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

This paper identifies and diagnoses the 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. 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 < delta_x) print(f"n_params={n_params}: within_delta={within:.1%}") Version 2 establishes a further structural consequence: the collusion does not merely facilitate agreement episode by episode, but generates a cumulative and irrecoverable debt of free parameters (k*) across theoretical generations. This debt is a mathematical necessity, not a historical contingency. Three lemmas ground the result: Representation Space Separation (Tier-3 operations cannot reach Nec(S)), Residual Persistence (any finite improvement of measurement resolution structurally generates truncation residuals that existing finite-parameter forms cannot absorb), and Tier-3 Closure (debt redemption is impossible within Tier-3). From these, the Debt Accumulation Theorem establishes k_m ≥ k_0 + m. Scientific progress and parameter debt are structurally identical processes: the expansion of explanatory domain, the improvement of predictive precision, and theoretical unification are each different descriptions of k* accumulation. The Epistemic Censorship Protocol ("Shut up and calculate") is the institutional norm that this debt structure makes necessary. The string theory landscape (k* → 10^500) is not an anomaly of overreach but the predicted endpoint of a trajectory established the moment calculus was applied to natural description (Leibniz, 1684). The following Python code illustrates the monotone growth of k* across Advance Events: import numpy as np import random random.seed(42) np.random.seed(42) k_star_0 = 6 # Newton's orbital elements per body n_steps = 30 p_ae = 0.6 # probability of Advance Event per step k_star = k_star_0 m = 0 print(f"{'Step':>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() < p_ae: delta = random.randint(1, 6) k_star += delta m += 1 print(f"{step:>5} {k_star:>6} {m:>4} AE +{delta}") else: print(f"{step:>5} {k_star:>6} {m:>4} RE (k* unchanged)") # Verify Theorem: k*_m >= k*_0 + m at every step 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) The replacement criterion of translatability, derived from Operatiology (操作原論) / Cognitional Mechanics (知能力学), is developed as the genuinely external standard for structural adequacy, immune to the collusion. Translatability is also the only mechanism by which structural debt is genuinely redeemed: a parameter established as the unique exclusive projection of Nec(S) is withdrawn from k*.