Canonical Pair Observables in Weil Blocks: Structure of the Residual Amplitude Factor and Pipeline-Independent Formulation

Abstract

The O12–O18 sequence identifies the pair-level quantity ${\sigma_{\mathrm{pair}}}(n) = \sigma_{c}(n)\,\sigma_{q-c}(n)$ as the structurally motivated candidate observable on ${\mathrm{Heis}_{3}(\mathbb{Z}/q\mathbb{Z})}$, with capacity exponent ${\delta_{\mathrm{pair}}} \approx 7.44$ consistent with the phenomenological target ${\delta_{\mathrm{pair}}} \in [7.4, 10.6]$. However, a residual amplitude factor ${r(c,q)} \in \{1,2,3,4\}$ persists in the current pipeline, reflecting a dependence on the Gram–Schmidt initialisation parameters $(b_{1}, b_{2})$ identified in O17 but not yet resolved. The present paper begins by making explicit the implicit normalisation factor $D(c, b_{1}, b_{2}, n)$ of the O12/O13 implementation, separating its dependence on the initial basis, the block instance, and the normalisation convention. We then determine the dependence structure of ${r(c,q)}$ by two independent tests: variation under basis change at fixed block instance (Test P2), and variation under block-instance change at rigidly fixed basis (Test P3). Three outcomes are possible: ${r(c,q)}$ is a pure basis artefact (Case A, eliminable by canonical choice), a discrete instance invariant (Case B, requiring a canonical quotient), or a mixed quantity with both components (Case C, requiring a decomposition $r = r_{\mathrm{base}}\,r_{\mathrm{inst}}$). We identify which case holds for the primes tested, relate the observed structure to the metaplectic central-phase coherence ${\ell_{\gamma}}(n)$ identified in O12, and construct the corresponding canonical observable ${\sigma_{\mathrm{pair}}^{\mathrm{can}}}(n)$ independent of pipeline conventions. The exponent ${\delta_{\mathrm{pair}}}$ is unaffected in all cases; the contribution of this paper concerns the amplitude structure, which is a prerequisite for the fibre-level derivation of the ${\delta_{\mathrm{pair}}} \to {\beta^{*}}$ relation deferred to O21.