valley-k-small
grid2d_two_target_double_peak

Grid2D Two-Target Double-Peak

This report studies two absorbing targets in 2D and maps when the total first-passage distribution develops visible double peaks. Under reflecting boundaries and corridor-style bias design, it provides phase maps over target-coupling parameters and isolates how competing destinations create multi-timescale structure.

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

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

Model

The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

Method

The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

Result

Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.

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 1: Core FPT Concepts

Also appears in chapter-1-core-fptchapter-2-grid2d-familychapter-5-cross-model-synthesischapter-6-repro-validationchapter-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.

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

Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.

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_target_double_peak-c1

The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

Evidence trail
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Model formula context in Grid2D Two-Target Double-Peak: The model places two absorbing targets in a reflecting lattice with fixed start

Linked reports Grid2D Rectangle BimodalityGrid2D Bimodality BaselineGrid2D Reflecting-Boundary BimodalityTwo-Target Lazy Ring Mechanics

METHOD

method grid2d_two_target_double_peak-c2

The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

Evidence trail
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Method formula context in Grid2D Two-Target Double-Peak: The pipeline combines exact/approximate first-passage diagnostics, truncation

  • dataset /data/v1/reports/grid2d_two_target_double_peak/series/method_comparison_c1-probability.json

    method_comparison_c1 [probability]: mfpt_truncation_scan_t_max -> mfpt_truncation_scan_mass_any [probability]

Linked reports Destination-Scan Valley ControlGrid2D Rectangle BimodalityLazy Ring Jump-Over Mechanism (K2 vs K4)Two-Target Lazy Ring Mechanics

RESULT

result grid2d_two_target_double_peak-c3

Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.

Evidence trail
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Result formula context in Grid2D Two-Target Double-Peak: Double-peak regions appear when direct-to-near-target and delayed-to-far-target

  • dataset /data/v1/reports/grid2d_two_target_double_peak/series/method_comparison_c1-probability.json

    method_comparison_c1 [probability]: mfpt_truncation_scan_t_max -> mfpt_truncation_scan_mass_any [probability]

Linked reports Final Multitimescale FPT and Encounter ReportGrid2D Rectangle BimodalityGrid2D Bimodality BaselineLazy Ring Jump-Over Mechanism (K2 vs K4)

FINDING

finding grid2d_two_target_double_peak-c4

Phase maps identify stable double-peak bands and transition zones to unimodal behavior.

Evidence trail
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Phase maps identify stable double-peak bands and transition zones to unimodal behavior.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Finding formula context in Grid2D Two-Target Double-Peak: Phase maps identify stable double-peak bands and transition zones to unimodal

  • dataset /data/v1/reports/grid2d_two_target_double_peak/series/method_comparison_c1-probability.json

    method_comparison_c1 [probability]: mfpt_truncation_scan_t_max -> mfpt_truncation_scan_mass_any [probability]

Linked reports Grid2D Blackboard Endpoint CaseFinal Multitimescale FPT and Encounter Report

finding grid2d_two_target_double_peak-c5

Two-target competition creates a controlled mechanism for multi-timescale first-passage behavior.

Evidence trail
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Two-target competition creates a controlled mechanism for multi-timescale first-passage behavior.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Finding formula context in Grid2D Two-Target Double-Peak: Two-target competition creates a controlled mechanism for multi-timescale

  • dataset /data/v1/reports/grid2d_two_target_double_peak/series/method_comparison_c1-probability.json

    method_comparison_c1 [probability]: mfpt_truncation_scan_t_max -> mfpt_truncation_scan_mass_any [probability]

Linked reports Grid2D Rectangle BimodalityGrid2D Membrane Near Target2D Two-Walker Encounter With ShortcutRing Derivation Backbone

Interactive Dataset

Plot controls
window=1

mfpt_truncation_scan_t_max mfpt_truncation_scan_mass_any [probability]

Loading plot data…

Provenance: research/reports/grid2d_two_target_double_peak/artifacts/data/method_comparison_c1.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_two_target_double_peak/artifacts/tables/case_mechanism.tex

Narrative Sections

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

Overview

Research report grid2d_two_target_double_peak.

Source

research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report grid2d_two_target_double_peak --lang en
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

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