The Impedance of Information — One Möbius Coordinate for Decoherence, Entropy and Extractable Work: Measured from a Superconducting Qubit to a Black-Hole Ringdown; Two Predictions Refutable Against Existing Literature; and a Self-Audit of the Framework That Produced Them

The coordinate is an impedance. This version proves that the framework's reduced coordinate and the Smith chart of transmission-line engineering are the same Möbius transformation, and that the binary entropy is its primitive. It also corrects three errors of fact in the deposited text, one of which inverted the reading of a 38-event table. 81 pages. Eloy René Becerra Daly Proved With Γ = 2x−1 and z = (1+Γ) / (1−Γ): z = x/ (1−x), ln z = logit x, z (1−x) z (x) = 1, 1−Γ² = 4x (1−x), and e^IQ = 1+z — six identities, symbolically. The binary entropy satisfies dH₂/dx = −ln z: the coordinate is the derivative of the entropy, sign-reversed, and its curvature diverges exactly where transmitted power vanishes. The one-bit threshold sits at Γ = 0, which is impedance matching. The chart is Smith's and the line physics Heaviside's; what is contributed is that the two coordinates are one. Measured The asymmetry separating the two channels of a qubit is not a free parameter: it equals tanh (ℏω/2kBT), verified symbolically, which is 1. 000000 at the 15 mK of the cryostat and 0. 030 at 4 K. The arrow of that channel is thermal and switches off on heating — a temperature sweep on the same apparatus is the declared falsifier. Corrected Three errors of fact in the deposited text. The body quoted the LIGO exponent as 0. 931 ± 0. 003 at 13. 8σ while the paper's own erratum had already corrected it to 0. 93113 ± 0. 00551 at 12. 5σ. The remnant table used the cosmological mean density for a terrestrial rate without saying so. And the sign convention of ΔAIC was stated inverted relative to the tabulated column, which reversed the reading of all 38 ringdown entries. Registered Entry D24 — the framework writes σ for the entropy and computes H₂ (D) — carried no number until now and now carries the worst available one: under amplitude damping the two separate by 0. 6931 nats, the whole range of a qubit entropy, and at full application the state is pure with S = 0 where the framework reports its maximum. D43 moves to partially answered: the ringdown fits are admissible (χ²/dof median 1. 023, none above 2), the hardware ones are not. D45 opens: the reduced coordinate is intensive and cannot represent an object whose entropy is extensive. Forty-six entries, nineteen open. What is not claimed The passive-state construction is Pusz–Woronowicz and Lenard's, the cycle efficiency Carnot's, and the kBT ln 2 per erased bit Landauer's. What this framework adds is the criterion for which channel satisfies the equation, never the equation.