Quantum non-commutativity, complex amplitudes, and the squared-modulus Born rule are usually taken as postulates. This paper shows that on the finite eight-tick recognition ledger they are forced consequences of integer arithmetic on the cycle ℤ/8. Occupation advance and cost-rate phase are the shift and clock generators of the finite Heisenberg-Weyl group; their failure to commute is exactly multiplication by a primitive eighth root of unity, CS = ωSC with ω = e^(2πi/8) and ω ≠ 1 because ω⁴ = −1 (Theorem, unconditional). The two-tick quarter-turn of the cycle squares to minus the identity, which forces the scalar field of the state space to be ℂ and not ℝ or ℍ (Theorem, under a named quadrature premise). Given the companion-forced geometric measure w(n) = φ⁻ⁿ on the discrete grading, the modulus is the unique half-density |ψ(n)| = φ^(−n/2), and the Born exponent is forced to q = 2 by the identity that |1 + e^(iΔ)|^q + |1 − e^(iΔ)|^q is phase-independent if and only if q = 2 -- a dimension-two statement, precisely where Gleason's theorem is silent (Theorem, under the single quantum premise of linear superposition). The clock and shift eigenbases are mutually unbiased with all overlaps exactly 1/8, the finite signature of position/momentum complementarity, and a native two-path recognition interferometer reproduces cosine fringes with probability conserved across complementary ports, with no continuum variable anywhere. The residual amplitude phase is identified as a U(1) symplectic-holonomy cocycle over the recognition history groupoid -- its home a theorem, its value a single named open frontier. The continuum commutator [x, p] = iℏ and the magnitude ℏ = φ⁻⁵ are stated openly as open, not as derived results. The stance is finite-first, not finite-only: finite Heisenberg-Weyl kinematics is a complete, standard quantum theory in its own right, and the continuum is a separate limit pursued in the sequel. Every headline claim is graded as theorem, identification, or conjecture.