CONFIRMED | [scaling-fx] Viazovska's g is scaled so that its cutoff sits at | https://ar5iv.labs.arxiv.org/html/1603.04246 | "Theorem 2 applied to the optimal function f(x)=g(sqrt2 x)" - CONFIRMED | [dim24-authors] Dimension 24 is a different paper. The Leech lattice result is Cohn, Kumar, Miller, Radchenko and Viazovska, five authors, a separate article in the same volume of the Annals, pages 1017–1033. | https://api.crossref.org/works/10.4007/annals.2017.185.3.8 | Crossref: Cohn, Kumar, Miller, Radchenko, Viazovska; Annals 185(3) 1017-1033, same issue as 991-1015 - CONFIRMED | [dim24-pages] Annals, pages 1017–1033. | https://arxiv.org/abs/1603.06518 | "Annals of Mathematics 185 (2017), 1017-1033" - CONFIRMED | [dim24-seven-days] Its first preprint went up | https://arxiv.org/abs/1603.06518 | v1 Mon, 21 Mar 2016 17:57:05 UTC against 14 Mar 2016 13:00:35 UTC: 7.2 days - CONFIRMED | [arxiv-v1-v2] arXiv:1603.04246, v1 dated 14 March 2016, v2 dated | https://arxiv.org/abs/1603.04246 | "[v1] Mon, 14 Mar 2016 ... [v2] Tue, 4 Apr 2017" - CONFIRMED | [arxiv-pages-figs] listed at 22 pages and | https://arxiv.org/abs/1603.04246 | Comments: "22 pages, 2 figures" - CONFIRMED | [journal-pages] The journal version spans pages 991 to 1015, which is | https://arxiv.org/abs/1603.04246 | "Pages 991-1015 from Volume 185 (2017), Issue 3"; 25 inclusive - CONFIRMED | [ce-arxiv] Cohn–Elkies paper is arXiv:math/0110009, v1 dated 1 October 2001, published in the Annals in 2003 at pages 689–714. | https://arxiv.org/abs/math/0110009 | v1 "Mon, 1 Oct 2001"; "Annals of Mathematics 157 (2003), 689-714" - CONFIRMED | [ce-v1-date] v1 dated 1 October 2001 | https://arxiv.org/abs/math/0110009 | "[v1] Mon, 1 Oct 2001 09:32:11 UTC" - MINOR | [thirteen-years-rounded] is therefore commonly rounded to | https://arxiv.org/abs/math/0110009 | Two problems. "commonly" is unsupported: two web searches for "thirteen years"/"13 years" with Cohn Elkies Viazovska found no such usage (positive control: the same searches returned pages dating Cohn-Elkies to 2003 and Viazovska to March 2016). And "therefore" does not follow: the method was public on 1 Oct 2001 (the page's previous sentence), 14 years 5 months before 14 Mar 2016; thirteen counts from the 2003 journal year [declined] fixer: tier B MINOR: index.html line 820 "therefore commonly rounded to thirteen years"; "commonly" is unsupported and thirteen counts from the 2003 journal year, but the sentence itself says the figure is not sharp and nothing depends on it. Loose wording; not in a sentence being fixed. - CONFIRMED | [improved-further] This bound can be improved even further by more extensive computer computations. | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 2 verbatim - MINOR | [her-sentence-continues] That factor is Viazovska's own description of the numerical | https://ar5iv.labs.arxiv.org/html/1603.04246 | the quoted words are the NEXT sentence, not a continuation: "... only 1.000001 times greater than the lower bound, which is given by the density of the E8 sphere packing. This bound can be improved even further by more extensive computer computations." The README calls it "her sentence continues, verbatim" [declined] fixer: tier B MINOR: index.html line 823 "her sentence continues"; the quoted words are exact but are her NEXT sentence (checker re-read ar5iv 1603.04246). The quotation itself is unaltered, so this is loose framing, not an altered quotation; not in a sentence being fixed. - MINOR | [numerical-optimisation] It was the output of numerical optimisation over admissible functions | https://ar5iv.labs.arxiv.org/html/math/0110009 | Sec. 7: the 1.000001 bound is the performance of one explicit function f = -g + h (Laguerre polynomials in 2 pi x^2 times a