valley-k-small
grid2d_two_walker_encounter_shortcut

2D Two-Walker Encounter With Shortcut

A robust numerical routine verifying equivalence between appendix Eq. Exact time-domain first-encounter computation for two independent walkers.

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

Updated: 9 Jun 2026, 21:57:42 UTC

Model

Research report grid2d_two_walker_encounter_shortcut.

Method

Research report grid2d_two_walker_encounter_shortcut.

Result

Research report grid2d_two_walker_encounter_shortcut.

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 6: Reproducibility & Validation

Also appears in chapter-6-repro-validation

← 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.

Narrative Arc Position

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

grid2d progression

grid2d track links 13 reports into one continuous argument.

Checkpoints: 13 Claims: 63

Open arc sequence
  1. Grid2D Bimodality Baseline
  2. Grid2D Reflecting-Boundary Bimodality
  3. Grid2D Blackboard Endpoint Case
  4. Grid2D Two-Target Double-Peak
  5. 2D Two-Walker Encounter With Shortcut
  6. Grid2D Rectangle Bimodality
  7. Grid2D Membrane Near Target
  8. Grid2D One Target — Base
  9. Grid2D One Target — Exit Timing
  10. Grid2D One Target — Valley/Peak Budget
  11. Grid2D One Target — Window Measures
  12. Grid2D One vs Two Target — Gating
  13. Grid2D Two-Target Bias-Radius Scaffold

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

Research report grid2d_two_walker_encounter_shortcut.

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 grid2d_two_walker_encounter_shortcut-c1

Research report grid2d_two_walker_encounter_shortcut.

Evidence trail
  • source_document research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Research report grid2d_two_walker_encounter_shortcut.

  • section_summary research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Research report grid2d_two_walker_encounter_shortcut.

  • math_block research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Model formula context in 2D Two-Walker Encounter With Shortcut: Research report grid2d_two_walker_encounter_shortcut.

Linked reports Grid2D Membrane Near TargetGrid2D One Target — Valley/Peak Budget1D Ring Two-Walker Encounter With Shortcut

METHOD

method grid2d_two_walker_encounter_shortcut-c2

[grid2d] Method chain: Research report grid2d_two_walker_encounter_shortcut.

Evidence trail
  • source_document research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    [grid2d] Method chain: Research report grid2d_two_walker_encounter_shortcut.

  • section_summary research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Research report grid2d_two_walker_encounter_shortcut.

  • math_block research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Method formula context in 2D Two-Walker Encounter With Shortcut: [grid2d] Method chain: Research report

  • dataset /data/v1/reports/grid2d_two_walker_encounter_shortcut/series/encounter_beta_scan-probability.json

    encounter_beta_scan [probability]: beta -> peak_ratio, valley_ratio [probability]

Linked reports Grid2D Membrane Near TargetGrid2D One Target — Valley/Peak Budget1D Ring Two-Walker Encounter With ShortcutReflecting Encounter Diagonal Decomposition

RESULT

result grid2d_two_walker_encounter_shortcut-c3

[grid2d] Result statement: Research report grid2d_two_walker_encounter_shortcut.

Evidence trail
  • source_document research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    [grid2d] Result statement: Research report grid2d_two_walker_encounter_shortcut.

  • section_summary research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Research report grid2d_two_walker_encounter_shortcut.

  • math_block research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Result formula context in 2D Two-Walker Encounter With Shortcut: [grid2d] Result statement: Research report

  • dataset /data/v1/reports/grid2d_two_walker_encounter_shortcut/series/encounter_beta_scan-probability.json

    encounter_beta_scan [probability]: beta -> peak_ratio, valley_ratio [probability]

Linked reports Grid2D Membrane Near TargetGrid2D One Target — Valley/Peak Budget1D Ring Two-Walker Encounter With ShortcutFinal Multitimescale FPT and Encounter Report

FINDING

finding grid2d_two_walker_encounter_shortcut-c4

Exact time-domain first-encounter computation for two independent walkers.

Evidence trail
  • source_document research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Exact time-domain first-encounter computation for two independent walkers.

  • section_summary research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Research report grid2d_two_walker_encounter_shortcut.

  • math_block research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Finding formula context in 2D Two-Walker Encounter With Shortcut: Exact time-domain first-encounter computation for two independent walkers.

  • dataset /data/v1/reports/grid2d_two_walker_encounter_shortcut/series/encounter_beta_scan-probability.json

    encounter_beta_scan [probability]: beta -> peak_ratio, valley_ratio [probability]

Linked reports 1D Ring Two-Walker Encounter With ShortcutGrid2D Two-Target Double-PeakLazy Ring Jump-Over Mechanism (K2 vs K4)Grid2D Bimodality Baseline

finding grid2d_two_walker_encounter_shortcut-c5

Key finding for grid2d_two_walker_encounter_shortcut: see the equation chain, interactive panels, and source-linked assets.

Evidence trail
  • source_document research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Key finding for grid2d_two_walker_encounter_shortcut: see the equation chain, interactive panels, and source-linked assets.

  • section_summary research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Research report grid2d_two_walker_encounter_shortcut.

  • math_block research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

    Finding formula context in 2D Two-Walker Encounter With Shortcut: Key finding for grid2d_two_walker_encounter_shortcut: see the equation

  • dataset /data/v1/reports/grid2d_two_walker_encounter_shortcut/series/encounter_beta_scan-probability.json

    encounter_beta_scan [probability]: beta -> peak_ratio, valley_ratio [probability]

Linked reports 1D Ring Two-Walker Encounter With ShortcutReflecting Encounter Diagonal DecompositionExact Recursion — Method GuideGrid2D One Target — Valley/Peak Budget

Interactive Dataset

Plot controls
window=1

beta peak_ratio, valley_ratio [probability]

Loading plot data…

Provenance: research/reports/grid2d_two_walker_encounter_shortcut/artifacts/data/encounter_beta_scan.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

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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/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

Narrative Sections

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

Overview

Research report grid2d_two_walker_encounter_shortcut.

Source

research/reports/grid2d_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report grid2d_two_walker_encounter_shortcut --lang en
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

Download Assets

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Figure Gallery

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