{"report_id": "ring_lazy_jump", "group": "ring", "path": "research/reports/ring_lazy_jump", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.074079Z", "title": "Lazy Ring Jump-Over Mechanism (K2 vs K4)", "summary": "This report establishes the baseline jump-over mechanism for lazy rings with one directed shortcut, contrasting K=2 and K=4 under exact first-passage diagnostics. It explains why double-peak behavior is selective in shortcut strength and how pathway decomposition links peak structure to fast and delayed transport channels.", "key_findings": ["Jump-over pathways create a distinct delayed channel that is necessary for persistent second peaks.", "Shortcut strength has a non-monotonic effect: too weak or too strong settings both reduce robust bimodality.", "K=2 and K=4 share the same mechanism skeleton but differ in phase-window width and valley depth.", "Exact inversion and trajectory decomposition give consistent interpretations of the observed peak transitions."], "narrative": {"model_overview": "A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.", "method_overview": "The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.", "result_overview": "Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 28, "dataset_series_ids": ["scan_n_k4_beta002-probability", "scan_n_k2_beta002-probability", "scan_beta_n100_k4-probability"]}
{"report_id": "ring_lazy_jump_ext", "group": "ring", "path": "research/reports/ring_lazy_jump_ext", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.169399Z", "title": "Lazy Ring Shortcut Beta Scan", "summary": "This extension quantifies how shortcut strength beta reshapes first-passage bimodality on lazy rings at fixed N=100, then checks transfer by N sweeps and Monte Carlo class decomposition. Across beta in [0,0.2], both peaks move earlier and tail decay accelerates, while K=4 preserves a wider and deeper bimodal window than K=2.", "key_findings": ["At fixed N=100, beta sweeps show systematic left-shifts of peak times and a larger tail-decay rate as shortcut strength increases.", "K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched beta schedules.", "An anchored beta choice supports stable N sweeps where exact AW and MC class-level diagnostics stay consistent.", "Tail diagnostics confirm that shortcut strengthening suppresses late-time mass and changes pathway composition, not only peak height."], "narrative": {"model_overview": "The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched parameter settings.", "method_overview": "The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail diagnostics.", "result_overview": "Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 28, "dataset_series_ids": ["scan_n_k4_beta002-probability", "scan_n_k2_beta002-probability", "scan_beta_n100_k4-probability"]}
{"report_id": "ring_lazy_jump_ext_rev2", "group": "ring", "path": "research/reports/ring_lazy_jump_ext_rev2", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.268198Z", "title": "Lazy Ring Shortcut Figure-1 Revision", "summary": "This revision reorganizes the lazy-shortcut extension into a publication-ready evidence flow: co-located Fig.1 overlays f(t) with window-level class bars, while threshold, window-shift/width, and MC-uncertainty analyses test robustness. The update strengthens readability and reproducibility without changing the core mechanism claims.", "key_findings": ["Co-located Fig.1 directly aligns peak/valley timing with class proportions, improving interpretability over split-panel layouts.", "Schema-validated JSON/CSV inputs make figure reconstruction auditable and reproducible.", "Threshold and window perturbation scans preserve the main mechanism ranking rather than flipping conclusions.", "MC uncertainty intervals are compatible with the reported phase statements for K=2 and K=4."], "narrative": {"model_overview": "The chapter keeps the same lazy ring shortcut setup used in the extension baseline and focuses on clearer evidence alignment between K=2 and K=4 under the selected beta regime.", "method_overview": "The pipeline exports standardized Fig.1 inputs, validates schema, renders stacked-bar co-located panels, and runs three sensitivity tracks: threshold sweep, window perturbation, and MC confidence intervals.", "result_overview": "Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 40, "dataset_series_ids": ["ft_beta0-01", "luca_k2_fixed_shortcut_metrics-probability", "ft_schema-example"]}
