valley-k-small
Ring Baseline

Chapter 3: Ring Family Baseline

Establish lazy and non-lazy ring baselines, then align inversion logic and reference behavior before shortcut variants.

Read time 12 min Reports 4 Interactive panels 4

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Chapter Guide

Ring models provide a compact setting for isolating drift, waiting probability, and spectral structure.

This baseline chapter is intentionally conservative: it first aligns lazy and non-lazy controls before adding shortcut perturbations.

The resulting baseline is reused in later chapters as a control for mechanism attribution.

Bridge from Chapter 2: Grid2D established boundary-sensitive diagnostics; this chapter keeps those diagnostics but shifts geometry to a ring so shortcut and lazy effects can be isolated.

Narrative Walkthrough

We begin by fixing the model premise: A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

We then move to an auditable method chain: The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.

Under the same diagnostic criterion, the chapter-level result and finding are: 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.…

Carry notation and verified claims from Chapter 3: Ring Family Baseline into Chapter 4: Shortcut Variants, then extend mechanism and evidence without resetting assumptions.

Concept Cards

AW inversion

Discrete Cauchy / FFT inversion from generating functions.

Reports 11

First-passage distribution

Core PMF/CDF/survival quantities used across the major report families.

Reports 26

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

Reports 13

Survival and hazard

Links between f(t), S(t), and hazard-style diagnostics.

Reports 14

Hazard interpretation

Peak/valley interpretation using hazard dynamics.

Reports 12

Spectral decomposition

Eigenvalue / resolvent based derivations.

Reports 3

Theory Chain

Distribution Setup ring_deriv_k2

Ring Derivation Backbone · Distribution Setup

Defines first-passage probability objects used by later diagnostics.

Pr⁡(Xt+1=n∣Xt=n′)=Bn,n′(defect-free ring),Pr⁡(Xt+1=n∣Xt=n′)=An,n′(with defect).\begin{aligned} \Pr(X_{t+1}=n\mid X_t=n') &= B_{n,n'} \qquad \text{(defect-free ring)},\\ \Pr(X_{t+1}=n\mid X_t=n') &= A_{n,n'} \qquad \text{(with defect)}. \end{aligned}

Spectral / Inversion Step ring_deriv_k2

Ring Derivation Backbone · Spectral / Inversion Step

Provides analytic inversion machinery for computing trajectories.

P(n,t+1)=∑n′Bn,n′P(n′,t)P(n,t+1)=\sum_{n'} B_{n,n'} P(n',t)

Derivation Link ring_lazy_flux

Lazy Ring Flux Baseline · Derivation Link

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

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

Derivation Link ring_lazy_flux

Lazy Ring Flux Baseline · Derivation Link

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

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

Distribution Setup ring_lazy_jump

Lazy Ring Jump-Over Mechanism (K2 vs K4) · 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)}

Derivation Link ring_valley

Ring Valley Regime Map · 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}

Derivation Link ring_valley

Ring Valley Regime Map · 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)

Interactive Evidence Panel

Ring Derivation Backbone · note_k2 [tabular probability]

Compare first/second peak prominence first, then adjust smoothing to test valley stability.

Interactive Dataset

Plot controls
window=1

N q, p [probability]

Loading plot data…

Provenance: research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Lazy Ring Flux Baseline · lazy_K2_equal4_paper_geometry_summary_cn [tabular metric]

Toggle series and tune smoothing to see how parameter shifts reweight fast versus delayed pathways.

Interactive Dataset

Plot controls
window=1

N v [metric]

Loading plot data…

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

Lazy Ring Jump-Over Mechanism (K2 vs K4) · scan_N_K4_beta002 [probability]

Compare first/second peak prominence first, then adjust smoothing to test valley stability.

Interactive Dataset

Plot controls
window=1

N q [probability]

Loading plot data…

Provenance: research/reports/ring_lazy_jump/artifacts/data/scan_N_K4_beta002.csv

Ring Valley Regime Map · bimodality_scan

Toggle series and tune smoothing to see how parameter shifts reweight fast versus delayed pathways.

Interactive Dataset

Plot controls
window=1

n bimodal

Loading plot data…

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

Evidence Trail

This chapter is presented as one coherent story. The underlying report artifacts are preserved as auditable evidence nodes.

Open evidence-node index
  • Ring Derivation Backbone (The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.)
  • Lazy Ring Flux Baseline (A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.)
  • Lazy Ring Jump-Over Mechanism (K2 vs K4) (Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.)
  • Ring Valley Regime Map (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.)

Claim Ledger

finding ring_deriv_k2-c5 ring_deriv_k2

Directed long-range links alter first-passage statistics through resolvent-level corrections rather than ad-hoc fitting.

Open evidence links
  • source_document research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Directed long-range links alter first-passage statistics through resolvent-level corrections rather than ad-hoc fitting.

  • section_summary research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.

  • section_summary research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n

  • math_block research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Finding formula context in Ring Derivation Backbone: Directed long-range links alter first-passage statistics through resolvent-level

  • dataset /data/v1/reports/ring_deriv_k2/series/note_k2-probability.json

    note_k2 [tabular probability]: N -> q, p [probability]

finding ring_lazy_flux-c5 ring_lazy_flux

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

Open evidence links
  • 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]

model ring_lazy_jump-c1 ring_lazy_jump

A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

Open evidence links
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Model formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): A lazy ring with one directed shortcut is used as the baseline setting

method ring_valley-c2 ring_valley

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

Open evidence links
  • 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

method ring_lazy_jump-c2 ring_lazy_jump

The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.

Open evidence links
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Method formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): The analysis combines AW inversion for exact first-passage series with

  • dataset /data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

model ring_valley-c1 ring_valley

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.

Open evidence links
  • 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

result ring_valley-c3 ring_valley

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.

Open evidence links
  • 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

result ring_lazy_jump-c3 ring_lazy_jump

Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.

Open evidence links
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.

  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Result formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Bimodality appears only in selected shortcut-strength intervals

  • dataset /data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

Chapter Summary

Establish lazy and non-lazy ring baselines, then align inversion logic and reference behavior before shortcut variants.

Open chapter glossary links
  • AW Inversion: Discrete Cauchy/FFT-based inversion from generating functions to time-domain FPT quantities.
  • Beta Scan: Parameter sweep over shortcut strength β to identify phase shifts and regime boundaries.
  • Bimodality Criterion: Operational criterion to separate true two-peak structure from noisy shoulders.
  • Claim Ledger: Structured mapping from statement to evidence paths and cross-report links.
  • Equal4 Baseline: Four-way equalized baseline used to compare shortcut effects under symmetric local movement.
  • First-Passage Time (FPT): Random time needed for the trajectory to hit an absorbing target for the first time.
  • Hazard Rate: Conditional probability of first passage at step t given survival up to t.
  • Renormalize Shortcut Mode: Base transition weights are rescaled after shortcut injection to preserve normalization constraints.
  • Selfloop Shortcut Mode: Shortcut probability mass is taken from self-loop probability without renormalizing other moves.
  • Survival Function: Probability that first passage has not happened by step t.
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