Why the de Sitter vacuum cannot fix the MOND acceleration scale: the required moment is the pole of the Riemann zeta function

Abstract

Preprint. This deposit exists to timestamp a NO-GO result and to carry its reproducibility scripts; it is not a journal publication. It does NOT derive the MOND coefficient and is not a partial derivation -- it closes routes. The result in one line. In a modified-inertia worldline action whose memory kernel is the de Sitter vacuum autocorrelation, the acceleration scale is fixed by that kernel's FIRST moment. All the moments have one closed form, Mp = ∫0∞ ds sp a2/sinh2(as) = 21−p Γ(p+1) ζ(p) a1−p, a = H/2 = π/β so the moment the theory needs, p = 1, is exactly the pole of ζ -- the only pole ζ has. And the same integer forbids the required √π: a half-integer power of π enters Mp only through Γ(3/2) = √π/2 at HALF-INTEGER p, while the rapidity gap θ = (s/c)|a| is LINEAR in s, which forces p = 1 and forbids every half-integer. One integer carries the whole obstruction, and the divergence and the missing √π are the same fact. The positive consequence. M1 = c/a0 is a RENORMALISATION CONDITION, not a computable number, and a0 is the subtraction point of the theory rather than one of its outputs. This explains, rather than merely restates, why the worldline kernel K = (N/λ)e−s/λ has two free numbers with only the PRODUCT Nλ = M1 fixed. What is verified rather than cited. That the dS geodesic correlator k(s) = a2/sinh2(as) is thermal at T = H/2π is established three ways: KMS periodicity k(s + iπ/a) = k(s) identically; coincidence with the flat thermal correlator symbol for symbol; and the short-distance structure k = 1/s2 − H2/12 + O(s2), so the Hadamard subtraction is exactly the FLAT 1/s2. The closed form for Mp is checked against quadrature to <10−20 at p = 1/2, 3/2, 2, 3, 4 (bare for p > 1, Hadamard-subtracted for p < 1 where the bare integral power-diverges), and at p = 0 the continuation gives EXACTLY −a while the PHYSICAL subtraction reproduces −a to 10−25 -- the agreement that licenses using the continuation at all. The pole has residue exactly 1: (p−1)Mp → 1.0000000066. A two-pronged no-go on the memory TIME. UNSUBTRACTED, M0 power-diverges as 1/δ while M1 only logs, so the coupling-free correlation time τc = M1/M0 → 0: the bare de Sitter correlator has ZERO memory. HADAMARD-SUBTRACTED, M0 = −a exactly (finite, and it DOES carry the thermal scale) but M1 then diverges in the INFRARED as −ln(aS). No scheme has both moments finite. Reported because exact and not because predictive: cutting the log at the horizon, δ = 1/H, gives the dimensionless first moment EXACTLY 1. TWO COMPANION NO-GOES, closed by the same π-parity mechanism. (i) The mode sum over the SO(1,3) generators: √6 = √(dim so(1,3)) does check out from the D2 root data, and D = 4 is the ONLY integer with D(D−1)/2 = 6 -- but the √π CANNOT be group-theoretic, because every Vol(Sn−1) = 2πn/2/Γ(n/2) has INTEGER π-weight (Γ at half-integer argument returns the compensating √π) and Vol(SU(2)) = 2π2, Vol(SO(4)) = 2π4 are π-even; the rational 8/9 sits in a menu of 33 rationals containing all four prespecified decoys; and C2(adjoint) is 4 or 2 by convention, a factor of 2 exactly the size of the thing to be explained. (ii) A WITHDRAWAL AGAINST INTEREST: the author's own earlier observation that κ = (2/3)(D−1)/D returns exactly 1/2 at D = 4 with no fitted quantity is WITHDRAWN AS A FORM. The framework's own D-dependence is derived here -- κD = ½√(6/((D−1)(D−2))) -- from the D-dimensional Friedmann coefficient computed off the FRW Einstein tensor (Gtt = 3, 6, 10, 15 H2 at D = 4..7), the D-independence of TdS (f(r) = 1 − H2r2 identically; control: Schwarzschild-de Sitter surface gravity IS D-dependent), and the worldline character of the memory kernel. The two agree at D = 4 and nowhere else (ratio 1.509 at D = 5, 4.157 at D = 10), and FIVE prespecified functions pass through exactly 1/2 at D = 4, so one point cannot select a form. The D = 4 VALUE stands; the FORM does not, and the original hedge ('but NOT a proof') was right. What this buys. With √6 traced to FRIEDMANN and √π to the odd-dimensional momentum measure, the unexplained part of the target is