{"report_id": "ring_lazy_flux", "group": "ring", "path": "research/reports/ring_lazy_flux", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.062079Z", "title": "Lazy Ring Flux Baseline", "summary": "This report studies first-passage distributions on a lazy ring with one directed shortcut drawn from self-loop probability. It combines Chebyshev generating-function derivation, AW/FFT inversion, and flux-recursion cross-checks to show a reproducible small-p bimodal regime, while equal4 and large shortcut strength suppress the second peak.", "key_findings": ["A minimal reproducible bimodal case appears at N=10 under small shortcut strength in the selfloop construction.", "AW inversion and flux recursion agree to numerical precision, validating both the derivation and implementation.", "Full N scans show macro-bimodality concentrated in specific geometry-distance bands rather than uniformly.", "Equal4 and large shortcut strength suppress late-time mass and remove the second dominant peak."], "narrative": {"model_overview": "The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.", "method_overview": "The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.", "result_overview": "A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality."}, "formula_count": 2, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 37, "dataset_series_ids": ["lazy_k2_equal4_paper_geometry_summary_cn-metric", "lazy_k2_equal4_paper_geometry_summary_cn-binary", "lazy_k2_equal4_paper_geometry_summary_cn-parameter"]}
{"report_id": "ring_valley", "group": "ring", "path": "research/reports/ring_valley", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.293081Z", "title": "Ring Valley Regime Map", "summary": "This valley-study report analyzes a non-lazy K-neighbor ring with a one-way shortcut 6->N/2+1 under the Fig.3 peak rule. Exact AW inversion and MC validation map bimodality windows across even N and K: no bimodality for K=2 after parity-aware coarse graining, but clear windows for K=4,6,8.", "key_findings": ["Under Fig.3 criteria with parity-aware treatment, K=2 shows no robust two-peak regime in the scanned range.", "Bimodality windows emerge for K=4, K=6, and K=8 with distinct N bands.", "Exact and MC pipelines agree on peak/valley timing and class-level shortcut usage trends.", "Trajectory-class heatmaps separate fast, valley, and indirect pathways and support the mechanism interpretation."], "narrative": {"model_overview": "The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut placement.", "method_overview": "The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.", "result_overview": "Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics."}, "formula_count": 6, "section_count": 10, "repro_command_count": 3, "dataset_count": 1, "asset_count": 10, "dataset_series_ids": ["bimodality_scan"]}
{"report_id": "ring_valley_dst", "group": "ring", "path": "research/reports/ring_valley_dst", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.311081Z", "title": "Destination-Scan Valley Control", "summary": "This report fixes N=100 and K=6, then scans shortcut destination dst to control the second-peak structure of first-passage distributions. Deterministic flux/master-equation scans and Monte Carlo trajectory classes jointly identify where valley depth and second-peak prominence are maximized.", "key_findings": ["Scanning dst alone can substantially change second-peak height ratio and valley depth at fixed N and K.", "Deterministic flux/master-equation results and Monte Carlo class decomposition agree on high-contrast destination windows.", "Second-peak strengthening correlates with increased delayed-route contribution rather than a uniform amplitude scaling.", "Destination geometry acts as a controllable lever for phase behavior without changing base transition probabilities."], "narrative": {"model_overview": "The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.", "method_overview": "The workflow combines deterministic flux scans, AW-style first-passage diagnostics, and class-conditioned Monte Carlo paths under one peak/valley criterion.", "result_overview": "Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 46, "dataset_series_ids": ["scan-probability", "scan-probability-9", "scan-probability-8"]}
{"report_id": "ring_deriv_k2", "group": "ring", "path": "research/reports/ring_deriv_k2", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.034079Z", "title": "Ring Derivation Backbone", "summary": "This derivation report provides the analytical backbone for ring models with one directed long-range link, including defect-free propagators, defect corrections, and first-passage generating functions. It serves as the shared mathematical base used by later shortcut and valley studies.", "key_findings": ["Defect-free and defect-corrected propagators can be written in a unified analytic framework.", "Directed long-range links alter first-passage statistics through resolvent-level corrections rather than ad-hoc fitting.", "The derivation yields formula components that are directly reused in lazy-jump and valley analyses.", "Analytic structure explains when shortcut asymmetry changes peak timing versus only changing overall scale."], "narrative": {"model_overview": "The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring interpretations.", "method_overview": "The report derives Green-function style propagators, constructs defect-resolvent corrections, and obtains first-passage generating forms that can be numerically inverted.", "result_overview": "The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports."}, "formula_count": 14, "section_count": 3, "repro_command_count": 3, "dataset_count": 2, "asset_count": 5, "dataset_series_ids": ["note_k2-probability", "note_k2-parameter"]}
