Prime wells: a formalization

Multiplication of positive integers, written additively through Φ(n) = (log₂ n, Ω(n)), puts the integers in the plane as a free commutative monoid generated by one step per prime. Primes are its indecomposable generators, composites are sums of steps, and gcd, lcm and coprimality are shared trunks and parallelograms. Evaluating the quantum-cohomology Frobenius algebra of ℙ¹ at q = n makes the log axis a flat coordinate of a Frobenius manifold, with a nilpotent point at q = 0 and an algebra that ramifies exactly at the prime factors of n.

Let N = (ℤ_>0, ·). By unique factorization, N is the free commutative monoid on the primes: n = ∏ p^v_p(n). Put Ω(n) = Σ_p v_p(n), the number of prime factors counted with multiplicity.

Define Φ : N → ℝ² by Φ(n) = (log₂ n, Ω(n)). Then Φ(ab) = Φ(a) + Φ(b) and Φ(1) = (0, 0), because both log and Ω turn products into sums. Φ is a monoid homomorphism. The image of a prime is the step s_p = (log₂ p, 1).

Φ is injective, since log₂ is injective on integers. So M = Φ(N) is a copy of N inside the plane, and M is the free commutative monoid on {s_p}. Ω is a function of log₂ n, so the depth coordinate adds no information about which integer it is. Its job is to make the monoid structure visible as geometry.

0 ≤ Ω(n) ≤ log₂ n, with equality on the right exactly when n is a power of 2, since each prime factor is at least 2. So M lies under the ray of 2. A prime power p^k lies on the ray through s_p. An integer with j distinct prime factors lies strictly inside the cone spanned by those j rays.

Primes as indecomposables. Write M₊ = M ∖ {0}. Then n is composite exactly when Φ(n) ∈ M₊ + M₊. So the primes are M₊ ∖ (M₊ + M₊): the points of M that no slide of M by a generator reaches. This is the Sieve board, and it is the same as Ω(n) = 1.

Relations between numbers. For all a, b: Φ(a) + Φ(b) = Φ(gcd(a, b)) + Φ(lcm(a, b)). The routes to a and to b share the trunk Φ(gcd), and swapping their branches closes a parallelogram at the lcm. Consecutive integers are coprime, so for n and n + 1 the trunk is {0}: their routes split at the origin.

For a window L > 0, let T_L = ℝ² / (Lℤ)² and Φ_L = Φ mod L, with coordinates φ = log₂ n / L and θ = Ω / L. The boards use L = 7, so that 2^L = 128.

Φ_L is a monoid homomorphism into the group T_L. For L = 7 it is injective on {1, …, 127} and sends 128 to the same point as 1. The helix of a prime p is the curve k ↦ (k·log₂ p, k) mod L, passing through Φ_L(p^k) at integer k.

For integer L, the image of Φ_L lies on the L rings θ = j/L, because Ω is an integer. Every ring is dense. By the prime number theorem p_k+1/p_k → 1, so log₂ p_k+1 − log₂ p_k → 0 while log₂ p_k → ∞, which makes the primes dense on ring 1. Multiplying by a fixed m with Ω(m) = j − 1 carries that density to ring j.

Consequence: on the torus, the set of angular positions of the primes carries no information beyond density. Whatever the construction says about where primes fall has to be read in the unwrapped coordinate or in extra structure, not in the point set on the torus.

The horn embedding E : T_L → ℝ³ is E(φ, θ) = ((R + R cos 2πθ) cos 2πφ, (R + R cos 2πθ) sin 2πφ, R sin 2πθ), with major radius equal to minor radius. The pulled-back metric is (2πR)² [ (1 + cos 2πθ)² dφ² + dθ² ].

E is injective except on the circle C = {θ = ½}, which it collapses to the single point O = (0, 0, 0), and the pulled-back metric degenerates exactly on C. The flat metric dφ² + dθ² on T_L has no singular circle. So the pinch is a feature of the embedding, not of the intrinsic torus.

