valley-k-small
grid2d_rect_bimodality

Grid2D Rectangle Bimodality

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.

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

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

Model

The model uses a rectangular reflecting lattice with controllable width/height and either one or two absorbing endpoint targets.

Method

The workflow scans geometry and bias parameters while keeping first-passage diagnostics fixed, then compares pathway composition across rectangular configurations.

Result

Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.

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 2: Grid2D Family

Also appears in chapter-2-grid2d-familychapter-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

Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.

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_rect_bimodality-c1

The model uses a rectangular reflecting lattice with controllable width/height and either one or two absorbing endpoint targets.

Evidence trail
  • source_document research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    The model uses a rectangular reflecting lattice with controllable width/height and either one or two absorbing endpoint targets.

  • section_summary research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Research report grid2d_rect_bimodality.

  • math_block research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Model formula context in Grid2D Rectangle Bimodality: The model uses a rectangular reflecting lattice with controllable width/height and

Linked reports Grid2D Two-Target Double-PeakGrid2D Bimodality BaselineGrid2D Blackboard Endpoint CaseGrid2D Reflecting-Boundary Bimodality

METHOD

method grid2d_rect_bimodality-c2

The workflow scans geometry and bias parameters while keeping first-passage diagnostics fixed, then compares pathway composition across rectangular configurations.

Evidence trail
  • source_document research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    The workflow scans geometry and bias parameters while keeping first-passage diagnostics fixed, then compares pathway composition across rectangular configurations.

  • section_summary research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Research report grid2d_rect_bimodality.

  • math_block research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Method formula context in Grid2D Rectangle Bimodality: The workflow scans geometry and bias parameters while keeping first-passage

  • dataset /data/v1/reports/grid2d_rect_bimodality/series/ot_scan_bias2d.json

    ot_scan_bias2d: bx_values -> by_values

Linked reports Destination-Scan Valley ControlGrid2D Two-Target Double-PeakGrid2D Bimodality BaselineLazy Ring Shortcut Beta Scan

RESULT

result grid2d_rect_bimodality-c3

Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.

Evidence trail
  • source_document research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.

  • section_summary research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Research report grid2d_rect_bimodality.

  • math_block research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Result formula context in Grid2D Rectangle Bimodality: Aspect ratio and endpoint arrangement shift the balance between direct and detour

  • dataset /data/v1/reports/grid2d_rect_bimodality/series/ot_scan_bias2d.json

    ot_scan_bias2d: bx_values -> by_values

Linked reports Final Multitimescale FPT and Encounter ReportGrid2D Two-Target Double-PeakCross-Model Luca Regime Map

FINDING

finding grid2d_rect_bimodality-c4

Rectangular anisotropy can suppress or recover bimodality depending on corridor alignment and target placement.

Evidence trail
  • source_document research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Rectangular anisotropy can suppress or recover bimodality depending on corridor alignment and target placement.

  • section_summary research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Research report grid2d_rect_bimodality.

  • math_block research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Finding formula context in Grid2D Rectangle Bimodality: Rectangular anisotropy can suppress or recover bimodality depending on corridor

  • dataset /data/v1/reports/grid2d_rect_bimodality/series/ot_scan_bias2d.json

    ot_scan_bias2d: bx_values -> by_values

Linked reports Grid2D Bimodality BaselineGrid2D Membrane Near TargetGrid2D One Target — Exit TimingRing Valley Regime Map

finding grid2d_rect_bimodality-c5

Two-target endpoint constructions offer a reproducible mechanism for separating fast and delayed channels.

Evidence trail
  • source_document research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Two-target endpoint constructions offer a reproducible mechanism for separating fast and delayed channels.

  • section_summary research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Research report grid2d_rect_bimodality.

  • math_block research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Finding formula context in Grid2D Rectangle Bimodality: Two-target endpoint constructions offer a reproducible mechanism for separating

  • dataset /data/v1/reports/grid2d_rect_bimodality/series/ot_scan_bias2d.json

    ot_scan_bias2d: bx_values -> by_values

Linked reports Two-Target Lazy Ring MechanicsGrid2D Bimodality BaselineGrid2D Two-Target Double-PeakGrid2D Membrane Near Target

Interactive Dataset

Plot controls
window=1

bx_values by_values

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Provenance: research/reports/grid2d_rect_bimodality/artifacts/data/ot_scan_bias2d.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

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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_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

Narrative Sections

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

Overview

Research report grid2d_rect_bimodality.

Source

research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report grid2d_rect_bimodality --lang en
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

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