valley-k-small
ring_two_target

Two-Target Lazy Ring Mechanics

This report builds an exact two-target lazy-ring framework and compares no-shortcut versus selfloop-shortcut regimes. By scanning N, K, and beta with peak/valley diagnostics, it shows how target geometry and shortcut routing jointly control the transition among unimodal, bimodal, and trimodal first-passage behavior.

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 on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.

Method

Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.

Result

No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter 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 0: Reading Guide & Notation

Also appears in chapter-0-reading-guidechapter-1-core-fptchapter-5-cross-model-synthesischapter-7-outlook

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.

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

No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter 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 ring_two_target-c1

The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.

Evidence trail
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Model formula context in Two-Target Lazy Ring Mechanics: The model places two absorbing targets on a lazy ring with optional directed

Linked reports Destination-Scan Valley ControlGrid2D Bimodality BaselineGrid2D Two-Target Double-PeakLazy Ring Flux Baseline

METHOD

method ring_two_target-c2

Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.

Evidence trail
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Method formula context in Two-Target Lazy Ring Mechanics: Exact generating-function/AW inversion is combined with parameter scans and

  • dataset /data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json

    small_scan_metrics [probability]: beta -> q [probability]

Linked reports Grid2D Bimodality BaselineRing Valley Regime MapDestination-Scan Valley ControlGrid2D Blackboard Endpoint Case

RESULT

result ring_two_target-c3

No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.

Evidence trail
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Result formula context in Two-Target Lazy Ring Mechanics: No-shortcut drift can already produce strong bimodality, while shortcut

  • dataset /data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json

    small_scan_metrics [probability]: beta -> q [probability]

Linked reports Lazy Ring Jump-Over Mechanism (K2 vs K4)Ring Valley Regime MapGrid2D Rectangle Bimodality

FINDING

finding ring_two_target-c4

Two-target geometry introduces competing fast and delayed channels, making multimodality a structural rather than numerical artifact.

Evidence trail
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Two-target geometry introduces competing fast and delayed channels, making multimodality a structural rather than numerical artifact.

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Finding formula context in Two-Target Lazy Ring Mechanics: Two-target geometry introduces competing fast and delayed channels, making

  • dataset /data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json

    small_scan_metrics [probability]: beta -> q [probability]

Linked reports Grid2D Rectangle BimodalityGrid2D Blackboard Endpoint CaseGrid2D Bimodality BaselineGrid2D Membrane Near Target

finding ring_two_target-c5

Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.

Evidence trail
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Finding formula context in Two-Target Lazy Ring Mechanics: Under no-shortcut drift, robust bimodality appears in reproducible parameter

  • dataset /data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json

    small_scan_metrics [probability]: beta -> q [probability]

Linked reports Lazy Ring Flux BaselineLazy Ring Jump-Over Mechanism (K2 vs K4)Final Multitimescale FPT and Encounter ReportRing Valley Regime Map

Interactive Dataset

Plot controls
window=1

beta N [metric]

Loading plot data…

Provenance: research/reports/ring_two_target/artifacts/data/scan_bimodality_K4.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/ring_two_target/artifacts/tables/case_configs.tex

Narrative Sections

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

Overview

Research report ring_two_target.

Source

research/reports/ring_two_target/artifacts/tables/case_configs.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report ring_two_target --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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