The Collusion of Significant Figures and Integration Constants: The Historical Origins of Falsifiability and Its Structural Transcendence

This paper identifies and diagnoses the \emph{collusion of significant figures and integration constants} (有効数字と積分定数の談合) as the structural mechanism underlying the historical formation and epistemological limits of falsifiability. The finite precision of any measurement (δx>0, irremovable) and the underdetermination introduced by integration constants (C∈R, free) together constitute a double degree of freedom that structurally facilitates agreement between theory and observation without guaranteeing it. Through a two-thousand-five-hundred-year genealogy from Babylonian arithmetic to matrix mechanics, the paper establishes that falsifiability operates exclusively within Tier-3 representation and cannot reach Tier-2 operational structure. The following Python code illustrates the core structural mechanism: as the number of free parameters increases, the absorptive capacity of the model increases monotonically, regardless of the data structure---the same operation performed by Ptolemy's epicycles. ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes *, ． denotes .] ◆◆◆ import numpy as np np．random．seed(42) x_obs = np．linspace(0, 10, 20) y_obs = np．random．uniform(0, 10, 20) delta_x = None for n_params in [1, 2, 3, 5, 10, 15, 19]: coeffs = np．polyfit(x_obs, y_obs, n_params) y_fit = np．polyval(coeffs, x_obs) resid = np．abs(y_fit - y_obs) if delta_x is None: delta_x = np．max(resid) within = np．mean(resid 5} {'k':>6} {'m':>4} {'Event'}") print("-" × 35) print(f"{'0':>5} {k_star:>6} {m:>4} INIT") for step in range(1, n_steps + 1): if random．random() 5} {k_star:>6} {m:>4} AE +{delta}") else: print(f"{step:>5} {k_star:>6} {m:>4} RE (k* unchanged)") print(f"\nFinal: k* = {k_star}, m = {m}, " f"lower bound k_0 + m = {k_star_0 + m}") print("k_m >= k*_0 + m:", k_star >= k_star_0 + m) ◆◆◆ [END] ◆◆◆