Minima of a Matchstick-Machine Amplitude on σ= 1/2: A Numerical Coincidence with the Riemann Zeros

A finite geometric game produces the primes, and a finite object built from those primes has its minima sitting on the imaginary parts of the non-trivial Riemann zeros. This paper documents that coincidence and develops the structure inside the machine that carries it. The game is played with matchsticks of one fixed length, laid one per step. Four rules, applied in fixed order, decide each move: close a shape if possible; else echo a previously closed template; else, if the configuration reads as an integer multiple of an earlier template, block the closure and echo instead; else open a new direction. The rules never mention divisibility, factorisation, or the word "prime". The positions at which a new shape closes for the first time are 2, 3, 5, 7, 11, 13, 17, 19, 23,. . . The primes are the output of the game, not an input. From the primes produced by the game we build a finite amplitude Zₖ (s) as a sum over the vertices of the sign hypercube (Z/2) ᵏ, using only the first k primes. The modulus |Zₖ| has isolated local minima along the vertical line sigma = 1/2, and the t-coordinates tₙ* (k) of these minima lie close to the imaginary parts gammaₙ of the non-trivial Riemann zeros. For k in 20, 30, 100, 300 and the first twenty positions, the mean distance is fitted by |tₙ* - gammaₙ| approx 0. 11 k^-1/2, reaching 0. 008 at k = 300. The two objects were built by entirely different means: tₙ* (k) comes from a finite sum of 2ᵏ terms over a discrete sign hypercube on the first k primes; gammaₙ comes from the analytic continuation of an infinite Dirichlet series. Their numerical alignment is what the paper documents at ranges of k from 20 to 5000 and across the first 120 zeros. The machine rests on four pillars, each with definite status inside the paper: - Completeness of the game. The four rules produce a well-defined step for every position, and the set of positions closing a new shape is exactly the sequence of primes. Proved by induction on templates and matchstick counts. - Symmetry. The amplitude satisfies the exact reflection identity Zₖ (1 - s) = Zₖ (s), so |Zₖ| is invariant under sigma 1 - sigma and the line sigma = 1/2 is an exact axis of reflection of the machine. Proved from the definition and made geometric by the Vieta product form. A direct corollary is that the temporal position of every on-line minimum is stationary under perturbations in sigma. - Self-similarity. Passing from stage k to stage k + 1 replaces the primorial by the next primorial, appends one new involution axis to the sign hypercube, and leaves the rules and the definition of Zₖ unchanged. The construction is uniform in k. - Convergence. The mean distance between the local minima of |Zₖ| and the imaginary parts gammaₙ follows the empirical law 0. 11 k^-1/2 across two orders of magnitude in k. The exponent 1/2 is the natural rate of a central-limit-style averaging over k contributions. The number 1/2 appears in four independent places inside the machine: as the exact axis of reflection, as the radial exponent pᵢ^-1/2 on the critical line, as the empirical convergence rate k^-1/2, and as the curvature ratio Tₖ'' / f'' = -1/2 (four decimals of numerical agreement at k = 100 and 300, where Tₖ is a machine-intrinsic remainder and f is the Vieta-product skeleton). The first two are exact identities. The third and fourth are numerical identities verified on the data. Around the coincidence, the paper develops the internal structure of Zₖ: - A modulus-phase decomposition Zₖ = M (s) exp (i S (s) ) that lifts Vieta's cotangent product to a general modular symmetric form. - A Vieta product identity and an additive Fourier expansion of log |Zₖ| with only odd harmonics of the primes. - A frequency-contrast test showing that log pᵢ is the frequency set that produces the alignment: replacing it with other arithmetic sequences destroys it. - A region-universality test across three windows of t and two prime sets. - A symmetrised companion Zₖ* (s) whose zeros lie on the critical line by construction and whose minima track the same gammaₙ to five decimals at k = 100. - Three internal diagnostics (A, B, C) that agree with the curvature identity Tₖ'' / f'' = -1/2 to four decimals in the tested regions. - An exact rank-one plus diagonal identity for the time-averaged Gram matrix of the odd-Fourier submoduli, proved as a closed-form theorem using Poisson-Fourier expansion, orthogonality of cosines under time-averaging, and multiplicative independence of the primes. One rank-one direction carries about 89% of the trace. - A machine-intrinsic depth score that ranks candidate matchings between minima and reference points; the Riemann zeros achieve the top score by a large margin against every alternative tested. The paper is standalone. Every claim is verified on the finite objects Zₖ, Zₖ*, and their explicit derivatives up to k = 5000 for selected diagnostics. All plots, tables, and derivations are reproducible from the definitions given inside the paper. The bridge to the classical zeros is a finite analytic statement about the curvature ratio between Tₖ and its Vieta-product skeleton. That is a single equation between explicit finite sums. It does not require the analytic continuation of anything. It can be attacked directly. Matchsticks in, minima out. The values are the Riemann zeros to within eight thousandths at k = 300. Keywords: Riemann zeta function, non-trivial zeros, critical line, prime numbers, combinatorial game, matchstick machine, finite Dirichlet amplitude, Vieta product, Fourier expansion, frequency contrast, symmetric companion, curvature identity, Gram matrix, rank-one asymptotics, geometric number theory, sigma = 1/2, k^-1/2 convergence, four pillars, completeness, self-similarity