States as Direction and Calibration: A Triadic Decomposition of the Quantum State

This paper presupposes three papers — the foundations paper, "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811); the unified-description paper, "A Unified Description via the Triadic Structure" (DOI: 10.5281/zenodo.21131636); and the Tomita–Takesaki paper, "Selection and Heat" (DOI: 10.5281/zenodo.21161282) — and decomposes the quantum state itself in the vocabulary of the triad: direction and calibration. The scope is the minimal case (the two-level system); d > 2 is explicitly reserved. The claims of this paper are the theorems of the decomposition and a dictionary; no unification of dynamics and no new predictions are claimed. The central results are the following. First, the three degrees of freedom of the two-level state ρ = ½(id + r n·σ) are exhausted, with nothing left over, by the direction n (2) and the strength r (1) (Proposition 1.1). This is the same counting 3 = 2 + 1 as Proposition 2.5 of the unified-description paper — the three degrees of freedom of the pointwise metric are exhausted by direction (the two null directions) and calibration (a positive factor) (Corollary 1.2; the counting is a theorem on both sides, the identification is a reading). Second, for 0 < r < 1 the modular flow of the state is a rotation of the Bloch sphere about the axis n with angular speed 2 artanh(r) (Proposition 2.1): the direction selects the axis of time, and the strength sets its speed (Remark 2.4). At r = 0 the flow vanishes; for the thermal state, r = tanh(βε) and the speed is 2βε — the calibration encodes the inverse temperature (Proposition 2.3). Third, the update between two selections is implemented by the Connes cocycle u_t = ρ_ω^{it} ρ_φ^{−it} (Proposition 3.1 and Remark 3.2; the properties are theorems, the identification carries the tag). Fourth, non-selective measurement projects the direction and shrinks the strength, while selective measurement jumps to r = 1 — outside faithfulness. The mechanism lives on 0 < r < 1 and degenerates at both ends (no selection; pure selection) (Remark 4.2, the naming of a condition). What is a theorem, and what is an identification, a reading, a citation, or a reservation, is stated explicitly in the demarcation table. Theorem-level claims are machine-verified (verify_bloch_2plus1.py 8/8, verify_flow_axis_speed.py 8/8, verify_update_cocycle.py 7/7; numpy/scipy).