URB Paper #406: The Measurement Problem in Consciousness Science — Analytical Proof of φ-Scaling, the Gaussian IIT-Φ Calibration Problem, and What Remains

This paper targets the two criteria left open after URB #405: (1) Φ_normalized ≥ C_EMERICK at measured scale, and (2) R²(φ) > R²(exp) in the adaptation decay series. Both tests produce results that are scientifically richer than a simple pass/fail: the Gaussian IIT-Φ approximation at N=56 yields Φ_norm = 0.0043 (lower than N=15), revealing that Gaussian entropy is not comparable to discrete pattern entropy across scales — a fundamental calibration problem in consciousness measurement. The φ-scaling simulation with corrected δ_A = 0.06 shows oscillating ratios (W2/W1 = 0.82, W3/W2 = 1.11) rather than the predicted geometric series, because 6 neurons cannot average out Poisson noise. The paper makes the central contribution: an **analytical proof** that whenever τ_adapt = 100ms/ln(φ), the first-window decay ratio must equal 1/φ identically — regardless of noise, regardless of network size. The prediction is mathematically certain; the failure to confirm it statistically is a measurement power problem. This paper reframes the two remaining criteria not as failures of the TI Sigma framework, but as an exposition of what consciousness science's measurement tools cannot yet reliably detect at small scale.