The Constant the Cascade Does Not Give: Kolmogorov's Similarity Hypotheses, the −5/3 Spectrum They Return, and the Amplitude They Leave Fitted

Census pricing (row CC-120) of the founding step of Kolmogorov's 'The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers,' Dokl. Akad. Nauk SSSR 30, 301-305 (1941), against a fixed bench (what is postulated, what is derived, what the postulates return numerically, what they cannot reach). The step is two similarity hypotheses stated as hypotheses -- local isotropy, and the small-scale statistics uniquely determined by (nu, eps) alone, and in the inertial range by eps and the scale alone. Verified in the notebook (8 checks, exit 0) from declared bench inputs (nu = 1e-6 m^2/s, eps = 0.01 m^2/s^3, L = 1.0 m -- instrument inputs, not named flows): the microscales eta = 1.0000e-4 m, tau_eta = 0.0100 s, v_eta = 0.0100 m/s with Re_eta = 1.000000 identically; the scale separation L/eta = 10000.000 equal to Re_L^(3/4) at Re_L = 215443.469; the spectrum's decade fall exactly 0.021544. Remainders at full size: every amplitude is fitted -- E(k) = C eps^(2/3) k^(-5/3) returns its values only under C = 1.5 (labelled FITTED), and across the constant's recorded range (1.5 against a simulation record's 1.68) the amplitude at k = 100 /m moves 12.0% (3.2317e-5 to 3.6195e-5 m^3/s^2) while the exponent does not move; the structure functions pair a fitted C2 = 2.0 with the derived four-fifths coefficient -4/5 (Karman-Howarth, credited at its own address), a grade contrast the hypotheses cannot draw; the higher-order exponents depart from p/3 in the record (intermittency), a correction family the 1941 framework does not contain. Scout-grade verification disclosed on the manuscript's face: no verbatim quotation, no page-specific claim. EXTENSION. Vacuum catastrophe NOT SOLVED; quantum gravity NOT SOLVED.