valley-k-small
cross_luca_regime_map

Cross-Model Luca Regime Map

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.

Book PositionReading PathConnected ReportsNarrative Arc PositionVerifiable ClaimsInteractive FiguresMathematical LogicFormula LibraryNarrative NotesAppendix & Assets

Updated: 9 Jun 2026, 21:57:41 UTC

Model

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

Runs use warm-up plus repeated timed executions, defect-pair routing rules, and pooled medians to stabilize solver-side variance before regime labeling.

Result

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.

Book Position

This report is part of the chapterized mainline. Use chapter links to keep continuity instead of reading reports in isolation.

Primary chapter Chapter 5: Cross-Model Synthesis

Also appears in chapter-5-cross-model-synthesischapter-7-outlook

← Previous chapter Next chapter →

Reading Path

  1. Scan key findings first to decide whether this report is relevant.
  2. Use the interactive panel to test parameter and shape sensitivity.
  3. Then read the mathematical chain and formula library for derivation details.

Key Findings

Connected Reports

This report sits inside a shared chain. Use links below to move upstream/downstream and across model families.

Track Continuity

  • No previous report in this track.
  • No next report in this track.

Same-Group Links

No same-group links detected.

Narrative Arc Position

This report appears in one or more global arcs. Use these checkpoints to keep reading continuity across pages.

cross progression

cross track links 1 reports into one continuous argument.

Checkpoints: 1 Claims: 5

Open arc sequence
  1. Cross-Model Luca Regime Map

Grid and ring mechanisms converge into cross-report synthesis.

Global storyline that connects all report families from mechanism to synthesis.

Checkpoints: 27 Claims: 130

Open arc sequence
  1. Lazy Ring Jump-Over Mechanism (K2 vs K4)
  2. Lazy Ring Shortcut Beta Scan
  3. Lazy Ring Shortcut Figure-1 Revision
  4. Lazy Ring Flux Baseline
  5. Ring Valley Regime Map
  6. Destination-Scan Valley Control
  7. Ring Derivation Backbone
  8. Two-Target Lazy Ring Mechanics
  9. 1D Ring Two-Walker Encounter With Shortcut
  10. Grid2D Bimodality Baseline
  11. Grid2D Reflecting-Boundary Bimodality
  12. Grid2D Blackboard Endpoint Case
  13. Grid2D Two-Target Double-Peak
  14. 2D Two-Walker Encounter With Shortcut
  15. Grid2D Rectangle Bimodality
  16. Grid2D Membrane Near Target
  17. Grid2D One Target — Base
  18. Grid2D One Target — Exit Timing
  19. Grid2D One Target — Valley/Peak Budget
  20. Grid2D One Target — Window Measures
  21. Grid2D One vs Two Target — Gating
  22. Grid2D Two-Target Bias-Radius Scaffold
  23. Cross-Model Luca Regime Map
  24. Final Multitimescale FPT and Encounter Report
  25. Exact Recursion — Method Guide
  26. Reflecting Encounter Diagonal Decomposition
  27. Reflecting Encounter Mean Validation

Verifiable Claims

Claims below are tied to explicit evidence paths so each statement can be audited.

Report Objective

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.

Verification Steps
  1. Read the key claims and their evidence references first.
  2. Verify at least one equation card and one dataset panel against source paths.
  3. Cross-check this report with upstream/downstream linked reports.

MODEL

model cross_luca_regime_map-c1

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.

Evidence trail
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    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.

  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Model formula context in Cross-Model Luca Regime Map: The comparison protocol aligns Grid2D and Ring instances under one fairness

Linked reports Final Multitimescale FPT and Encounter ReportDestination-Scan Valley ControlRing Derivation Backbone

METHOD

method cross_luca_regime_map-c2

Runs use warm-up plus repeated timed executions, defect-pair routing rules, and pooled medians to stabilize solver-side variance before regime labeling.

Evidence trail
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Runs use warm-up plus repeated timed executions, defect-pair routing rules, and pooled medians to stabilize solver-side variance before regime labeling.

  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Method formula context in Cross-Model Luca Regime Map: Runs use warm-up plus repeated timed executions, defect-pair routing rules, and

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

Linked reports Grid2D Bimodality BaselineGrid2D Blackboard Endpoint CaseRing Derivation BackboneLazy Ring Shortcut Beta Scan

RESULT

result cross_luca_regime_map-c3

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.

Evidence trail
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    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

  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Result formula context in Cross-Model Luca Regime Map: Across the scanned workload, sparse exact remains the dominant full-FPT baseline

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

Linked reports Ring Valley Regime MapFinal Multitimescale FPT and Encounter ReportGrid2D Blackboard Endpoint CaseGrid2D One vs Two Target — Gating

FINDING

finding cross_luca_regime_map-c4

Pooled timing medians indicate sparse exact dominates most scanned regimes in full-FPT mode.

Evidence trail
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Pooled timing medians indicate sparse exact dominates most scanned regimes in full-FPT mode.

  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Finding formula context in Cross-Model Luca Regime Map: Pooled timing medians indicate sparse exact dominates most scanned regimes in

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

Linked reports Ring Valley Regime MapGrid2D Bimodality BaselineGrid2D Blackboard Endpoint CaseGrid2D Reflecting-Boundary Bimodality

finding cross_luca_regime_map-c5

The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with MFPT linear systems separated as reference only.

Evidence trail
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with MFPT linear systems separated as reference only.

  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Finding formula context in Cross-Model Luca Regime Map: The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

Linked reports Grid2D One vs Two Target — GatingDestination-Scan Valley ControlGrid2D Rectangle BimodalityFinal Multitimescale FPT and Encounter Report

Interactive Dataset

Plot controls
window=1

t_max defect_pairs, local_bias_sites, sparse_seconds

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Provenance: research/reports/cross_luca_regime_map/artifacts/data/runtime_raw.csv

Mathematical Logic Chain

From model assumptions to interpretation in a short, ordered chain.

Distribution Setup

Defines first-passage probability objects used by later diagnostics.

f(t)=Pr⁡[T=t],S(t)=Pr⁡[T>t],h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=\Pr[T>t],\quad h(t)=\frac{f(t)}{S(t-1)}

Fallback

Mathematical Principles

Showing 1 / 1

Fallback EN

f(t)=Pr⁡[T=t],S(t)=Pr⁡[T>t],h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=\Pr[T>t],\quad h(t)=\frac{f(t)}{S(t-1)}
Formula source

research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

Narrative Sections

Cleaned chapter summaries are shown first; low-value placeholders are hidden.

Overview

Research report cross_luca_regime_map.

Source

research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report cross_luca_regime_map --lang en
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

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Figure Gallery

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regime estimation error anchor (figures/smoke)

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regime winner heatmap reflecting (figures/smoke)

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regime winner heatmap two target (figures/smoke)

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