Precision Quotient 1: Noise-Weighted Identification on a Non-Abelian Network Fibre

An observation protocol can identify a model algebraically and still fail to identify it at any usable precision. This report makes that distinction explicit on a six-vertex SO(3) network carrier. Gaussian block noise turns each protocol into a Fisher geometry and gives two complementary precision-dependent reductions: a local tangent-space quotient determined by whitened singular values, and an exact finite quotient of the two-point residual fibre. The structural part is proved rather than sampled. A single spectral response map is 2 by 3 and therefore has a forced null direction. For the selected two-probe protocol, four norm-compatible branches reduce to exactly two rotations; the mixed branches violate inner-product preservation by 0.9534915. The local and global notions then separate sharply. The two-probe protocol has full local Fisher rank, yet its alternatives induce identical distributions: their Kullback-Leibler divergence is zero, so no estimator can beat chance at any noise level or sample size. Nevertheless their held-out response at a fourth hidden vertex differs by 0.00488535. A spectral extension separates the pair only through a quadratically opening direction, whereas a third spatial probe opens linearly. A fixed-direction audit over two decades gives an added-probe advantage proportional to the inverse symmetry-breaking amplitude, with fitted exponent -1.0057. A separate 5,000-carrier audit varies direction at nearly fixed amplitude; its heavy upper tail is caused principally by a small quadratic spectral denominator. These are protocol-design results, not claims of universal optimality or thermodynamic irreversibility. This deposit contains the manuscript in PDF and DOCX form together with the complete evidence archive: all source code, generated CSV and JSON tables, 600 dpi figures, pinned environment versions, an independent verifier and SHA-256 checksums.