Standardized-cell Z-ratio build and Cauchy-slab constants#162
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The between-graph normalizer ratio is invariant under the diagonal congruence Theta = A K A, so it depends on (delta, eta) with eta = pairwise_scale * scale_rate, and the estimator's fixed quadrature grids are sized for the sigma = 1 frame. All call sites now resolve the constants at (delta, sigma = 1, beta = eta) via zratio_cell_constants(), and the calibration oracle receives the same cell. At the previous bare-scale build a scale-2.5 fit read the saddle 0.10 low in log Z(Gamma-)/Z(Gamma+) and the q = 6 hierarchical graph marginal sat at 0.292 for a 0.30 edge prior; unit-scale cells are bit-identical before and after.
Under the hierarchical specification a Cauchy interaction prior reused the Normal-slab Z-ratio constants at the base scale, which broke the graph-law identity (q = 6 marginal 0.247 for a 0.30 edge prior). The per-graph normalizer for the Cauchy slab is the marginal-Cauchy constant Z_C(Gamma) = E_omega[Z_N(Gamma; sigma sqrt(omega))], which keeps the omega updates conjugate and the between-move structure unchanged; only the constants and the calibration oracle are family-specific. zratio_constants() gains a slab argument. The Cauchy builders swap the slab density in the pair integral and mix each channel over the IG(1/2, 1/2) leg weights through the exact resolvent-form rule E[omega^k (A + B omega)^-(k+1/2)] = sqrt(2/pi) sum w_i (2 t_i A + B)^-(k+1/2) on the plain Gauss-Laguerre grid; the clique-2 chain runs omega-augmented with leg-dressed block moments. The C++ block-Gibbs oracle gets the same augmentation (conjugate block-edge weights, fresh per-sweep leg weights) behind a slab flag carried in zratio_spec and the audit interfaces. Verification: node, bridge, and psi0 constants within 0.14% of direct Monte Carlo of the same F-measure moments (standing tests in test-zratio-cauchy.R); the omega-augmented oracle sits at the additive prediction on the exact no-edge baseline block; the q = 6 Cauchy hierarchical graph marginal reads +0.0005 (adaptive-metropolis) and -0.0038 (gibbs) against the 0.30 target; the with-data Cauchy battery (three update methods, calibration window, alarm suite) passes.
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The hierarchical Z-ratio depends only on (delta, eta): the diagonal congruence factors the slab scale out of every between-graph ratio, and the quadrature grids are sized for the standardized (unit slab scale) frame. zratio_constants() now takes (delta, eta) and fixes sigma = 1 internally, so an off-frame build is unrepresentable rather than merely avoided by convention (Fix A already pinned every call site). The spec list, the C++ oracle params (set_oracle_params/enable_calibration), and the audit/test interfaces carry eta in place of (sigma, beta). The reference-builder parity test now matches the two sigma = 1 fixture cells (the sigma = 2 cells are the same eta = 1 physical cell up to grid error and are still covered by the engine-level reproduction test); the online-calibrator test runs at eta = 1 and still beats additive-only. Also removes the orphaned laguerre_half quadrature kind, left unused when the omega mixture was rewritten to the exact resolvent-form rule on the plain Gauss-Laguerre grid.
Slow-gated standing check that the hierarchical Cauchy spec reproduces the edge prior in the coupling regime (q = 15, 20; p_inc = 0.5, 0.7; both update methods), where the single-edge memo flags the mixture slab as coupling-sensitive. The deployed system (additive saddle + warm-up OLS correction, default window engaging at p >= 15) holds the marginal to max |dev| = 0.0033. Asserts the graph-law marginal; the per-block alarm verdict is expected to flag at dense high q (it does for the Normal slab too) and is not asserted.
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Summary
Two coupled fixes to the hierarchical-spec per-edge Z-ratio estimator, both surfaced by the Cauchy-slab graph-law miscalibration.
Fix A — build the constants in the standardized cell. The between-graph normalizer ratio is invariant under the diagonal congruence
Theta = A K A, so it depends only on(delta, eta = pairwise_scale * scale_rate); the estimator's quadrature grids are sized for thesigma = 1frame. Every call site (and the calibration oracle) now resolves the constants at(delta, sigma = 1, beta = eta)viazratio_cell_constants(). At the previous bare-scale build anormal_prior(scale = 2.5)fit read the saddle 0.10 low inlog Z(Gamma-)/Z(Gamma+), and the q = 6 hierarchical graph marginal sat at 0.292 for a 0.30 edge prior. Unit-scale cells are bit-identical before/after.Fix B — slab-family-specific constants for the Cauchy prior. The Cauchy slab reused the Normal-slab constants at the base scale, which broke the graph-law identity (q = 6 marginal 0.247 for a 0.30 edge prior — the omega scale-mixture weight never entered the normalizer path). The hierarchical Cauchy normalizer is the marginal-Cauchy constant
Z_C(Gamma) = E_omega[Z_N(Gamma; sigma sqrt(omega))], which keeps the omega updates conjugate and the chain machinery unchanged; only the constants and the calibration oracle are family-specific. Each channel is mixed over the IG(1/2, 1/2) leg weights through an exact resolvent-form rule; the C++ block-Gibbs oracle runs omega-augmented with leg-dressed moments.Validation
tests/testthat/test-zratio-cauchy.R.tests/testthat/test-hier-zratio-identity.R.R CMD check --as-cranclean.Analysis of record:
dev/plans/active/2026-07-06_zratio-standardized-cell-cauchy_ANALYSIS.md.