valley-k-small
grid2d_bimodality

Grid2D Bimodality Baseline

This foundational Grid2D report establishes how biased/lazy random walks generate first-passage bimodality on a square lattice. It unifies model constraints, defect-aware propagators, AW inversion, and candidate-case scans into one auditable chain that separates genuine two-channel mechanisms from plotting artifacts.

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

Updated: 9 Jun 2026, 21:57:41 UTC

Model

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

Method

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

Result

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

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 0: Reading Guide & Notation

Also appears in chapter-0-reading-guidechapter-1-core-fptchapter-2-grid2d-familychapter-6-repro-validation

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.

grid2d progression

grid2d track links 13 reports into one continuous argument.

Checkpoints: 13 Claims: 63

Open arc sequence
  1. Grid2D Bimodality Baseline
  2. Grid2D Reflecting-Boundary Bimodality
  3. Grid2D Blackboard Endpoint Case
  4. Grid2D Two-Target Double-Peak
  5. 2D Two-Walker Encounter With Shortcut
  6. Grid2D Rectangle Bimodality
  7. Grid2D Membrane Near Target
  8. Grid2D One Target — Base
  9. Grid2D One Target — Exit Timing
  10. Grid2D One Target — Valley/Peak Budget
  11. Grid2D One Target — Window Measures
  12. Grid2D One vs Two Target — Gating
  13. Grid2D Two-Target Bias-Radius Scaffold

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

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

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 grid2d_bimodality-c1

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

Evidence trail
  • 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

Linked reports Grid2D Two-Target Double-PeakTwo-Target Lazy Ring MechanicsGrid2D Rectangle BimodalityGrid2D Reflecting-Boundary Bimodality

METHOD

method grid2d_bimodality-c2

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

Evidence trail
  • 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]

Linked reports Two-Target Lazy Ring MechanicsRing Derivation BackboneGrid2D Blackboard Endpoint CaseGrid2D Rectangle Bimodality

RESULT

result grid2d_bimodality-c3

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

Evidence trail
  • 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]

Linked reports Grid2D Two-Target Double-PeakGrid2D Rectangle BimodalityLazy Ring Jump-Over Mechanism (K2 vs K4)Grid2D Blackboard Endpoint Case

FINDING

finding grid2d_bimodality-c4

A unified FPT criterion distinguishes structural bimodality from numerical or visualization artifacts.

Evidence trail
  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    A unified FPT criterion distinguishes structural bimodality from numerical or visualization artifacts.

  • 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

    ： 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.

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

    Finding formula context in Grid2D Bimodality Baseline: A unified FPT criterion distinguishes structural bimodality from numerical or

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

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

Linked reports Two-Target Lazy Ring MechanicsCross-Model Luca Regime MapFinal Multitimescale FPT and Encounter ReportGrid2D Blackboard Endpoint Case

finding grid2d_bimodality-c5

Candidate corridor/bias settings show that delayed channels can be amplified without changing the target definition.

Evidence trail
  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

    Candidate corridor/bias settings show that delayed channels can be amplified without changing the target definition.

  • 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

    ： 、 ； 、 （ p pass ）、 sticky ； local bias； (g x,g y) 。 ， 、 P(n,t) 、FPT ， ； B valley。

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

    Finding formula context in Grid2D Bimodality Baseline: Candidate corridor/bias settings show that delayed channels can be amplified

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

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

Linked reports Grid2D Rectangle BimodalityTwo-Target Lazy Ring MechanicsGrid2D Membrane Near Target

Interactive Dataset

Plot controls
window=1

hv_over_max h1, h2, h2_over_h1

Loading plot data…

Provenance: research/reports/grid2d_bimodality/artifacts/data/scan_candidate_C_bias.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.

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

结论与关键图示

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}

模型定义与坐标约定

Derivation Link

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

Δ=δpstay\Delta=\delta p_{\text{stay}}

模型定义与坐标约定

Derivation Link

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

qsite=factor⋅qq_{\text{site}}=\text{factor}\cdot q

模型定义与坐标约定

Distribution Setup

Defines first-passage probability objects used by later diagnostics.

