The wave of a prime as a closed string on the modulo-30 wheel
Year: 2026
Type: preprint
Abstract: In our paper on prime gaps every prime p ≥ 7 is a wave on the eight vectors of the modulo-30wheel. The present work treats this wave as a closed string wound on the cylinder whose circumferenceis the cycle of 30 and whose height is the cycle number n. Unlike the strings of physics, which movein a continuum of 10 or 26 dimensions with no preferred direction and carry a tension, a coupling andmoduli, these strings live on eight explicit vectors Vr = r + 30n, each with an offset, an angle 12◦r andthree characters, and have no parameter but the prime; we tabulate the vectors, draw them and draw thestrings on them. On each vector the string of p has one node every p cycles, at a phase given in closedform; eight consecutive partners carry it once around all eight vectors, so that it winds as a helix of pitchp; two strings p < q cross on a vector once every pq cycles; and the node pattern is the superpositionof the harmonics k/p, k = 0, . . . , p − 1. The composite entries are the points touched by at least onestring with p ≤√E, the semiprimes the crossings of exactly two strings, and the primes the free points.These statements are checked for all primes up to 1999 (phases and winding), for all pairs below 100(crossings), and on a window of 30,000 integers at 1012, where the free points are exactly the 1,084primes. On that window the number of free points agrees with the prime number theorem (1,085.7expected) and not with independent strings (1,219.1): the ratio is 1.1229 = 2e−γ, the Mertens factor,so the strings are correlated. Counted with the strings of the powers pkand followed along its winding,the string of p produces exactly the local factor ζp(s) = (1 − p−s)−1, the building block of the p-adicstrings; the product of the local operators of the strings over the eight vectors gives the eight sector sumsthrough the Dirichlet L-functions modulo 30. The phase law of a wave, χ(p)p−ρ, is the first harmonicof this local factor; the strings of pkcarry its k-th power, which the zeros of ζ and of L(s, ψ5) confirmwithin 5.8◦and 3.2◦for p = 7, 11, 13 and k = 1, 2, 3. The strings generate an arithmetic quantumfield: the occupation of the mode p is the number of strings pkthrough a point, the energy is ln E, thecharge is the vector, the eight sectors share the pole at β = 1 equally, and bosons and fermions cancelto the vacuum by the Möbius function. Within complete periods the strings are independent; theirinteraction lives in the crossings longer than the window, which carry the whole deficit of free pointsagainst independent strings. As the strings are added up to X1/uthe interaction follows Buchstab’sfunction, eγω(u), measured on windows at 109, 1012 and 1015 across its whole oscillation; its value atu = 2, eγ/2, is the Mertens factor, and in spectral form every string is a line −1/(p − 1) and everycrossing a line µ(q)/φ(q). The interaction of a prime with the semiprimes turns them: on the eightvectors the character ψ5 of order 4 is a double cover of the character χ5 of the two halves, ψ25 = χ5,so that one turn of χ5 reverses ψ5 and two turns restore it, the finite form of the two-valuedness of aspin 12; the double node p2carries the sign −1, every division by 2 or by 3 in the constructions of thewheel is a quarter turn of ψ5 and every division by 5 a half turn of χ3, which the zeros of L(s, ψ5)show on the string of 49, and the square roots of the factors of a semiprime turn by half its hyperbolicangle. The two factors of a semiprime turn on their helices by the same angle in opposite directions,one rising from p2and the other descending from q2, and exchange their vectors; the pair of factorsis a maximally entangled state whose entropy is log2τ (N), prepared by measuring a product. Thephases of the wheel are additive and cannot entangle, so the entanglement lives in the constraint on theproduct, the arithmetic counterpart of the 1/r coupling in the proposed tests of the quantum nature ofgravity; no quantisation of gravity is claimed. Finally we connect the strings with our later research onthe directional-product model: the product N = S2 − d2read with momenta and with energies givesthe family of acceleration laws of disk galaxies, whose energy reading the rotation curves select; everysuch law is a dome whose feet are the total and the dark acceleration; and the minimum quantum of theacceleration ladder is the scale at which the 1/r coupling of the factor-pair section reaches one radian.
Institution: University of Basrah
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225813578
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