Gaussian, m = 11 forced double roots, exact rational arithmetic), whose root locations were chosen by "a computer search". "output of numerical optimisation" is right; "not the performance of any single closed-form test function" overstates the contrast [declined] fixer: tier B MINOR: index.html line 825 "not the performance of any single closed-form test function"; Cohn-Elkies Sec. 7 gives one explicit polynomial-times-Gaussian with computer-searched roots. "output of numerical optimisation" is right; the contrast overstates. Loose wording; not in a sentence being fixed. - CONFIRMED | [paper-factor-quote] Viazovska 2017, section 1: “only 1.000001 times greater than the lower bound” | https://ar5iv.labs.arxiv.org/html/1603.04246 | Sec. 2 "only 1.000001 times greater than the lower bound" verbatim; NOT FOUND is the ar5iv doubled rendering - CONFIRMED | [arxiv-22-quote] arXiv:1603.04246v2 abstract page, comments field: “22 pages, 2 figures” | https://arxiv.org/abs/1603.04246 | "22 pages, 2 figures" - CONFIRMED | [paper-factor-meaning] the factor by which the numerical Cohn-Elkies bound in dimension 8 | https://ar5iv.labs.arxiv.org/html/math/0110009 | "too high by factors of 1.000001" - CONFIRMED | [doesnt-reproduce] It does not reproduce the Schwartz-class decay estimates | https://ar5iv.labs.arxiv.org/html/1603.04246 | the paper has the decay estimates (Props. 1, 5); the page does not reproduce them - CONFIRMED | [cite-viazovska] Maryna S. Viazovska, “The sphere packing problem in dimension 8,” Annals of Mathematics 185 (2017), 991–1015 | https://api.crossref.org/works/10.4007/annals.2017.185.3.7 | title, Annals 185(3), 991-1015, 2017 match; the tool's DOI MISMATCH is the middle initial (Crossref "Maryna", arXiv byline "Maryna S. Viazovska") - CONFIRMED | [cite-cohn-elkies] Henry Cohn and Noam Elkies, “New upper bounds on sphere packings I,” Annals of Mathematics 157 (2003), 689–714 | https://api.crossref.org/works/10.4007/annals.2003.157.689 | title, authors, 157, 689-714, 2003 match - CONFIRMED | [cite-dim24] Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska, “The sphere packing problem in dimension 24,” Annals of Mathematics 185 (2017), 1017–1033 | https://api.crossref.org/works/10.4007/annals.2017.185.3.8 | matches; DOI MISMATCH is the initial D. (Crossref "Stephen", arXiv "Stephen D. Miller") - CONFIRMED | [cite-ce-linear-programming] for the linear programming theorem and the conjecture that it is sharp in dimensions 8 and 24. | https://ar5iv.labs.arxiv.org/html/math/0110009 | "We conjecture that our techniques can be used to prove sharp bounds in 8 and 24 dimensions." - MINOR | [notes-in-research-dir] Working notes, primary-source quotations and the scope statement: | research/viazovska-magic/README.md | the notes exist in the repository, but a reader cannot open them: the page prints research/viazovska-magic/ as plain text, https://artwaste.land/checks/research/viazovska-magic/README.md and /research/viazovska-magic/ return 404, and the GitHub repository is private (404). Positive control: /checks/research/viazovska-magic/verify-viazovska-magic.mjs returns 200 [declined] fixer: tier B MINOR: the page names research/viazovska-magic/ as where the working notes are; they exist in the repository but a reader cannot open the README (404 on /checks/). Metadata about the page's own notes; publishing the README needs a shared /checks/ change, outside this layer. - WRONG | [check-placard-two-files] It is published, and it runs from 2 files this site serves | https://artwaste.land/checks/#run-research-viazovska-magic-verify-viazovska-magic-mjs | The /checks/ recipe (2 curls + node) run in an empty directory on 2026-10-18 exits 1: 107 PASS, 1 FAIL "the shipped page makes no third-party request: