valley-k-small
ring_valley

Ring Valley Regime Map

This valley-study report analyzes a non-lazy K-neighbor ring with a one-way shortcut 6->N/2+1 under the Fig.3 peak rule. Exact AW inversion and MC validation map bimodality windows across even N and K: no bimodality for K=2 after parity-aware coarse graining, but clear windows for K=4,6,8.

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

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

Model

The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut placement.

Method

The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.

Result

Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.

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-baseline

← 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

Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.

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-c1

The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut placement.

Evidence trail
  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex

    The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut placement.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Model and graph construction (one-way shortcut): The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality ranges detected under the Fig. 3 rule are printed to stdout.

  • math_block research/reports/ring_valley/manuscript/ring_valley_en.tex

    Classification of trajectories

  • math_block research/reports/ring_valley/manuscript/ring_valley_en.tex

    Directed-shortcut transition kernel

Linked reports Destination-Scan Valley ControlLazy Ring Flux BaselineTwo-Target Lazy Ring MechanicsRing Derivation Backbone

METHOD

method ring_valley-c2

The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.

Evidence trail
  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex

    The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality ranges detected under the Fig. 3 rule are printed to stdout.

  • math_block research/reports/ring_valley/manuscript/ring_valley_en.tex

    Classification of trajectories

  • math_block research/reports/ring_valley/manuscript/ring_valley_en.tex

    Directed-shortcut transition kernel

  • dataset /data/v1/reports/ring_valley/series/bimodality_scan.json

    bimodality_scan: n -> bimodal

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

RESULT

result ring_valley-c3

Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.

Evidence trail
  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex

    Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality ranges detected under the Fig. 3 rule are printed to stdout.

  • math_block research/reports/ring_valley/manuscript/ring_valley_en.tex

    Classification of trajectories

  • dataset /data/v1/reports/ring_valley/series/bimodality_scan.json

    bimodality_scan: n -> bimodal

Linked reports Lazy Ring Shortcut Beta ScanCross-Model Luca Regime MapTwo-Target Lazy Ring MechanicsGrid2D Rectangle Bimodality

FINDING

finding ring_valley-c4

Bimodality windows emerge for K=4, K=6, and K=8 with distinct N bands.

Evidence trail
  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality windows emerge for K=4, K=6, and K=8 with distinct N bands.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality ranges detected under the Fig. 3 rule are printed to stdout.

  • math_block research/reports/ring_valley/manuscript/ring_valley_en.tex

    Classification of trajectories

  • dataset /data/v1/reports/ring_valley/series/bimodality_scan.json

    bimodality_scan: n -> bimodal

Linked reports Grid2D Reflecting-Boundary BimodalityTwo-Target Lazy Ring Mechanics

finding ring_valley-c5

Under Fig.3 criteria with parity-aware treatment, K=2 shows no robust two-peak regime in the scanned range.

Evidence trail
  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex

    Under Fig.3 criteria with parity-aware treatment, K=2 shows no robust two-peak regime in the scanned range.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.

  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

    Let the first two peaks be at t1<t2, with valley time t valley between them, and. Classes by first-passage time T (valley + second-peak based split): Note: the usual "second-peak

  • math_block research/reports/ring_valley/manuscript/ring_valley_en.tex

    Classification of trajectories

  • dataset /data/v1/reports/ring_valley/series/bimodality_scan.json

    bimodality_scan: n -> bimodal

Linked reports Final Multitimescale FPT and Encounter ReportGrid2D Blackboard Endpoint CaseGrid2D Membrane Near Target

Interactive Dataset

Plot controls
window=1

n bimodal

Loading plot data…

Provenance: research/reports/ring_valley/artifacts/data/bimodality_scan.json

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.

direct:T<tvalley−Δ,valley:∣T−tvalley∣≤Δ,intermediate:tvalley+Δ<T≤t2+Δ,indirect:T>t2+Δ.\begin{aligned} \text{direct:}\quad & T < t_{\text{valley}}-\Delta,\\ \text{valley:}\quad & |T-t_{\text{valley}}|\le \Delta,\\ \text{intermediate:}\quad & t_{\text{valley}}+\Delta < T \le t_{2}+\Delta,\\ \text{indirect:}\quad & T > t_{2}+\Delta. \end{aligned}

Classification of trajectories

Derivation Link

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

Pi→j=1K1[j∈NK(i)]+1[i=6](1K+11 ⁣[j=N2+1]−1K(K+1)1[j∈NK(6)])P_{i\to j}=\frac{1}{K}\mathbf{1}[j\in\mathcal{N}_K(i)]+\mathbf{1}[i=6]\left(\frac{1}{K+1}\mathbf{1}\!\left[j=\frac{N}{2}+1\right]-\frac{1}{K(K+1)}\mathbf{1}[j\in\mathcal{N}_K(6)]\right)

Directed-shortcut transition kernel

Distribution Setup

Defines first-passage probability objects used by later diagnostics.

f(t)=Pr⁡[T=t],S(t)=1−∑u=1tf(u),h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=1-\sum_{u=1}^{t}f(u),\quad h(t)=\frac{f(t)}{S(t-1)}

FPT-survival-hazard consistency

Derivation Link

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

A~(m)=A(2m−1)+A(2m),m=1,…,⌊Tmax⁡2⌋\tilde{A}(m)=A(2m-1)+A(2m),\quad m=1,\dots,\left\lfloor\frac{T_{\max}}{2}\right\rfloor

Parity-aware coarse graining for K=2

Derivation Link

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

A(t1−1)<A(t1)>A(t1+1), A(t2−1)<A(t2)>A(t2+1), A(t2)≥0.01max⁡tA(t)A(t_1-1)<A(t_1)>A(t_1+1),\ A(t_2-1)<A(t_2)>A(t_2+1),\ A(t_2)\ge 0.01\max_{t}A(t)

Fig.3 two-peak admissibility criterion

Derivation Link

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

tvalley=arg⁡min⁡t1<t<t2A(t),Δ=max⁡{1,⌊0.05 (t2−t1)⌋}t_{\text{valley}}=\arg\min_{t_1<t<t_2}A(t),\quad \Delta=\max\left\{1,\left\lfloor0.05\,(t_2-t_1)\right\rfloor\right\}

Valley location and adaptive width

Mathematical Principles

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Classification of trajectories EN

direct:T<tvalley−Δ,valley:∣T−tvalley∣≤Δ,intermediate:tvalley+Δ<T≤t2+Δ,indirect:T>t2+Δ.\begin{aligned} \text{direct:}\quad & T < t_{\text{valley}}-\Delta,\\ \text{valley:}\quad & |T-t_{\text{valley}}|\le \Delta,\\ \text{intermediate:}\quad & t_{\text{valley}}+\Delta < T \le t_{2}+\Delta,\\ \text{indirect:}\quad & T > t_{2}+\Delta. \end{aligned}
Formula source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Directed-shortcut transition kernel EN

Pi→j=1K1[j∈NK(i)]+1[i=6](1K+11 ⁣[j=N2+1]−1K(K+1)1[j∈NK(6)])P_{i\to j}=\frac{1}{K}\mathbf{1}[j\in\mathcal{N}_K(i)]+\mathbf{1}[i=6]\left(\frac{1}{K+1}\mathbf{1}\!\left[j=\frac{N}{2}+1\right]-\frac{1}{K(K+1)}\mathbf{1}[j\in\mathcal{N}_K(6)]\right)
Formula source

research/reports/ring_valley/manuscript/ring_valley_en.tex

FPT-survival-hazard consistency EN

f(t)=Pr⁡[T=t],S(t)=1−∑u=1tf(u),h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=1-\sum_{u=1}^{t}f(u),\quad h(t)=\frac{f(t)}{S(t-1)}
Formula source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Parity-aware coarse graining for K=2 EN

A~(m)=A(2m−1)+A(2m),m=1,…,⌊Tmax⁡2⌋\tilde{A}(m)=A(2m-1)+A(2m),\quad m=1,\dots,\left\lfloor\frac{T_{\max}}{2}\right\rfloor
Formula source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Fig.3 two-peak admissibility criterion EN

A(t1−1)<A(t1)>A(t1+1), A(t2−1)<A(t2)>A(t2+1), A(t2)≥0.01max⁡tA(t)A(t_1-1)<A(t_1)>A(t_1+1),\ A(t_2-1)<A(t_2)>A(t_2+1),\ A(t_2)\ge 0.01\max_{t}A(t)
Formula source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Valley location and adaptive width EN

tvalley=arg⁡min⁡t1<t<t2A(t),Δ=max⁡{1,⌊0.05 (t2−t1)⌋}t_{\text{valley}}=\arg\min_{t_1<t<t_2}A(t),\quad \Delta=\max\left\{1,\left\lfloor0.05\,(t_2-t_1)\right\rfloor\right\}
Formula source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Narrative Sections

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

Classification of trajectories

Let the first two peaks be at t1<t2, with valley time t valley between them, and. Classes by first-passage time T (valley + second-peak based split): Note: the usual "second-peak window" is T [t2-,t2+], which is contained in the intermediate class.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Data appendix (key values)

Scan summary : K=2 none; Detailed per- N records in data/bimodality scan.json.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Data exports and seeds

Scan raw data: data/bimodality scan.json (details for each K, N, peaks, valley). Case metrics: data/N100K6 metrics.json (peaks, valley, MC class counts, shortcut usage);

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Exact dynamics

Analytical AW inversion of the generating function is the default solver (max steps fixed to 2000 for stability). The numerical master- iterator is kept only as a validation baseline.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Model and graph construction (one-way shortcut)

Model and graph construction (one-way shortcut): this section connects Ring Valley assumptions to measurable observables, verification commands, and downstream chapter claims.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Monte Carlo (joblib)

Vectorized batch simulator; batches spawned with independent seeds.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Show more sections

Peak and valley detection (Fig. 3 rule)

Peak at t if A(t - 1)<A(t)>A(t + 1) and A(t)>1e-7. If a second peak exists, it must be at least 1 % of the highest peak.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Scan setup

Bimodality ranges detected under the Fig. 3 rule are printed to stdout.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Scan summary (directed shortcut)

Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Single case (N=100, K=6, with fixed control parameter)

First-passage distribution Class definitions and proportions Peaks: t1=14, t2=101; Classes by T: direct <t valley -, valley |T-t valley |, intermediate t valley + < T t2+, indirect >t2+.

Source

research/reports/ring_valley/manuscript/ring_valley_en.tex

Reproducibility Commands

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