valley-k-small
ring_valley_dst

Destination-Scan Valley Control

This report fixes N=100 and K=6, then scans shortcut destination dst to control the second-peak structure of first-passage distributions. Deterministic flux/master-equation scans and Monte Carlo trajectory classes jointly identify where valley depth and second-peak prominence are maximized.

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

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

Model

The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.

Method

The workflow combines deterministic flux scans, AW-style first-passage diagnostics, and class-conditioned Monte Carlo paths under one peak/valley criterion.

Result

Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.

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-variantschapter-5-cross-model-synthesischapter-6-repro-validationchapter-7-outlook

← 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

Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.

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_valley_dst-c1

The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.

Evidence trail
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.

  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Research report ring_valley_dst.

  • math_block research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Model formula context in Destination-Scan Valley Control: The model uses a K=6 ring with one directed shortcut src->dst and an absorbing

Linked reports Two-Target Lazy Ring MechanicsLazy Ring Flux BaselineLazy Ring Shortcut Beta ScanRing Valley Regime Map

METHOD

method ring_valley_dst-c2

The workflow combines deterministic flux scans, AW-style first-passage diagnostics, and class-conditioned Monte Carlo paths under one peak/valley criterion.

Evidence trail
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    The workflow combines deterministic flux scans, AW-style first-passage diagnostics, and class-conditioned Monte Carlo paths under one peak/valley criterion.

  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Research report ring_valley_dst.

  • math_block research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Method formula context in Destination-Scan Valley Control: The workflow combines deterministic flux scans, AW-style first-passage

  • dataset /data/v1/reports/ring_valley_dst/series/scan-probability.json

    scan [probability]: steps -> mass, remaining [probability]

Linked reports Lazy Ring Shortcut Beta ScanGrid2D Rectangle BimodalityGrid2D Two-Target Double-PeakTwo-Target Lazy Ring Mechanics

RESULT

result ring_valley_dst-c3

Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.

Evidence trail
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining

  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Research report ring_valley_dst.

  • math_block research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Result formula context in Destination-Scan Valley Control: Destination scanning reveals structured dst windows where the second peak is

  • dataset /data/v1/reports/ring_valley_dst/series/scan-probability.json

    scan [probability]: steps -> mass, remaining [probability]

Linked reports Lazy Ring Shortcut Beta ScanTwo-Target Lazy Ring MechanicsGrid2D Blackboard Endpoint CaseLazy Ring Jump-Over Mechanism (K2 vs K4)

FINDING

finding ring_valley_dst-c4

Deterministic flux/master-equation results and Monte Carlo class decomposition agree on high-contrast destination windows.

Evidence trail
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Deterministic flux/master-equation results and Monte Carlo class decomposition agree on high-contrast destination windows.

  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Research report ring_valley_dst.

  • math_block research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Finding formula context in Destination-Scan Valley Control: Deterministic flux/master-equation results and Monte Carlo class decomposition

  • dataset /data/v1/reports/ring_valley_dst/series/scan-probability.json

    scan [probability]: steps -> mass, remaining [probability]

Linked reports Lazy Ring Flux BaselineLazy Ring Shortcut Beta ScanGrid2D Reflecting-Boundary BimodalityGrid2D Blackboard Endpoint Case

finding ring_valley_dst-c5

Scanning dst alone can substantially change second-peak height ratio and valley depth at fixed N and K.

Evidence trail
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Scanning dst alone can substantially change second-peak height ratio and valley depth at fixed N and K.

  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Research report ring_valley_dst.

  • math_block research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    Finding formula context in Destination-Scan Valley Control: Scanning dst alone can substantially change second-peak height ratio and

  • dataset /data/v1/reports/ring_valley_dst/series/scan-probability.json

    scan [probability]: steps -> mass, remaining [probability]

Linked reports Cross-Model Luca Regime MapGrid2D Blackboard Endpoint CaseLazy Ring Shortcut Beta ScanLazy Ring Shortcut Figure-1 Revision

Interactive Dataset

Plot controls
window=1

steps mass, remaining [probability]

Loading plot data…

Provenance: research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/scan.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

Showing 1 / 1

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_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

Narrative Sections

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

Overview

Research report ring_valley_dst.

Source

research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report ring_valley_dst --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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bimodality scan (N100K2_n0_1_target_50_src_6/latest)

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example curves (N100K2_n0_1_target_50_src_6/latest)

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peak times (N100K2_n0_1_target_50_src_6/latest)

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bimodality scan (N100K6_n0_1_target_50_src_1/latest)

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example curves (N100K6_n0_1_target_50_src_1/latest)

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peak times (N100K6_n0_1_target_50_src_1/latest)

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aw vs flux (N100K6_n0_1_target_50_src_6/latest)

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bimodality scan (N100K6_n0_1_target_50_src_6/latest)

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example curves (N100K6_n0_1_target_50_src_6/latest)

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peak times (N100K6_n0_1_target_50_src_6/latest)

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bimodality scan (N101K2_n0_1_target_30_src_1/latest)

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example curves (N101K2_n0_1_target_30_src_1/latest)

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peak times (N101K2_n0_1_target_30_src_1/latest)

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bimodality scan (N101K2_n0_1_target_40_src_1/latest)

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example curves (N101K2_n0_1_target_40_src_1/latest)

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peak times (N101K2_n0_1_target_40_src_1/latest)

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bimodality scan (N101K2_n0_1_target_51_src_1/latest)

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example curves (N101K2_n0_1_target_51_src_1/latest)

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peak times (N101K2_n0_1_target_51_src_1/latest)

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bimodality scan (N101K2_n0_1_target_60_src_1/latest)

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