valley-k-small
ring_lazy_jump

Lazy Ring Jump-Over Mechanism (K2 vs K4)

This report establishes the baseline jump-over mechanism for lazy rings with one directed shortcut, contrasting K=2 and K=4 under exact first-passage diagnostics. It explains why double-peak behavior is selective in shortcut strength and how pathway decomposition links peak structure to fast and delayed transport channels.

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

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

Model

A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

Method

The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.

Result

Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.

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 3: Ring Family Baseline

Also appears in chapter-3-ring-baselinechapter-4-shortcut-variants

← 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.

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

Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.

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_lazy_jump-c1

A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

Evidence trail
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Model formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): A lazy ring with one directed shortcut is used as the baseline setting

Linked reports Lazy Ring Shortcut Beta ScanLazy Ring Flux BaselineRing Derivation BackboneLazy Ring Shortcut Figure-1 Revision

METHOD

method ring_lazy_jump-c2

The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.

Evidence trail
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Method formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): The analysis combines AW inversion for exact first-passage series with

  • dataset /data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Grid2D Two-Target Double-PeakTwo-Target Lazy Ring MechanicsRing Valley Regime MapDestination-Scan Valley Control

RESULT

result ring_lazy_jump-c3

Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.

Evidence trail
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Result formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Bimodality appears only in selected shortcut-strength intervals

  • dataset /data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Two-Target Lazy Ring MechanicsGrid2D Two-Target Double-PeakLazy Ring Flux BaselineFinal Multitimescale FPT and Encounter Report

FINDING

finding ring_lazy_jump-c4

Jump-over pathways create a distinct delayed channel that is necessary for persistent second peaks.

Evidence trail
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Jump-over pathways create a distinct delayed channel that is necessary for persistent second peaks.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Finding formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Jump-over pathways create a distinct delayed channel that is

  • dataset /data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Grid2D Blackboard Endpoint CaseGrid2D Reflecting-Boundary BimodalityGrid2D Bimodality Baseline

finding ring_lazy_jump-c5

Shortcut strength has a non-monotonic effect: too weak or too strong settings both reduce robust bimodality.

Evidence trail
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Shortcut strength has a non-monotonic effect: too weak or too strong settings both reduce robust bimodality.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Finding formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Shortcut strength has a non-monotonic effect: too weak or too strong

  • dataset /data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Two-Target Lazy Ring MechanicsLazy Ring Flux BaselineLazy Ring Shortcut Beta ScanFinal Multitimescale FPT and Encounter Report

Interactive Dataset

Plot controls
window=1

N q [probability]

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Provenance: research/reports/ring_lazy_jump/artifacts/data/scan_N_K4_beta002.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_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

Narrative Sections

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

Overview

Research report ring_lazy_jump.

Source

research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

Reproducibility Commands

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

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Figure Gallery

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N100 K4 beta002 jumpover control.compare f t

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