valley-k-small
ring_two_walker_encounter_shortcut

1D Ring Two-Walker Encounter With Shortcut

Robust numerical verification of appendix Eq. Exact first-encounter computation on 1D ring with directed shortcut.

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

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

Model

Research report ring_two_walker_encounter_shortcut.

Method

Research report ring_two_walker_encounter_shortcut.

Result

Research report ring_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

Show remaining findings
  • rebuild CN + EN PDFs,

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.

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 ring_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 ring_two_walker_encounter_shortcut-c1

Research report ring_two_walker_encounter_shortcut.

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

    Research report ring_two_walker_encounter_shortcut.

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

    Research report ring_two_walker_encounter_shortcut.

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

    Model formula context in 1D Ring Two-Walker Encounter With Shortcut: Research report ring_two_walker_encounter_shortcut.

Linked reports Grid2D Membrane Near TargetGrid2D One Target — Valley/Peak Budget2D Two-Walker Encounter With Shortcut

METHOD

method ring_two_walker_encounter_shortcut-c2

[ring] Method chain: Research report ring_two_walker_encounter_shortcut.

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

    [ring] Method chain: Research report ring_two_walker_encounter_shortcut.

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

    Research report ring_two_walker_encounter_shortcut.

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

    Method formula context in 1D Ring Two-Walker Encounter With Shortcut: [ring] Method chain: Research report

  • dataset /data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json

    case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]

Linked reports Grid2D Membrane Near TargetGrid2D One Target — Valley/Peak Budget2D Two-Walker Encounter With ShortcutReflecting Encounter Diagonal Decomposition

RESULT

result ring_two_walker_encounter_shortcut-c3

[ring] Result statement: Research report ring_two_walker_encounter_shortcut.

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

    [ring] Result statement: Research report ring_two_walker_encounter_shortcut.

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

    Research report ring_two_walker_encounter_shortcut.

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

    Result formula context in 1D Ring Two-Walker Encounter With Shortcut: [ring] Result statement: Research report

  • dataset /data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json

    case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]

Linked reports Grid2D Membrane Near TargetGrid2D One Target — Valley/Peak Budget2D Two-Walker Encounter With ShortcutFinal Multitimescale FPT and Encounter Report

FINDING

finding ring_two_walker_encounter_shortcut-c4

Exact first-encounter computation on 1D ring with directed shortcut.

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

    Exact first-encounter computation on 1D ring with directed shortcut.

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

    Research report ring_two_walker_encounter_shortcut.

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

    Finding formula context in 1D Ring Two-Walker Encounter With Shortcut: Exact first-encounter computation on 1D ring with directed shortcut.

  • dataset /data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json

    case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]

Linked reports 2D Two-Walker Encounter With ShortcutRing Derivation BackboneLazy Ring Flux BaselineLazy Ring Jump-Over Mechanism (K2 vs K4)

finding ring_two_walker_encounter_shortcut-c5

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

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

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

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

    Research report ring_two_walker_encounter_shortcut.

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

    Finding formula context in 1D Ring Two-Walker Encounter With Shortcut: Key finding for ring_two_walker_encounter_shortcut: see the

  • dataset /data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json

    case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]

Linked reports 2D Two-Walker Encounter With ShortcutReflecting Encounter Diagonal DecompositionExact Recursion — Method GuideGrid2D One Target — Valley/Peak Budget

Interactive Dataset

Plot controls
window=1

detector_compare_beta detector_compare_peak_ratio_legacy [probability]

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Provenance: research/reports/ring_two_walker_encounter_shortcut/artifacts/data/case_summary.json

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/ring_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 ring_two_walker_encounter_shortcut.

Source

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

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report ring_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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ring two walker encounter shortcut addendum cn

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