valley-k-small
Reading Guide

Chapter 0: Reading Guide & Notation

How to read this atlas, verify claims, and reuse symbols consistently across all reports.

Read time 10 min Reports 3 Interactive panels 3

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Chapter Guide

This chapter defines the reading protocol: start from claims, verify evidence paths, then inspect formula chains and interactive traces.

Notation is aligned across grid and ring families, so symbol reuse does not introduce hidden semantic drift.

After this chapter, every subsequent page can be read as one continuous argument instead of isolated report fragments.

Narrative Walkthrough

We begin by fixing the model premise: The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring interpretations.

We then move to an auditable method chain: The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.

Under the same diagnostic criterion, the chapter-level result and finding are: No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.

Carry notation and verified claims from Chapter 0: Reading Guide & Notation into Chapter 1: Core FPT Concepts, then extend mechanism and evidence without resetting assumptions.

Concept Cards

First-passage distribution

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

Reports 26

AW inversion

Discrete Cauchy / FFT inversion from generating functions.

Reports 11

Spectral decomposition

Eigenvalue / resolvent based derivations.

Reports 3

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

Theory Chain

Derivation Link grid2d_bimodality

Grid2D Bimodality Baseline · Derivation Link

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

(gx,gy,δ)=(−0.25,0.40,0.70)(g_x,g_y,\delta)=(-0.25,0.40,0.70)

Derivation Link grid2d_bimodality

Grid2D Bimodality Baseline · Derivation Link

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

pleft=q4(1+gx),pright=q4(1−gx),pdown=q4(1+gy),pup=q4(1−gy),pstay=1−q.\begin{aligned} p_{\text{left}} &= \frac{q}{4}(1+g_x),\quad p_{\text{right}} = \frac{q}{4}(1-g_x),\\ p_{\text{down}} &= \frac{q}{4}(1+g_y),\quad p_{\text{up}} = \frac{q}{4}(1-g_y),\\ p_{\text{stay}} &= 1-q. \end{aligned}

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)

Distribution Setup ring_two_target

Two-Target Lazy Ring Mechanics · 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)}

Interactive Evidence Panel

Grid2D Bimodality Baseline · scan_candidate_B_corridor [probability]

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

Interactive Dataset

Plot controls
window=1

l mass [probability]

Loading plot data…

Provenance: research/reports/grid2d_bimodality/artifacts/data/scan_candidate_B_corridor.json

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

Two-Target Lazy Ring Mechanics · small_scan_metrics [probability]

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

Interactive Dataset

Plot controls
window=1

beta q [probability]

Loading plot data…

Provenance: research/reports/ring_two_target/artifacts/data/small_scan_metrics.csv

Evidence Trail

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

Open evidence-node index
  • Grid2D Bimodality Baseline (Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.)
  • Ring Derivation Backbone (The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.)
  • Two-Target Lazy Ring Mechanics (No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.)

Claim Ledger

method grid2d_bimodality-c2 grid2d_bimodality

The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.

Open evidence links
  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.

  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    ： propagator -> propagator -> FPT -> AW 。 。 A ， A v,u = ， 1， p t+1 =A p t 。 P n0 (n,t)= e n^->p A^t e n0, P (z)= t0 A^t z^, P n0 (n,z)= e n^->p (I-zA)e-1 e n0.

  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    reflecting / ： “ ”。 periodic / ， ， ， （ A）。

  • math_block research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    Method formula context in Grid2D Bimodality Baseline: The method links defect-free and defect-corrected propagators to generating-function

  • dataset /data/v1/reports/grid2d_bimodality/series/scan_candidate_b_corridor-probability.json

    scan_candidate_B_corridor [probability]: l -> mass [probability]

method ring_deriv_k2-c2 ring_deriv_k2

The report derives Green-function style propagators, constructs defect-resolvent corrections, and obtains first-passage generating forms that can be numerically inverted.

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

    The report derives Green-function style propagators, constructs defect-resolvent corrections, and obtains first-passage generating forms that can be numerically inverted.

  • 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

  • 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.

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

    Method formula context in Ring Derivation Backbone: The report derives Green-function style propagators, constructs defect-resolvent

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

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

method ring_two_target-c2 ring_two_target

Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.

Open evidence links
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Method formula context in Two-Target Lazy Ring Mechanics: Exact generating-function/AW inversion is combined with parameter scans and

  • dataset /data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json

    small_scan_metrics [probability]: beta -> q [probability]

model ring_deriv_k2-c1 ring_deriv_k2

The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring interpretations.

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

    The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring

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

    We consider a discrete-time random walk on a 1D ring (cycle) of size N with periodic boundary conditions (PBC). Transition matrices and convention.

  • 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.

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

    Model formula context in Ring Derivation Backbone: The setting is a finite ring random walk with periodic indexing and one directed

model grid2d_bimodality-c1 grid2d_bimodality

The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary assumptions.

Open evidence links
  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary assumptions.

  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    " y " ， g y>0 。 、 （ x 、 y ）

  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    reflecting / ： “ ”。 periodic / ， ， ， （ A）。

  • math_block research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    Model formula context in Grid2D Bimodality Baseline: The model is a two-dimensional N×N lattice with an absorbing target, anisotropic

model ring_two_target-c1 ring_two_target

The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.

Open evidence links
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Model formula context in Two-Target Lazy Ring Mechanics: The model places two absorbing targets on a lazy ring with optional directed

result ring_two_target-c3 ring_two_target

No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.

Open evidence links
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

    No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and

  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Result formula context in Two-Target Lazy Ring Mechanics: No-shortcut drift can already produce strong bimodality, while shortcut

  • dataset /data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json

    small_scan_metrics [probability]: beta -> q [probability]

result ring_deriv_k2-c3 ring_deriv_k2

The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.

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

    The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.

  • 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

    Result formula context in Ring Derivation Backbone: The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and

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

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

Chapter Summary

How to read this atlas, verify claims, and reuse symbols consistently across all reports.

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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