valley-k-small
Core Concepts

Chapter 1: Core FPT Concepts

Build the common vocabulary of f(t), survival, hazard, and practical bimodality diagnostics.

Read time 12 min Reports 4 Interactive panels 4

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Chapter Guide

The first-passage distribution is interpreted through complementary views: density, cumulative survival, and hazard-style turning points.

Bimodality is treated as an evidence-backed diagnosis, not a visual impression, and every chapter keeps this constraint.

The same diagnostic language is reused in both lattice and ring settings to keep conclusions comparable.

Bridge from Chapter 0: now that notation and audit protocol are fixed, we formalize f(t), S(t), and h(t) as the common diagnostic language.

Narrative Walkthrough

We begin by fixing the model premise: The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

We then move to an auditable method chain: The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

Under the same diagnostic criterion, the chapter-level result and finding are: Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion. Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.

Carry notation and verified claims from Chapter 1: Core FPT Concepts into Chapter 2: Grid2D Family, 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

Survival and hazard

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

Reports 14

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

Reports 13

Hazard interpretation

Peak/valley interpretation using hazard dynamics.

Reports 12

Spectral decomposition

Eigenvalue / resolvent based derivations.

Reports 3

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 grid2d_two_target_double_peak

Grid2D Two-Target Double-Peak · 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_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_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

Grid2D Two-Target Double-Peak · method_comparison_c1 [probability]

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

Interactive Dataset

Plot controls
window=1

mfpt_truncation_scan_t_max mfpt_truncation_scan_mass_any [probability]

Loading plot data…

Provenance: research/reports/grid2d_two_target_double_peak/artifacts/data/method_comparison_c1.json

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

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.)
  • Grid2D Two-Target Double-Peak (Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.)
  • 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.)
  • 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

result grid2d_bimodality-c3 grid2d_bimodality

Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.

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

    Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.

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

    A （ + ）：FPT ； / " " " " ；AW exact recursion ，MC （ 、 ）。 B （ + ）： (g x,g y,)=(-0.25,0.40,0.70) ， x ， fast ； （ ）。

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

    。 B （ g x,g y ）； C door + sticky， wrap-around 。

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

    Result formula context in Grid2D Bimodality Baseline: Bimodality emerges when fast direct routes and delayed wrap-around/detour routes

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

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

model grid2d_two_target_double_peak-c1 grid2d_two_target_double_peak

The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

Open evidence links
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Model formula context in Grid2D Two-Target Double-Peak: The model places two absorbing targets in a reflecting lattice with fixed start

model ring_lazy_flux-c1 ring_lazy_flux

The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.

Open evidence links
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.

  • 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

    Model formula context in Lazy Ring Flux Baseline: The model is a lazy nearest-neighbor ring with one directed shortcut u->v

finding ring_two_target-c5 ring_two_target

Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.

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

    Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.

  • 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

    Finding formula context in Two-Target Lazy Ring Mechanics: Under no-shortcut drift, robust bimodality appears in reproducible parameter

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

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

method grid2d_two_target_double_peak-c2 grid2d_two_target_double_peak

The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

Open evidence links
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Method formula context in Grid2D Two-Target Double-Peak: The pipeline combines exact/approximate first-passage diagnostics, truncation

  • dataset /data/v1/reports/grid2d_two_target_double_peak/series/method_comparison_c1-probability.json

    method_comparison_c1 [probability]: mfpt_truncation_scan_t_max -> mfpt_truncation_scan_mass_any [probability]

method ring_lazy_flux-c2 ring_lazy_flux

The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.

Open evidence links
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.

  • 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

    Method formula context in Lazy Ring Flux Baseline: The pipeline derives the generating function analytically, inverts it via AW/FFT, and

  • 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]

result ring_lazy_flux-c3 ring_lazy_flux

A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.

Open evidence links
  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

    A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.

  • 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

    Result formula context in Lazy Ring Flux Baseline: A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger

  • 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]

result grid2d_two_target_double_peak-c3 grid2d_two_target_double_peak

Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.

Open evidence links
  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.

  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Research report grid2d_two_target_double_peak.

  • math_block research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

    Result formula context in Grid2D Two-Target Double-Peak: Double-peak regions appear when direct-to-near-target and delayed-to-far-target

  • dataset /data/v1/reports/grid2d_two_target_double_peak/series/method_comparison_c1-probability.json

    method_comparison_c1 [probability]: mfpt_truncation_scan_t_max -> mfpt_truncation_scan_mass_any [probability]

Chapter Summary

Build the common vocabulary of f(t), survival, hazard, and practical bimodality diagnostics.

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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