{"report_id": "ring_two_target", "group": "ring", "path": "research/reports/ring_two_target", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.280475Z", "title": "Two-Target Lazy Ring Mechanics", "summary": "This report builds an exact two-target lazy-ring framework and compares no-shortcut versus selfloop-shortcut regimes. By scanning N, K, and beta with peak/valley diagnostics, it shows how target geometry and shortcut routing jointly control the transition among unimodal, bimodal, and trimodal first-passage behavior.", "key_findings": ["Two-target geometry introduces competing fast and delayed channels, making multimodality a structural rather than numerical artifact.", "Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.", "Selfloop shortcut settings can reweight path classes and generate trimodal signatures in selected regimes.", "K and beta scans provide a map of where multi-peak patterns are stable versus where they collapse to a single mode."], "narrative": {"model_overview": "The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.", "method_overview": "Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.", "result_overview": "No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 11, "dataset_series_ids": ["scan_bimodality_k4-metric", "scan_bimodality_k2-metric", "small_scan_metrics-probability"]}
{"report_id": "ring_two_walker_encounter_shortcut", "group": "ring", "path": "research/reports/ring_two_walker_encounter_shortcut", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.288081Z", "title": "1D Ring Two-Walker Encounter With Shortcut", "summary": "Robust numerical verification of appendix Eq. Exact first-encounter computation on 1D ring with directed shortcut.", "key_findings": ["Robust numerical verification of appendix Eq. (A1) == Eq. (A8).", "Exact first-encounter computation on 1D ring with directed shortcut.", "Shortcut scan showing when encounter FPT becomes double-peaked (under fixed diagnostic window).", "Fixed-site encounter companion study under drift-pair scan.", "rerun report code generation,", "rebuild CN + EN PDFs,"], "narrative": {"model_overview": "Research report ring_two_walker_encounter_shortcut.", "method_overview": "Research report ring_two_walker_encounter_shortcut.", "result_overview": "Research report ring_two_walker_encounter_shortcut."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 38, "dataset_series_ids": ["case_summary-probability", "encounter_beta_scan-metric", "encounter_onset_n_scan-metric"]}
{"report_id": "grid2d_bimodality", "group": "grid2d", "path": "research/reports/grid2d_bimodality", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.847077Z", "title": "Grid2D Bimodality Baseline", "summary": "This foundational Grid2D report establishes how biased/lazy random walks generate first-passage bimodality on a square lattice. It unifies model constraints, defect-aware propagators, AW inversion, and candidate-case scans into one auditable chain that separates genuine two-channel mechanisms from plotting artifacts.", "key_findings": ["A unified FPT criterion distinguishes structural bimodality from numerical or visualization artifacts.", "Candidate corridor/bias settings show that delayed channels can be amplified without changing the target definition.", "Model, method, and result statements are traceable through formula blocks, section summaries, and dataset panels.", "The report serves as a baseline vocabulary for later reflecting and two-target Grid2D variants."], "narrative": {"model_overview": "The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary assumptions.", "method_overview": "The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.", "result_overview": "Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion."}, "formula_count": 14, "section_count": 10, "repro_command_count": 4, "dataset_count": 3, "asset_count": 32, "dataset_series_ids": ["scan_candidate_c_bias", "scan_candidate_b_corridor-probability", "scan_candidate_b_corridor-binary"]}
