The Half-Line of a Prime Machine: Four Coincident Halves from One Antipodal Involution

There is a small matchstick game with four rules and a finite board. The rules do not mention primes, complex analysis, or the Riemann zeta function. Yet the game generates all primes and only primes. This paper starts from that fact and asks: what must be true of the machine for it to succeed? The answer is that the machine carries a single geometric axis, and this axis expresses the number one half in four independent ways — all traceable to one underlying operator. In the complex plane, the axis is the line Re(s) = 1/2. In the limit, all minima of the machine's natural complex function accumulate onto that line and nowhere else. The argument is fully internal to the machine. It uses no zeta function, no classical analytic input, no numerical curve-fitting. It reads the four rules, identifies the antipodal reflection as the load-bearing rule, and shows that four different notions of one half — a reflection axis, a radial exponent, a saddle-balance signature, and a recursive concentration rate — are four faces of the same idempotent operator. The paper closes with the statement that the half-line is the unique locus on which the minima can concentrate in the limit, and that dropping any one of the four halves would break the machine. What the paper is:- A self-contained geometric statement about one concrete prime-producing machine.- A structural argument that this machine cannot succeed without the four coincident halves.- A demonstration that the half-line and the four halves are not accidents but necessary conditions. What the paper is not:- A proof of the classical Riemann Hypothesis. The paper studies the machine's own complex function, not zeta directly.- A claim that nature realises this machine. Whether the identity between the machine and reality holds is an open question. The author believes, firmly and personally, that this identity is the right reading of what has been shown: that the primes are what they are because a structure with four coincident halves generates them, and that the half-line of the Riemann zeta function is the same axis seen from outside. Turning this belief into theorem is the direction of continued work. The paper is written to be read on its own. All definitions, all proofs, and all numerical diagnostics are self-contained. Keywords:prime numbers, critical line, Riemann Hypothesis, matchstick machine, primorial, involution group, sixteen-cell, cross-polytope, geometric number theory, coprime residues, antipodal reflection, structural forcing