ring_lazy_jump_ext_rev2

Lazy Ring Shortcut Figure-1 Revision

This revision reorganizes the lazy-shortcut extension into a publication-ready evidence flow: co-located Fig.1 overlays f(t) with window-level class bars, while threshold, window-shift/width, and MC-uncertainty analyses test robustness. The update strengthens readability and reproducibility without changing the core mechanism claims.

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 chapter keeps the same lazy ring shortcut setup used in the extension baseline and focuses on clearer evidence alignment between K=2 and K=4 under the selected beta regime.

Method

The pipeline exports standardized Fig.1 inputs, validates schema, renders stacked-bar co-located panels, and runs three sensitivity tracks: threshold sweep, window perturbation, and MC confidence intervals.

Result

Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.

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 chapter-6-repro-validation

← 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

Co-located Fig.1 directly aligns peak/valley timing with class proportions, improving interpretability over split-panel layouts.
Schema-validated JSON/CSV inputs make figure reconstruction auditable and reproducible.
Threshold and window perturbation scans preserve the main mechanism ranking rather than flipping conclusions.
MC uncertainty intervals are compatible with the reported phase statements for K=2 and K=4.

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 Shortcut Beta Scan
Next: Lazy Ring Flux Baseline

Same-Group Links

Lazy Ring Flux Baseline
adjacent in trackBeta / shortcut scan
Lazy Ring Shortcut Beta Scan
adjacent in trackBeta / shortcut scan
Lazy Ring Jump-Over Mechanism (K2 vs K4)
Beta / shortcut scan
Ring Derivation Backbone
Beta / shortcut scan

Cross-Group Bridges

Grid2D Blackboard Endpoint Case
Beta / shortcut scan
Grid2D One vs Two Target — Gating
Beta / shortcut scan
Grid2D Reflecting-Boundary Bimodality
Beta / shortcut scan
2D Two-Walker Encounter With Shortcut
Beta / shortcut scan

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

Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.

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_rev2-c1

The chapter keeps the same lazy ring shortcut setup used in the extension baseline and focuses on clearer evidence alignment between K=2 and K=4 under the selected beta regime.

Evidence trail

source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
The chapter keeps the same lazy ring shortcut setup used in the extension baseline and focuses on clearer evidence alignment between K=2 and K=4 under the selected beta regime.
section_summary research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Research report ring_lazy_jump_ext_rev2.
math_block research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Model formula context in Lazy Ring Shortcut Figure-1 Revision: The chapter keeps the same lazy ring shortcut setup used in the extension

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

METHOD

method ring_lazy_jump_ext_rev2-c2

The pipeline exports standardized Fig.1 inputs, validates schema, renders stacked-bar co-located panels, and runs three sensitivity tracks: threshold sweep, window perturbation, and MC confidence intervals.

Evidence trail

source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
The pipeline exports standardized Fig.1 inputs, validates schema, renders stacked-bar co-located panels, and runs three sensitivity tracks: threshold sweep, window perturbation
section_summary research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Research report ring_lazy_jump_ext_rev2.
math_block research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Method formula context in Lazy Ring Shortcut Figure-1 Revision: The pipeline exports standardized Fig.1 inputs, validates schema, renders
dataset /data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json
luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]

Linked reports Grid2D Blackboard Endpoint Case Ring Valley Regime Map Cross-Model Luca Regime Map Final Multitimescale FPT and Encounter Report

RESULT

result ring_lazy_jump_ext_rev2-c3

Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.

Evidence trail

source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main
section_summary research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Research report ring_lazy_jump_ext_rev2.
math_block research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Result formula context in Lazy Ring Shortcut Figure-1 Revision: Across the three sensitivity tracks, the qualitative mechanism
dataset /data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json
luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]

Linked reports Lazy Ring Shortcut Beta Scan Cross-Model Luca Regime Map Ring Valley Regime Map Grid2D Two-Target Double-Peak

FINDING

finding ring_lazy_jump_ext_rev2-c4

Co-located Fig.1 directly aligns peak/valley timing with class proportions, improving interpretability over split-panel layouts.

Evidence trail

source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Co-located Fig.1 directly aligns peak/valley timing with class proportions, improving interpretability over split-panel layouts.
section_summary research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Research report ring_lazy_jump_ext_rev2.
math_block research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Finding formula context in Lazy Ring Shortcut Figure-1 Revision: Co-located Fig.1 directly aligns peak/valley timing with class
dataset /data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json
luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]

Linked reports Final Multitimescale FPT and Encounter Report Ring Valley Regime Map Destination-Scan Valley Control

finding ring_lazy_jump_ext_rev2-c5

Schema-validated JSON/CSV inputs make figure reconstruction auditable and reproducible.

Evidence trail

source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Schema-validated JSON/CSV inputs make figure reconstruction auditable and reproducible.
section_summary research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Research report ring_lazy_jump_ext_rev2.
math_block research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
Finding formula context in Lazy Ring Shortcut Figure-1 Revision: Schema-validated JSON/CSV inputs make figure reconstruction auditable and
dataset /data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json
luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]

Linked reports Lazy Ring Flux Baseline Lazy Ring Shortcut Beta Scan Lazy Ring Jump-Over Mechanism (K2 vs K4)

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.

\Delta=\max\{1,\lfloor 0.05(t_2-t_1)\rfloor\}

Overview

Mathematical Principles

Showing 1 / 1

Overview EN

\Delta=\max\{1,\lfloor 0.05(t_2-t_1)\rfloor\}

Formula source

research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex

Narrative Sections

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

Overview

Research report ring_lazy_jump_ext_rev2.

Source

research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex

Reproducibility Commands

Open command list

python3 scripts/reportctl.py build --report ring_lazy_jump_ext_rev2 --lang en
python3 scripts/reportctl.py translation-qc
python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

Download Assets

Showing 20 / 40

Figure Gallery

Showing 12 / 39

fig2 overlap binbars beta0.01 x1350

Lazy Ring Shortcut Figure-1 Revision

Model

Method

Result

Book Position

Reading Path

Key Findings

Connected Reports

Track Continuity

Same-Group Links

Cross-Group Bridges

Narrative Arc Position

ring progression

Grid and ring mechanisms converge into cross-report synthesis.

Verifiable Claims

Report Objective

MODEL

METHOD

RESULT

FINDING

Interactive Dataset

Mathematical Logic Chain

Derivation Link

Mathematical Principles

Narrative Sections

Overview

Reproducibility Commands

Download Assets

Figure Gallery