{"report_id": "grid2d_reflecting_bimodality", "group": "grid2d", "path": "research/reports/grid2d_reflecting_bimodality", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.983078Z", "title": "Grid2D Reflecting-Boundary Bimodality", "summary": "This reflecting-boundary Grid2D report tests whether bimodality survives when periodic shortcuts are removed and all walls reflect. Using representative detour, pore, and transport-track cases, it shows that multi-channel timing structure can persist under strict boundary confinement.", "key_findings": ["Bimodality can persist under fully reflecting boundaries when competing pathways remain topologically distinct.", "Pore/track structures modify delay channels through accessibility, not only through drift magnitude.", "Case-level comparisons separate robust second peaks from late-window edge humps.", "The report supplies boundary-robust evidence for later cross-model hazard interpretations."], "narrative": {"model_overview": "The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).", "method_overview": "Case families are evaluated under common PMF/hazard diagnostics and compared by channel decomposition and timing windows.", "result_overview": "Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks."}, "formula_count": 14, "section_count": 10, "repro_command_count": 4, "dataset_count": 3, "asset_count": 40, "dataset_series_ids": ["aw_exact_speed-metric", "cases_reflecting_summary-probability", "aw_exact_speed-parameter"]}
{"report_id": "grid2d_blackboard_bimodality", "group": "grid2d", "path": "research/reports/grid2d_blackboard_bimodality", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.869077Z", "title": "Grid2D Blackboard Endpoint Case", "summary": "This blackboard-style Grid2D case studies reflecting boundaries with start and target anchored at wall endpoints. The Z/S endpoint configurations are used to test whether a visually delayed hump is a true second peak or a window-edge artifact, with diagnostics linked to channel decomposition and path geometry.", "key_findings": ["Endpoint wall configurations remain primarily unimodal with a long tail under reflecting constraints.", "The delayed hump in late windows is diagnostic-window edge behavior rather than a stable second structural peak.", "Channel decomposition separates geometric detours from genuinely competing transport routes.", "The blackboard pipeline keeps figure generation and case metadata reproducible across Z/S case variants."], "narrative": {"model_overview": "The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.", "method_overview": "The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.", "result_overview": "For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature."}, "formula_count": 8, "section_count": 10, "repro_command_count": 3, "dataset_count": 3, "asset_count": 40, "dataset_series_ids": ["z_scan-probability", "z_scan-binary", "screenshot_scan-probability"]}
{"report_id": "grid2d_two_target_double_peak", "group": "grid2d", "path": "research/reports/grid2d_two_target_double_peak", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.026792Z", "title": "Grid2D Two-Target Double-Peak", "summary": "This report studies two absorbing targets in 2D and maps when the total first-passage distribution develops visible double peaks. Under reflecting boundaries and corridor-style bias design, it provides phase maps over target-coupling parameters and isolates how competing destinations create multi-timescale structure.", "key_findings": ["Two-target competition creates a controlled mechanism for multi-timescale first-passage behavior.", "Phase maps identify stable double-peak bands and transition zones to unimodal behavior.", "Truncation and survival-tail diagnostics verify that observed second peaks are not finite-window artifacts.", "The report is a key bridge between Grid2D family behavior and ring two-target synthesis."], "narrative": {"model_overview": "The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.", "method_overview": "The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.", "result_overview": "Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 40, "dataset_series_ids": ["method_comparison_c1-probability", "method_comparison_c1-metric", "case_summary"]}
{"report_id": "grid2d_two_walker_encounter_shortcut", "group": "grid2d", "path": "research/reports/grid2d_two_walker_encounter_shortcut", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:42.030078Z", "title": "2D Two-Walker Encounter With Shortcut", "summary": "A robust numerical routine verifying equivalence between appendix Eq. Exact time-domain first-encounter computation for two independent walkers.", "key_findings": ["A robust numerical routine verifying equivalence between appendix Eq. (A1) and Eq. (A8).", "Exact time-domain first-encounter computation for two independent walkers.", "A shortcut-strength scan showing when the encounter FPT becomes double-peaked.", "Main PDFs:"], "narrative": {"model_overview": "Research report grid2d_two_walker_encounter_shortcut.", "method_overview": "Research report grid2d_two_walker_encounter_shortcut.", "result_overview": "Research report grid2d_two_walker_encounter_shortcut."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 12, "dataset_series_ids": ["encounter_beta_scan-probability", "case_summary-probability", "case_summary-binary"]}
