Hypergraph Lumpability and Bayesian Refinement

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1 \begin{longtable}{@{}p{0.08\linewidth}p{0.43\linewidth}p{0.12\linewidth}p{0.12\linewidth}p{0.08\linewidth}@{}}
2 \caption{Executed deterministic finite certificate ledger.}\label{tab:executed-ledger}\\
3 \toprule
4 ID & Check & Value & Threshold & Status\\
5 \midrule
6 \endfirsthead
7 \toprule
8 ID & Check & Value & Threshold & Status\\
9 \midrule
10 \endhead
11 Q1D & order 1 quotient defect & $1.19\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
12 Q1K & order 1 hidden input residual & $5.38\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
13 Q1R & order 1 coarse tensor recovery & $8.41\times 10^{-17}$ & $\leq 10^{-10}$ & PASS\\
14 Q2D & order 2 quotient defect & $3.76\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
15 Q2K & order 2 hidden input residual & $5.71\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
16 Q2R & order 2 coarse tensor recovery & $5.55\times 10^{-17}$ & $\leq 10^{-10}$ & PASS\\
17 Q3D & order 3 quotient defect & $8.39\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
18 Q3K & order 3 hidden input residual & $1.15\times 10^{-15}$ & $\leq 10^{-10}$ & PASS\\
19 Q3R & order 3 coarse tensor recovery & $1.11\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
20 QNEG & a perturbation of one coefficient violates fiber constancy & $0.125$ & $\geq 10^{-1}$ & PASS\\
21 QSYM & symmetric interaction remains symmetric after exact lift & $0$ & $\leq 10^{-10}$ & PASS\\
22 QHID & projected hidden tensor has zero coarse output & $0$ & $\leq 10^{-10}$ & PASS\\
23 QTRN & exact quotient transitivity across two aggregation levels & $4.84\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
24 QMLT & interaction defect identity across two levels & $9.46\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
25 QFLD & fine vector field factors through the coarse field & $2.22\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
26 QPRD & exact aggregate prediction commutation & $7.22\times 10^{-16}$ & $\leq 5\times 10^{-10}$ & PASS\\
27 QCOC & exact commutation of the prediction and update cocycle & $3.89\times 10^{-16}$ & $\leq 5\times 10^{-10}$ & PASS\\
28 QEVD & exact evidence aggregation at one step & $6.66\times 10^{-16}$ & $\leq 5\times 10^{-10}$ & PASS\\
29 QMAS & fine nonlinear trajectory preserves total mass & $7.77\times 10^{-16}$ & $\leq 5\times 10^{-10}$ & PASS\\
30 QPOS & fine nonlinear trajectory preserves nonnegativity & $0$ & $\leq 5\times 10^{-12}$ & PASS\\
31 QBAY & a likelihood constant within each fiber gives exact Bayesian refinement & $2.78\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
32 QBEV & a likelihood constant within each fiber preserves evidence & $2.22\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
33 QSCL & Bayesian update is invariant under global likelihood scale & $1.53\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
34 QFDB & controlled dynamical perturbation respects the flow certificate & $0$ & $\leq 2\times 10^{-9}$ & PASS\\
35 QFSL & flow discrepancy has first order perturbation scaling & $3.37\times 10^{-3}$ & $\leq 0.08$ & PASS\\
36 QLPS & perturbed likelihood remains strictly positive & $0.8$ & $\geq 10^{-12}$ & PASS\\
37 QOBB & variation of the likelihood within a fiber respects the Bayesian bound & $0$ & $\leq 10^{-12}$ & PASS\\
38 QBSC & the Bayesian perturbation sweep remains below the likelihood certificate & $0$ & $\leq 10^{-12}$ & PASS\\
39 QBSL & Bayesian discrepancy has first order scaling for small fiber mismatch & $4.63\times 10^{-5}$ & $\leq 0.08$ & PASS\\
40 QNMB & symmetric normalization bound uses the larger positive mass & $0$ & $\leq 10^{-12}$ & PASS\\
41 QPST & recursive posterior certificate dominates every joint perturbation run & $0$ & $\leq 10^{-10}$ & PASS\\
42 QPSL & joint posterior discrepancy has first order scaling for small perturbations & $1.23\times 10^{-4}$ & $\leq 0.12$ & PASS\\
43 QLEV & the evidence certificate at each step dominates the accumulated discrepancy & $0$ & $\leq 10^{-10}$ & PASS\\
44 QPAR & hyperedge energy satisfies the scalar Parseval identity & $8.88\times 10^{-16}$ & $\leq 10^{-10}$ & PASS\\
45 QBST & the residual from retaining the largest energy hyperedges decreases to zero & $0$ & $\leq 10^{-10}$ & PASS\\
46 \bottomrule
47 \end{longtable}

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