a pure rational: ξ2 = (M1HΛ)2 = 27π/33 = 2π(4/3)3, i.e. M1 = (4/3)tΛ, and 4/3 = 2 × (2/3) with the 2/3 the DERIVED memory-force renormalisation. The entire residue is ONE FACTOR OF 2 -- the same one already banked as Z2 = 4(8π/3), so that part is a re-presentation and is labelled as such. STATED AGAINST THE AUTHOR'S OWN FRAMING. The cutoff table is NOT a kill: because the coupling C in K = C k is free, a Planck-scale subtraction gives τc = 7.6×10−42 s, which PASSES the ephemeris bound λ ≤ 39 yr at kernel weight N = 2.8×1059, and N is free. Nothing in that table excludes anything -- and nothing in it predicts anything. The kill is the pole together with the linearity of the rapidity gap and the π-parity of the moments. Also stated: the CONCLUSION of the moment theorem is standard QFT in a costume (any kernel with 1/s2 short-distance behaviour has a log-divergent first moment -- the same log that makes mass renormalisation necessary), so a referee will not find the divergence surprising; what is new is WHERE it lands, on the unique moment the action is permitted to use. WHAT IS NOT CLAIMED. κ = ½ is NOT derived and this is not a partial derivation. It remains FITTED. No claim is made regarding particle physics, the Standard Model, or unification. The author publicly withdrew earlier statements of that kind and does not restate them. Nothing here is a theory of everything and nothing here is evidence for one. The π-parity theorem is about THIS class of moment integrals, not a general claim that √π cannot arise in physics. The identification of K with a vacuum autocorrelation is the paper's one physical postulate and is what a referee should attack first. The obvious alternative is closed by a control: for a free field the commutator (1+n) − n = 1 is exactly temperature-independent, so a RETARDED kernel carries no H and could never have supplied a memory scale -- the temperature must live in the noise kernel. A minimally coupled massless scalar in de Sitter is infrared-pathological and was not used; whether the physical kernel is conformally coupled is not established here. The interpolating function ν = √(1+1/y) is IDENTICAL to Eq. (9) of Milgrom, Phys. Lett. A 253, 273 (1999) -- not a variant of it. PRIOR WORK, and it is closer than expected. Milgrom 1999 (Phys. Lett. A 253, 273) is the nearest prior art: the same physical picture, but the route goes through the TEMPERATURE rather than a kernel moment, so it never encounters ζ's pole. Milgrom 1994 (Ann. Phys. 229, 384) already established that modified-inertia theories are generically time-nonlocal, so the memory-kernel framing is his. The √(a2+H2) structure is Deser & Levin 1997 (CQG 14, L163). Gibbons & Hawking 1977, Bunch & Davies 1978, Narnhofer-Peter-Thirring 1996 for the dS thermal correlator. The Mellin transform of sinh−2, the KMS condition, so(4) = su(2)+su(2), the D2 root system, Vol(Sn−1) and Tangherlini 1963 are classical. A literature check was performed and found no prior work computing the moments of the dS worldline correlator, expressing them as Γ(p+1)ζ(p), or connecting p = 1 to an acceleration scale -- unsurprising, since the question only exists once one posits M1 = c/a0. AI-assistance disclosure. Portions of the analysis, numerical verification and drafting were carried out with the assistance of a large language model (Anthropic Claude). The author directed the work, specified and reviewed every load-bearing calculation, and takes full responsibility including for any errors. No AI system satisfies the criteria for authorship and none is listed as an author. Several intermediate claims produced during the work were found to be incorrect and were withdrawn before this deposit -- among them a 'hundreds of rationals' figure corrected to the measured 33, and the (2/3)(D−1)/D form withdrawn outright. Reproducibility. Every quantitative claim is produced by a committed script that exits non-zero if any internal consistency check fails, and each carries negative controls that must trip. The three scripts are included (30/30, 23/23 and 23/23 checks). Both a0 footings are carried on every dimensionful number; the central target M1/β = √(32/27π) = 0.61421182 is footing-INVARIANT.