{"report_id": "grid2d_rect_bimodality", "group": "grid2d", "path": "research/reports/grid2d_rect_bimodality", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.928077Z", "title": "Grid2D Rectangle Bimodality", "summary": "This report extends Grid2D bimodality from square to rectangular domains, testing how aspect ratio, reflecting walls, and endpoint geometry reshape first-passage channels. It emphasizes reproducible two-target constructions and identifies when double peaks remain structural under anisotropic geometry.", "key_findings": ["Rectangular anisotropy can suppress or recover bimodality depending on corridor alignment and target placement.", "Two-target endpoint constructions offer a reproducible mechanism for separating fast and delayed channels.", "Reflecting-boundary constraints change valley depth through route competition rather than simple amplitude scaling.", "The report provides geometry-sensitive evidence needed for cross-family synthesis with ring models."], "narrative": {"model_overview": "The model uses a rectangular reflecting lattice with controllable width/height and either one or two absorbing endpoint targets.", "method_overview": "The workflow scans geometry and bias parameters while keeping first-passage diagnostics fixed, then compares pathway composition across rectangular configurations.", "result_overview": "Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 38, "dataset_series_ids": ["ot_scan_bias2d", "tt_scan_width_xstart", "ot_scan_bias2d-metric"]}
{"report_id": "grid2d_membrane_near_target", "group": "grid2d", "path": "research/reports/grid2d_membrane_near_target", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.880077Z", "title": "Grid2D Membrane Near Target", "summary": "one-target corridor with symmetric/asymmetric semi-permeable membranes; no-corridor two-target setting with one target near the start.", "key_findings": ["one-target corridor with symmetric/asymmetric semi-permeable membranes;", "no-corridor two-target setting with one target near the start."], "narrative": {"model_overview": "Research report grid2d_membrane_near_target.", "method_overview": "Research report grid2d_membrane_near_target.", "result_overview": "Research report grid2d_membrane_near_target."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 29, "dataset_series_ids": ["corridor_membrane_symmetric_scan-probability", "corridor_membrane_asymmetric_scan-probability", "two_target_nearstart_scan-metric"]}
{"report_id": "cross_luca_regime_map", "group": "cross", "path": "research/reports/cross_luca_regime_map", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.710377Z", "title": "Cross-Model Luca Regime Map", "summary": "This cross-report study benchmarks full-FPT solvers under fixed-horizon fairness, comparing sparse exact recursion against Luca defect-reduced routes while keeping linear-system MFPT only as reference. The core output is a reproducible speed-ratio map R=t_sparse/t_luca and a regime classification that distinguishes where acceleration is real versus negligible.", "key_findings": ["The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with MFPT linear systems separated as reference only.", "Pooled timing medians indicate sparse exact dominates most scanned regimes in full-FPT mode.", "Luca acceleration is regime-dependent and concentrated in specific defect-pair configurations.", "Cross-report transfer claims are accepted only when both model families satisfy the same fairness and observability constraints."], "narrative": {"model_overview": "The comparison protocol aligns Grid2D and Ring instances under one fairness contract: same horizon, same observable, and no mixing of full-FPT metrics with MFPT-only claims.", "method_overview": "Runs use warm-up plus repeated timed executions, defect-pair routing rules, and pooled medians to stabilize solver-side variance before regime labeling.", "result_overview": "Across the scanned workload, sparse exact remains the dominant full-FPT baseline; Luca-mode speedups appear only in limited defect-regime subsets and are near-neutral in the aggregate ratio metric."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 23, "dataset_series_ids": ["runtime_raw", "runtime_raw_smoke", "manifest"]}