Every helix crosses C at k = L/2, whatever the prime. An integer reaches C exactly when Ω(n) ≡ L/2 (mod L), which can happen only if L is even. For L = 7 no integer reaches O. For L = 8, every n with Ω(n) = 4 (16, 24, 36, 40, …) is sent to O. So "no integer sits at the center" is a property of the odd window, not of the construction.

Dubrovin's 2-dimensional Frobenius manifold for the quantum cohomology of ℙ¹ has flat coordinates (t₀, t₁), potential F = ½ t₀² t₁ + e^t₁, metric η(∂₀, ∂₁) = 1, unit ∂₀, and product ∂₁∘∂₁ = e^t₁ ∂₀. Its fibre over q = e^t₁ is the algebra A_q = ℂ[x] / (x² − q), with η(a, b) the coefficient of x in ab. The WDVV equations hold automatically in dimension 2, so they impose no constraint here.

Dictionary: evaluate the family at q = n for n in N. Then t₁ = ln n = (ln 2)·log₂ n, so the horizontal axis of the plane is a flat coordinate of this Frobenius manifold, and the integer n is the point q = n.

(a) A_q is semisimple for every q ≠ 0 and nilpotent, ℂ[x]/(x²), at q = 0. So the degeneration locus is q = 0, the end log₂ n → −∞ of the axis.

(b) Over ℚ, A_n ≅ ℚ × ℚ exactly when n is a perfect square, which holds exactly when Φ(n) ∈ 2M, that is, when Φ(n) is twice a point of M. Otherwise A_n ≅ ℚ(√n), a field.

Proved: for squarefree n > 1, the odd primes that ramify in ℚ(√n) are exactly the odd primes dividing n. For n = p prime, the only odd ramified prime is p itself, and in ℤ[√p] we have p = (√p)², so the prime reappears as the square of an element, which is x∘x = n.

Proposed: this is the precise form of "a prime re-enters only at itself". The algebra at q = n has x∘x = n and is ramified exactly at the prime factors of n.

Tensor products of Frobenius manifolds are Frobenius manifolds (the quantum cohomology of a product, by Künneth). For a finite set S of primes, take one copy of the ℙ¹ model per prime with q_p = p. The fibre is ⊗_p∈S ℚ[x_p]/(x_p² − p) = ℚ(√p : p ∈ S), a multiquadratic field of degree 2^|S| with Galois group (ℤ/2)^S. Proposed: this is the right higher-dimensional version of "one step per prime", since the free monoid on the primes becomes a product of one factor per prime.

For an odd prime ℓ not in S, the Frobenius element of ℓ in (ℤ/2)^S is the vector of Legendre symbols ((p/ℓ))_p∈S. By quadratic reciprocity, (p/ℓ) = (ℓ/p)·(−1)^{((p−1)/2)((ℓ−1)/2)}, so where ℓ falls is read off from its residues modulo 4 and modulo each p in S. This is where the position of a prime acquires actual content (Dirichlet, Chebotarev). Proposed: this is the mechanism that connects the construction to where primes fall. It is not visible in the current boards.

Depth. Nothing yet places Ω in the Frobenius structure. A candidate is the Euler field E = t₀∂₀ + 2∂₁, which gives q degree 2, but nothing derived from Ω supports it.

Which 0. One candidate is q = 0 (the nilpotent algebra, at the end of the log axis, 4.3a). The other is the pinch circle θ = L/2 (an embedding feature that depends on L, 3.2 and 3.3). A formal statement has to choose between them or relate them, for instance by letting L → ∞, which sends the pinch to infinity.

Periodicity. q = e^t₁ is periodic in the imaginary direction of t₁ (the phase of q), while the torus wraps the real direction (log n). Matching them needs complex n or a different identification of the torus with the base.

A picture of 5.2: primes placed by their Frobenius vector instead of by size.