>[!warning] >This content has not been peer reviewed. # Methodology protocol — Graph rule search A priori protocol for scientifically rigorous search. Defined before execution. --- ## 1. Success criteria ### Primary (full robustness) - **d_B** in [1.19, 1.25] (1.22 ± 0.03) - **d_s** in [1.31, 1.35] (4/3 ± 0.02) - **Multi-seed:** N ≥ 3 seeds, all converged, all d_s in [1.29, 1.36], mean d_s in band, variance < 0.15 - **Success claim:** Only when all above are met ### Secondary (documented single-seed) - If no config achieves primary within resource budget: report best config with explicit caveat - **Caveat text:** "Single-seed result (seed X); not robust across seeds. See robustness results." - Do not claim "found" without this qualification --- ## 2. Search procedure ### Phases (order fixed) 1. baseline 2. dw_diagnostic 3. ledger 4. ledger_balanced 5. ledger_d_s 6. ledger_p_init 7. ledger_hybrid_I 8. ledger_anneal 9. ledger_fine 10. ledger_search 11. ledger_quasicrystal 12. extended_search, biconnected, two_phase, spectral_fit, alt_rules, relaxed_long ### Rules - No post-hoc addition of phases to "rescue" a run - Log all phase results (positive and negative) - Multi-seed robustness is final validation before any unqualified "found" claim --- ## 3. Measurement settings | Param | Value | Notes | |-------|-------|-------| | L | 24 | Lattice side (finite-size effects apply) | | T | 2000 | Evolution steps | | measure_interval | 150 | Steps between dimension measurement | | n_walks | 800 | Spectral dimension random walks | | max_steps | 80 | Max steps per walk | | converged_n | 2 | Consecutive measurements within threshold | | eta_mode | ledger | μ from solve_fidelity_equilibrium | --- ## 4. Theoretical baseline **3D bond percolation at p_c:** - **d_B** ≈ 1.22 (Stauffer & Aharony) - **d_s** = 4/3 (Alexander-Orbach; proven for d≥6; ≈1.33 for d=3) Our target bands match this theoretical baseline. See [[expanded theory/knowledge/The backbone dimension at the percolation threshold]]. --- ## 5. Train / validation - **Search seeds:** 42, 123, 456 (tuning / phase selection) - **Holdout seeds:** 789, 2024, 3141 (final validation, never used for tuning) - **Report:** "Config selected on seeds {42, 123, 456}; validated on holdout {789, 2024, 3141}" --- ## 6. Scientific verification workflow (FSS) For a scientifically relevant result, use finite-size scaling with a **scale-invariant** rule: 1. **Rule:** `ledger_scale_invariant` — intensive $I_{ij}$ and $N$ only (no $L$, $n$, $|E|$). See [[Graph rule verification results]] for the extensive-noise bug diagnosis. 2. **FSS:** `python rst_graph_rule_fss.py --rule ledger_scale_invariant --d-method chemical` 3. **Evaluate:** $d_B(L)$ should flatten with $L$; PASS if $d_B(\infty) \in [1.17, 1.27]$. Outputs: `fss_extrapolation.png`, `fss_summary.txt`, `fss_data.csv`. --- ## 7. Limitations (stated explicitly) 1. **Finite L:** L=24 is far below literature (L~100–1000); dimensions may shift at larger L 2. **Seed dependence:** Some seeds land in bad attractors; no config yet achieves full robustness across all tested seeds 3. **Estimator bias:** Box-counting and spectral fits depend on ℓ-range and t-range; recursive μ can couple to estimator 4. **Initial condition:** Bond percolation at p_c; real universe may have different init 5. **No holdout (current):** If resource-limited, all seeds used for both search and report; document this