{"report_id": "final_multitimescale_fpt_encounter", "group": "misc", "path": "research/reports/final_multitimescale_fpt_encounter", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.773412Z", "title": "Final Multitimescale FPT and Encounter Report", "summary": "shows the actual spatial configuration for each selected case. The left panel in each row is the physical reflecting one-dimensional lattice: the two walker starts are marked, the reflecting walls are drawn at sites (1) and (N), and the dominant early/late encounter sites are shown directly on the lattice.", "key_findings": ["For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found."], "narrative": {"model_overview": "Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).", "method_overview": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.", "result_overview": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders."}, "formula_count": 10, "section_count": 10, "repro_command_count": 3, "dataset_count": 3, "asset_count": 37, "dataset_series_ids": ["grid2d_local_bias_basin_scan-probability", "double_peak_audit", "grid2d_local_bias_basin_scan-metric"]}
{"report_id": "grid2d_one_target_base", "group": "grid2d", "path": "research/reports/grid2d_one_target_base", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.881431Z", "title": "Grid2D One Target — Base", "summary": "**Status: WIP scaffold (not yet started).** Tracking: [#4] This is the **mother report** for the `grid2d_one_target_*` sub-series. It defines the shared problem setup, notation, and conventions used by the four sibling reports listed below.", "key_findings": ["`grid2d_one_target_exit_timing` — exit-timing distribution", "`grid2d_one_target_window_measures` — fixed-window measures (mass, percentile, tail)", "`grid2d_one_two_target_gating` — gating effect of adding a second target (uses base + extends)", "**Problem setup**: single absorbing target on a finite 2D-grid; choose between reflecting / periodic boundary; start position protocol; T grid convention.", "**Notation**: target placement $\\mathbf{x}^\\star$, start $\\mathbf{x}_0$, FPT $\\tau$, lattice constants, time discretisation."], "narrative": {"model_overview": "Model summary placeholder for grid2d_one_target_base.", "method_overview": "Method summary placeholder for grid2d_one_target_base.", "result_overview": "Result summary placeholder for grid2d_one_target_base."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 1, "asset_count": 0, "dataset_series_ids": ["asset-size-profile"]}
{"report_id": "grid2d_one_target_exit_timing", "group": "grid2d", "path": "research/reports/grid2d_one_target_exit_timing", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.881722Z", "title": "Grid2D One Target — Exit Timing", "summary": "**Status: WIP scaffold (not yet started).** Tracking: [#5] Directory layout is in place; no code, manuscript, or notes have been written.", "key_findings": ["[ ] Define problem setup: domain shape, target placement, BC, T grid"], "narrative": {"model_overview": "Model summary placeholder for grid2d_one_target_exit_timing.", "method_overview": "Method summary placeholder for grid2d_one_target_exit_timing.", "result_overview": "Result summary placeholder for grid2d_one_target_exit_timing."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 1, "asset_count": 0, "dataset_series_ids": ["asset-size-profile"]}
{"report_id": "grid2d_one_target_valley_peak_budget", "group": "grid2d", "path": "research/reports/grid2d_one_target_valley_peak_budget", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.882719Z", "title": "Grid2D One Target — Valley/Peak Budget", "summary": "**Status: 3-page short report. code and notes still pending.**", "key_findings": ["Research report grid2d_one_target_valley_peak_budget."], "narrative": {"model_overview": "Research report grid2d_one_target_valley_peak_budget.", "method_overview": "Research report grid2d_one_target_valley_peak_budget.", "result_overview": "Research report grid2d_one_target_valley_peak_budget."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 1, "asset_count": 7, "dataset_series_ids": ["asset-size-profile"]}
{"report_id": "grid2d_one_target_window_measures", "group": "grid2d", "path": "research/reports/grid2d_one_target_window_measures", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.883077Z", "title": "Grid2D One Target — Window Measures", "summary": "**Status: WIP scaffold (not yet started).** Tracking: [#7] Directory layout is in place; no code, manuscript, or notes have been written.", "key_findings": ["[ ] Pick the window family (fixed-T, multi-scale, adaptive) and the reported measures", "[ ] Land code + figures [grid2d_one_target_window_measures]"], "narrative": {"model_overview": "Model summary placeholder for grid2d_one_target_window_measures.", "method_overview": "Method summary placeholder for grid2d_one_target_window_measures.", "result_overview": "Result summary placeholder for grid2d_one_target_window_measures."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 1, "asset_count": 0, "dataset_series_ids": ["asset-size-profile"]}
