ring_lazy_jump_ext

Lazy Ring Shortcut Beta Scan

This extension quantifies how shortcut strength beta reshapes first-passage bimodality on lazy rings at fixed N=100, then checks transfer by N sweeps and Monte Carlo class decomposition. Across beta in [0,0.2], both peaks move earlier and tail decay accelerates, while K=4 preserves a wider and deeper bimodal window than K=2.

Book Position Reading Path Connected Reports Narrative Arc Position Verifiable Claims Interactive Figures Mathematical Logic Formula Library Narrative Notes Appendix & Assets

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

Model

The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched parameter settings.

Method

The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail diagnostics.

Result

Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.

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 4: Shortcut Variants

Also appears in chapter-4-shortcut-variants

← Previous chapter Next chapter →

Reading Path

Scan key findings first to decide whether this report is relevant.
Use the interactive panel to test parameter and shape sensitivity.
Then read the mathematical chain and formula library for derivation details.

Key Findings

At fixed N=100, beta sweeps show systematic left-shifts of peak times and a larger tail-decay rate as shortcut strength increases.
K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched beta schedules.
An anchored beta choice supports stable N sweeps where exact AW and MC class-level diagnostics stay consistent.
Tail diagnostics confirm that shortcut strengthening suppresses late-time mass and changes pathway composition, not only peak height.

Connected Reports

This report sits inside a shared chain. Use links below to move upstream/downstream and across model families.

Track Continuity

Previous: Lazy Ring Jump-Over Mechanism (K2 vs K4)
Next: Lazy Ring Shortcut Figure-1 Revision

Same-Group Links

Lazy Ring Jump-Over Mechanism (K2 vs K4)
adjacent in trackAW inversionBeta / shortcut scanFirst-passage distribution
Two-Target Lazy Ring Mechanics
AW inversionBeta / shortcut scanFirst-passage distribution
Ring Valley Regime Map
AW inversionBeta / shortcut scanFirst-passage distribution
Destination-Scan Valley Control
AW inversionBeta / shortcut scanFirst-passage distribution

Cross-Group Bridges

Grid2D Reflecting-Boundary Bimodality
AW inversionBeta / shortcut scanFirst-passage distribution
Cross-Model Luca Regime Map
AW inversionFirst-passage distributionHazard interpretation
2D Two-Walker Encounter With Shortcut
Beta / shortcut scanFirst-passage distributionHazard interpretation
Grid2D Membrane Near Target
First-passage distributionHazard interpretationSurvival and hazard

Narrative Arc Position

This report appears in one or more global arcs. Use these checkpoints to keep reading continuity across pages.

ring progression

ring track links 9 reports into one continuous argument.

Checkpoints: 9 Claims: 45

Open arc sequence

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

Verifiable Claims

Claims below are tied to explicit evidence paths so each statement can be audited.

Report Objective

Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.

Verification Steps

Read the key claims and their evidence references first.
Verify at least one equation card and one dataset panel against source paths.
Cross-check this report with upstream/downstream linked reports.

MODEL

model ring_lazy_jump_ext-c1

The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched parameter settings.

Evidence trail

source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched parameter settings.
section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Research report ring_lazy_jump_ext.
math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Model formula context in Lazy Ring Shortcut Beta Scan: The model keeps the lazy ring baseline with one directed shortcut under the

Linked reports Lazy Ring Flux Baseline Lazy Ring Jump-Over Mechanism (K2 vs K4)Lazy Ring Shortcut Figure-1 Revision Two-Target Lazy Ring Mechanics

METHOD

method ring_lazy_jump_ext-c2

The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail diagnostics.

Evidence trail

source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail
section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Research report ring_lazy_jump_ext.
math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Method formula context in Lazy Ring Shortcut Beta Scan: The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N
dataset /data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json
scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Destination-Scan Valley Control Two-Target Lazy Ring Mechanics Grid2D Rectangle Bimodality Lazy Ring Jump-Over Mechanism (K2 vs K4)

RESULT

result ring_lazy_jump_ext-c3

Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.

Evidence trail

source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.
section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Research report ring_lazy_jump_ext.
math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Result formula context in Lazy Ring Shortcut Beta Scan: Increasing beta advances both peaks and steepens tail decay
dataset /data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json
scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Ring Valley Regime Map Lazy Ring Shortcut Figure-1 Revision Cross-Model Luca Regime Map Grid2D Blackboard Endpoint Case

FINDING

finding ring_lazy_jump_ext-c4

At fixed N=100, beta sweeps show systematic left-shifts of peak times and a larger tail-decay rate as shortcut strength increases.

Evidence trail

source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
At fixed N=100, beta sweeps show systematic left-shifts of peak times and a larger tail-decay rate as shortcut strength increases.
section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Research report ring_lazy_jump_ext.
math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Finding formula context in Lazy Ring Shortcut Beta Scan: At fixed N=100, beta sweeps show systematic left-shifts of peak times and a
dataset /data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json
scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Grid2D One Target — Base Lazy Ring Flux Baseline Lazy Ring Jump-Over Mechanism (K2 vs K4)Destination-Scan Valley Control

finding ring_lazy_jump_ext-c5

K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched beta schedules.

Evidence trail

source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched beta schedules.
section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Research report ring_lazy_jump_ext.
math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
Finding formula context in Lazy Ring Shortcut Beta Scan: K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched
dataset /data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json
scan_N_K4_beta002 [probability]: N -> q [probability]

Linked reports Lazy Ring Shortcut Figure-1 Revision Grid2D Blackboard Endpoint Case Grid2D Reflecting-Boundary Bimodality Lazy Ring Jump-Over Mechanism (K2 vs K4)