The Forced Amplitude The Born Modulus is the Square Root of the Recognition Measure; the Phase is the Residual Freedom

The Born rule, that the probability of an outcome is the squared modulus of its amplitude, is the last independent postulate of quantum mechanics. The standard derivations fix the exponent from the amplitude: Gleason from the lattice of projections (in dimension at least three), Zurek from entanglement symmetry, the decision-theoretic program from rational preference. All of them take the amplitudes, and the Hilbert space they live in, as given. This paper inverts the order inside the recognition framework, where a companion result forces the probability measure on the discrete recognition grading to be the geometric weight w(n) = φ⁻ⁿ, φ the golden ratio, with no free parameter. Given a measure that is itself derived, I ask what amplitude sits beneath it and how much of the Born rule is forced. Three things are. First, the half-density lift |ψ(n)| = φ^(−n/2) is the unique modulus, fixing the amplitude up to phase. Second, and the central result, the squaring exponent is forced: on the renewal tree of recognition histories, requiring that a symmetric probability-conserving measurement exist over a fiber of equal-cost histories, for every phase configuration, forces the exponent to be exactly two. The condition |1 + e^(iΔ)|^q + |1 − e^(iΔ)|^q is independent of the relative phase Δ if and only if q = 2; this is a dimension-two statement, where Gleason is silent, and its only quantum input is linear superposition of history amplitudes. Third, probability is conserved at every recognition branch by the grammar's Kraft equality φ⁻¹ + φ⁻² = 1, which is identically the normalization of the posting grammar, and the forced measure is the exact no-interference diagonal of the amplitude calculus. What is not forced is the phase. I locate it precisely as a U(1) symplectic-holonomy cocycle over the history groupoid, the oriented-area companion of the even cost, and state its computation as the single open problem. I separate theorem, identification, and conjecture throughout.