From Gauge Structure to Coupling Strength: Projection Density and the No-Fit Criterion for Dimensionless Gauge Parameters: A Conditional Application to the Quantized Dimensional Ledger

The Standard Model gauge group SU(3)_C × SU(2)_L × U(1)_Y specifies the structural form of the strong, weak, and hypercharge interactions, but it does not by itself determine the numerical gauge-coupling strengths. This paper formulates a no-fit criterion for claims that move from gauge-sector structure to the derivation of dimensionless interaction strengths. Let S_gauge denote structural gauge information, including the gauge group, matter representations, hypercharge assignments, and anomaly constraints. Let C(mu) = (alpha_3(mu), alpha_2(mu), alpha_1(mu)) denote the gauge-coupling vector at a specified renormalization scale, scheme, field content, threshold convention, and U(1)_Y normalization. The central result is S_gauge does not imply C(mu) unless an additional independently justified rule supplies coupling boundary data or a coupling-density map. The no-fit criterion has two levels. Input-level non-circularity requires that observed gauge couplings, coupling ratios, inverse couplings, fitted unification scales, fitted threshold corrections, or algebraically equivalent target data not appear as inputs to a claimed derivation. Justification-level non-circularity requires that auxiliary assumptions, weights, incidence rules, normalization factors, or coupling maps not be selected solely because they reproduce the observed coupling vector. The paper distinguishes three epistemic achievements: structural representation; empirical calibration; non-circular derivation. A theory may successfully represent the gauge-sector structure while leaving the numerical interaction strengths undetermined. A calibrated construction may reproduce observed couplings, but it does not become a derivation unless the relevant mapping rules are fixed independently of those observed values. To make the problem mathematically explicit, the paper defines a normalized gauge-sector projection density over a finite admissible ledger. For configurations omega in a finite set Omega, non-negative weights w(omega), and sector-incidence functions n_a(omega), the projection density is Pi_a =[sum over omega of w(omega)n_a(omega)]/[sum over sectors b and configurations omega of w(omega)n_b(omega)]. The vector Pi = (Pi_3, Pi_2, Pi_1) is non-negative, normalized, and lies in the two-simplex. Once Omega, w, and n_a are independently specified, Pi_a is a finite computable object rather than an interpretive metaphor. A projection density is not itself a gauge coupling. A complete proposal must additionally specify a coupling map g_a(mu_0) = F_a(Pi_a, C_a, R), where C_a denotes independently fixed group-theoretic information and R denotes scale, renormalization, matching, and normalization conventions. Both the projection density and the coupling map must be fixed before comparison with observed coupling data. The paper illustrates the method using benchmark M_Z-scale Standard Model couplings and a simple generator-multiplicity projection based on the adjoint dimensions 8, 3, and 1. The resulting projection vector does not reproduce the benchmark direct coupling-density target. This mismatch is treated as a useful rejection of a naive structural hypothesis, not as a failure to be hidden by additional fitted factors. The analysis emphasizes that gauge-coupling predictions must specify the renormalization scale, renormalization scheme, active field content, threshold assumptions, matching conventions, and hypercharge normalization. There is no convention-free numerical triple that can simply be called the Standard Model gauge couplings. The final section applies the criterion conditionally to the Quantized Dimensional Ledger. QDL is not assumed to have derived the gauge couplings. To pass the no-fit test, a future QDL construction must independently derive a finite closure ledger, configuration weights, gauge-sector incidences, a projection-density vector Pi_a^QDC, and a fixed map from that vector to g_a(mu_0), with the observed couplings used only for posterior comparison. The paper does not derive g_3, g_2, or g_1. It does not derive gauge coupling unification, beta functions, threshold corrections, or the Standard Model gauge group. It supplies a computable bridge object, a falsifiable comparison procedure, and a non-circular methodological standard for evaluating future coupling-strength derivations. This record forms part of the QDL/QDC Completion Series and defines the dimensionless gauge-parameter and no-fit gate of the completion program.