Situation. The trilogy *Identifying 3* gave three layers of the same object. L1 (https: //doi. org/10. 5281/zenodo. 20336872) supplied the native simplicial chart: keep the redundant coordinate, work in the mean-zero space, and read probability through the square-root sphere. L2 (https: //doi. org/10. 5281/zenodo. 20338126) supplied the generating functional: the categorical log-partition \ (= Z\), whose derivatives generate position, metric, cubic vertex, and the closure tower. L3 (https: //doi. org/10. 5281/zenodo. 20339049) supplied the arithmetic substrate: when the outcome labels carry \ (Z/N\) -structure, the character basis decomposes the mean-zero space into conductor packets and the cubic vertex obeys the exact selection rule \ a+b+c 0 N. \ Surprise. After the trilogy, the dimensional ladder itself reveals another hidden structure. If one descends an \ (N\) -slot simplicial ledger by arithmetic mean removal, then climbs back by appending the balancing coordinate, the process is lossless. The resulting coordinates are the unnormalized Helmert balances, with \ (k\) leading ones followed by the balancing entry \ (-k\): \ hₖ= (1, , 1, -k, 0, , 0), 1 k N-1. \ This is the metric trick in ladder form: keep the integer incidence skeleton, and store length in the metric instead of leaking square roots into the coordinates. Action. We formalize the reversible ladder, identify it with the unnormalized Helmert system, then deliberately remove the metric trick by orthonormalizing the ladder. The normalizers k (k+1) generate a multiquadratic field \ KN= Q (sf (k (k+1) ): 1 k N), \ where \ (sf\) denotes squarefree kernel. Result. The field generated by the normalized ladder up to rung \ (N\) is exactly \ KN= Q (p: p N, \ p prime). \ Hence the field grows precisely on prime rungs: \ KN K₍-₁ N is prime. \ The statement is structural: primes are the rungs at which orthonormal measurement of the simplicial ladder forces a new quadratic square-root generator. The metric trick hides this arithmetic in the metric; unit normalization exposes it as radical residue. Harvest. L4 therefore opens a fourth door after *Identifying 3*: the same mean-zero space that supports Fisher-Rao geometry, cumulant tensors, and conductor packets also supports a reversible balance ladder whose metric normalization externalizes number-theoretic square classes. The positive-cone version is the geometric-mean gauge, with multiplicative gauge removal corresponding to additive gauge removal under. This connects simplicial coordinates, Aitchison log-ratio geometry, Fisher-Rao information geometry, and the squarefree arithmetic of prime-generated quadratic fields.
Leonardo Murillo Montero (Mon,) studied this question.