valley-k-small
grid2d_two_target_bias_radius

Grid2D Two-Target Bias-Radius Scaffold

State space: rectangular grid `{0, Lx-1} x {0, Ly-1}`. Boundary rule: reflecting attempted-outside-stays. Any move that would leave

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

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

Model

Model summary placeholder for grid2d_two_target_bias_radius.

Method

Method summary placeholder for grid2d_two_target_bias_radius.

Result

Result summary placeholder for grid2d_two_target_bias_radius.

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
  • `theta` is the angle of `a_near - x0` measured relative to the bias direction.

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

Result summary placeholder for grid2d_two_target_bias_radius.

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.

FINDING

finding grid2d_two_target_bias_radius-c4

Boundary rule: reflecting attempted-outside-stays.

Evidence trail
  • source_document research/reports/grid2d_two_target_bias_radius

    Boundary rule: reflecting attempted-outside-stays.

  • section_summary research/reports/grid2d_two_target_bias_radius

    Overview: Boundary rule: reflecting attempted-outside-stays.

  • math_block research/reports/grid2d_two_target_bias_radius

    Finding formula context in Grid2D Two-Target Bias-Radius Scaffold: Boundary rule: reflecting attempted-outside-stays.

  • dataset /data/v1/reports/grid2d_two_target_bias_radius/series/grid2d_bias_radius_scan_config-probability.json

    grid2d_bias_radius_scan_config [probability]: near_x -> q [probability]

Linked reports Grid2D Blackboard Endpoint CaseFinal Multitimescale FPT and Encounter ReportGrid2D Bimodality BaselineGrid2D Rectangle Bimodality

finding grid2d_two_target_bias_radius-c5

State space: rectangular grid {0, Lx-1} x {0, Ly-1}.

Evidence trail
  • source_document research/reports/grid2d_two_target_bias_radius

    State space: rectangular grid {0, Lx-1} x {0, Ly-1}.

  • section_summary research/reports/grid2d_two_target_bias_radius

    Overview: State space: rectangular grid {0, Lx-1} x {0, Ly-1}.

  • math_block research/reports/grid2d_two_target_bias_radius

    Finding formula context in Grid2D Two-Target Bias-Radius Scaffold: State space: rectangular grid {0, Lx-1} x {0, Ly-1}.

  • dataset /data/v1/reports/grid2d_two_target_bias_radius/series/grid2d_bias_radius_scan_config-probability.json

    grid2d_bias_radius_scan_config [probability]: near_x -> q [probability]

Linked reports Grid2D Rectangle Bimodality

Interactive Dataset

Plot controls
window=1

near_x q [probability]

Loading plot data…

Provenance: research/reports/grid2d_two_target_bias_radius/artifacts/data/grid2d_bias_radius_scan_config.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]f(t)=\Pr[T=t]

Fallback

Mathematical Principles

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Fallback EN

f(t)=Pr⁡[T=t]f(t)=\Pr[T=t]
Formula source

research/reports/grid2d_two_target_bias_radius

Narrative Sections

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

Overview

Overview: this section connects Grid2D Two Target Bias Radius assumptions to measurable observables, verification commands, and downstream chapter claims.

Source

research/reports/grid2d_two_target_bias_radius

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report grid2d_two_target_bias_radius --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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grid2d representative fpt decomp theta0 r04 b008

Open Figure

grid2d representative fpt decomp theta0 r04 b012

Open Figure

grid2d representative fpt decomp theta0 r05 b012

Open Figure