valley-k-small
ring_lazy_flux

Lazy Ring Flux Baseline

This report studies first-passage distributions on a lazy ring with one directed shortcut drawn from self-loop probability. It combines Chebyshev generating-function derivation, AW/FFT inversion, and flux-recursion cross-checks to show a reproducible small-p bimodal regime, while equal4 and large shortcut strength suppress the second peak.

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

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

Model

The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.

Method

The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.

Result

A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.

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-3-ring-baselinechapter-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.

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

A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.

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_flux-c1

The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.

Evidence trail
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.

  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Research report ring_lazy_flux.

  • math_block research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Model formula context in Lazy Ring Flux Baseline: The model is a lazy nearest-neighbor ring with one directed shortcut u->v

Linked reports Lazy Ring Shortcut Beta ScanLazy Ring Jump-Over Mechanism (K2 vs K4)Two-Target Lazy Ring MechanicsDestination-Scan Valley Control

METHOD

method ring_lazy_flux-c2

The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.

Evidence trail
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.

  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Research report ring_lazy_flux.

  • math_block research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Method formula context in Lazy Ring Flux Baseline: The pipeline derives the generating function analytically, inverts it via AW/FFT, and

  • dataset /data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json

    lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]

Linked reports Ring Derivation BackboneTwo-Target Lazy Ring MechanicsGrid2D Bimodality BaselineDestination-Scan Valley Control

RESULT

result ring_lazy_flux-c3

A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.

Evidence trail
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.

  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Research report ring_lazy_flux.

  • math_block research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Result formula context in Lazy Ring Flux Baseline: A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger

  • dataset /data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json

    lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]

Linked reports Grid2D Reflecting-Boundary BimodalityLazy Ring Jump-Over Mechanism (K2 vs K4)Ring Valley Regime MapLazy Ring Shortcut Beta Scan

FINDING

finding ring_lazy_flux-c4

A minimal reproducible bimodal case appears at N=10 under small shortcut strength in the selfloop construction.

Evidence trail
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    A minimal reproducible bimodal case appears at N=10 under small shortcut strength in the selfloop construction.

  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Research report ring_lazy_flux.

  • math_block research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Finding formula context in Lazy Ring Flux Baseline: A minimal reproducible bimodal case appears at N=10 under small shortcut strength in

  • dataset /data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json

    lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]

Linked reports Two-Target Lazy Ring MechanicsLazy Ring Jump-Over Mechanism (K2 vs K4)Lazy Ring Shortcut Beta Scan

finding ring_lazy_flux-c5

AW inversion and flux recursion agree to numerical precision, validating both the derivation and implementation.

Evidence trail
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    AW inversion and flux recursion agree to numerical precision, validating both the derivation and implementation.

  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Research report ring_lazy_flux.

  • math_block research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    Finding formula context in Lazy Ring Flux Baseline: AW inversion and flux recursion agree to numerical precision, validating both the

  • dataset /data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json

    lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]

Linked reports Destination-Scan Valley ControlLazy Ring Jump-Over Mechanism (K2 vs K4)Lazy Ring Shortcut Beta ScanTwo-Target Lazy Ring Mechanics

Interactive Dataset

Plot controls
window=1

N v [metric]

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Provenance: research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

Mathematical Logic Chain

From model assumptions to interpretation in a short, ordered chain.

Derivation Link

Adds a relation that links neighboring steps in the derivation chain.

target=⌊N/2⌋\text{target}=\lfloor N/2\rfloor

Overview

Derivation Link

Adds a relation that links neighboring steps in the derivation chain.

v=target+1v=\text{target}+1

Overview

Mathematical Principles

Showing 2 / 2

Overview EN

target=⌊N/2⌋\text{target}=\lfloor N/2\rfloor
Formula source

research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

Overview EN

v=target+1v=\text{target}+1
Formula source

research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

Narrative Sections

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

Overview

Research report ring_lazy_flux.

Source

research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

Reproducibility Commands

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

Showing 12 / 27

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lazy K2 paper geometry selfloop vs equal4 N100

Open Figure

lazy K2 selfloop smallp gallery compact beta0.02

Open Figure