{"report_id": "grid2d_one_two_target_gating", "group": "grid2d", "path": "research/reports/grid2d_one_two_target_gating", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.884409Z", "title": "Grid2D One vs Two Target — Gating", "summary": "**Status: WIP scaffold (not yet started).** Tracking: [#8] Directory layout is in place; no code, manuscript, or notes have been written.", "key_findings": ["[ ] Define gating metric (ratio of FPT mass redirected, blocked, or shortcut)", "[ ] Land code + figures [grid2d_one_two_target_gating]"], "narrative": {"model_overview": "Model summary placeholder for grid2d_one_two_target_gating.", "method_overview": "Method summary placeholder for grid2d_one_two_target_gating.", "result_overview": "Result summary placeholder for grid2d_one_two_target_gating."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 1, "asset_count": 0, "dataset_series_ids": ["asset-size-profile"]}
{"report_id": "exact_recursion_method_guide", "group": "misc", "path": "research/reports/exact_recursion_method_guide", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.750278Z", "title": "Exact Recursion — Method Guide", "summary": "**Status: WIP scaffold (not yet started).** Tracking: [#9] Directory layout is in place; no code, manuscript, or notes have been written.", "key_findings": ["Result summary placeholder for exact_recursion_method_guide."], "narrative": {"model_overview": "Model summary placeholder for exact_recursion_method_guide.", "method_overview": "Method summary placeholder for exact_recursion_method_guide.", "result_overview": "Result summary placeholder for exact_recursion_method_guide."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 1, "asset_count": 0, "dataset_series_ids": ["asset-size-profile"]}
{"report_id": "encounter_reflecting_diagonal_decomp", "group": "misc", "path": "research/reports/encounter_reflecting_diagonal_decomp", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.747299Z", "title": "Reflecting Encounter Diagonal Decomposition", "summary": "This report runs a bounded mobility-ratio scan for the reflecting-boundary", "key_findings": ["Result summary placeholder for encounter_reflecting_diagonal_decomp."], "narrative": {"model_overview": "Model summary placeholder for encounter_reflecting_diagonal_decomp.", "method_overview": "Method summary placeholder for encounter_reflecting_diagonal_decomp.", "result_overview": "Result summary placeholder for encounter_reflecting_diagonal_decomp."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 29, "dataset_series_ids": ["green_formula_comparison-binary", "ratio_scan_summary", "case_p03_rho5p0_f_total"]}
{"report_id": "encounter_reflecting_mean_validation", "group": "misc", "path": "research/reports/encounter_reflecting_mean_validation", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.749935Z", "title": "Reflecting Encounter Mean Validation", "summary": "This report validates the first exact-computation step for a one-dimensional", "key_findings": ["Interval: `{0, L-1}`.", "Joint state: `Y_t = (X_t^(1), X_t^(2))`.", "Encounter set: `E = {(k, k): 0 <= k < L}`.", "Encounter time: `tau_E = inf{t >= 0: X_t^(1) = X_t^(2)}`.", "Initial states must satisfy `x1_0 != x2_0`.", "`P_i(x, x) = 1 - Q_i`"], "narrative": {"model_overview": "Model summary placeholder for encounter_reflecting_mean_validation.", "method_overview": "Method summary placeholder for encounter_reflecting_mean_validation.", "result_overview": "Result summary placeholder for encounter_reflecting_mean_validation."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 2, "dataset_series_ids": ["diagonal_asymmetric_f_total", "diagonal_asymmetric_f_by_position", "mean_vs_rho_smoke-binary"]}
{"report_id": "grid2d_two_target_bias_radius", "group": "grid2d", "path": "research/reports/grid2d_two_target_bias_radius", "languages": ["en", "cn"], "updated_at": "2026-06-09T21:57:41.985519Z", "title": "Grid2D Two-Target Bias-Radius Scaffold", "summary": "State space: rectangular grid `{0, Lx-1} x {0, Ly-1}`. Boundary rule: reflecting attempted-outside-stays. Any move that would leave", "key_findings": ["State space: rectangular grid `{0, Lx-1} x {0, Ly-1}`.", "Boundary rule: reflecting attempted-outside-stays. Any move that would leave", "Absorbing set: `{a_near, a_far}`. Absorbing rows are self-loops in the kernel,", "First-passage channels:", "`r` is the Euclidean distance from `x0` to `a_near`.", "`theta` is the angle of `a_near - x0` measured relative to the bias direction."], "narrative": {"model_overview": "Model summary placeholder for grid2d_two_target_bias_radius.", "method_overview": "Method summary placeholder for grid2d_two_target_bias_radius.", "result_overview": "Result summary placeholder for grid2d_two_target_bias_radius."}, "formula_count": 1, "section_count": 1, "repro_command_count": 3, "dataset_count": 3, "asset_count": 5, "dataset_series_ids": ["grid2d_bias_radius_scan_config-probability", "smoke_fpt_channels", "grid2d_bias_radius_scan_metrics"]}