Av,u=Pr⁡(u→v)A_{v,u}=\Pr(u\to v)

解析推导：从无缺陷 propagator 到时域 FPT

Derivation Link

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

pt+1=Apt\mathbf{p}_{t+1}=A\mathbf{p}_t

解析推导：从无缺陷 propagator 到时域 FPT

Mathematical Principles

Showing 6 / 14

结论与关键图示 EN

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

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

模型定义与坐标约定 EN

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}
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

模型定义与坐标约定 EN

Δ=δpstay\Delta=\delta p_{\text{stay}}
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

模型定义与坐标约定 EN

qsite=factor⋅qq_{\text{site}}=\text{factor}\cdot q
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

Av,u=Pr⁡(u→v)A_{v,u}=\Pr(u\to v)
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

pt+1=Apt\mathbf{p}_{t+1}=A\mathbf{p}_t
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

Show remaining formulas

解析推导：从无缺陷 propagator 到时域 FPT EN

Pn0(n,t)=en⊤Aten0,P~(z)=∑t≥0Atzt=(I−zA)−1,P_{n_0}(n,t)=\mathbf{e}_n^\top A^t \mathbf{e}_{n_0}, \qquad \tilde{\mathbf{P}}(z)=\sum_{t\ge 0} A^t z^t=(I-zA)^{-1},
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

P~n0(n,z)=en⊤(I−zA)−1en0.\tilde P_{n_0}(n,z)=\mathbf{e}_n^\top (I-zA)^{-1}\mathbf{e}_{n_0}.
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

sk(γ)=q (λk(γ)−1),Λk1,k2=1+sk1(γx)+sk2(γy)2.s_k^{(\gamma)} = q\,\bigl(\lambda_k^{(\gamma)}-1\bigr),\qquad \Lambda_{k_1,k_2}=1+\frac{s_{k_1}^{(\gamma_x)}+s_{k_2}^{(\gamma_y)}}{2}.
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

Pn0(γ)(n,t)=∑k1∑k2hk1(γx)(x,x0) hk2(γy)(y,y0) Λk1,k2t.P^{(\gamma)}_{n_0}(n,t)=\sum_{k_1}\sum_{k_2} h_{k_1}^{(\gamma_x)}(x,x_0)\,h_{k_2}^{(\gamma_y)}(y,y_0)\,\Lambda_{k_1,k_2}^t.
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

P~n0(γ)(n,z)=∑k1∑k2hk1(γx)(x,x0) hk2(γy)(y,y0)1−zΛk1,k2.\tilde P^{(\gamma)}_{n_0}(n,z)=\sum_{k_1}\sum_{k_2} \frac{h_{k_1}^{(\gamma_x)}(x,x_0)\,h_{k_2}^{(\gamma_y)}(y,y_0)}{1-z\Lambda_{k_1,k_2}}.
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

θk=2πkN,λk(per)=1+g2e−iθk+1−g2eiθk,hk(per)(x,x0)=1Neiθk(x−x0).\theta_k=\frac{2\pi k}{N},\quad \lambda_k^{(\text{per})}=\frac{1+g}{2}e^{-i\theta_k}+\frac{1-g}{2}e^{i\theta_k},\quad h_k^{(\text{per})}(x,x_0)=\frac{1}{N}e^{i\theta_k(x-x_0)}.
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

pℓ=(1+g)/2,  pr=(1−g)/2,  ρ=pr/pℓp_\ell=(1+g)/2,\;p_r=(1-g)/2,\;\rho=p_r/p_\ell
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

λ0(ref)=1,λk(ref)=2pℓprcos⁡kπN (k≥1),hk(ref)(x,x0)=uk(x)uk(x0)πxπx0,\lambda_0^{(\text{ref})}=1,\quad \lambda_k^{(\text{ref})}=2\sqrt{p_\ell p_r}\cos\frac{k\pi}{N}\ (k\ge 1),\quad h_k^{(\text{ref})}(x,x_0)=u_k(x)u_k(x_0)\sqrt{\frac{\pi_x}{\pi_{x_0}}},
Formula source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

Narrative Sections

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

三类构造与机制

、 （ 、FPT ， ）。 A： + PhysRevE 102, 062124 "periodic + bias" 。 ， 。

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

与参考文献的对应

PhysRevE 102, 062124： + bias （1D ）， A C wrap-around 。 arXiv:2311.00464v2 Fig.3： bias、permeable barrier sticky （20 % stay, 80/20 door, sticky factor 0.2） B/C 。

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

参数表

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

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

图形阅读指南

： 、 ； 、 （ p pass ）、 sticky ； local bias； (g x,g y) 。 ， 、 P(n,t) 、FPT ， ； B valley。

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

数值基线：Exact/flux 递推与 MC

Exact transient f(t)=r^->p Q^ t-1 p 0 （ eq:exact-recursion ）， ， AW MC （ -- ）。MC " / " ， 。 f(t) ， t_p1,t_p2 t v ， h 1=f, h 2=f, peak ratio =((h 1,h 2))/((h 1,h 2)), valley ratio =(f(t v))/((h 1,h 2)).

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

模型定义与坐标约定

" y " ， g y>0 。 、 （ x 、 y ）

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

Show more sections

直观机制图解：两通道混合 + 时间尺度分离

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

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

结论与关键图示

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

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT

： 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.

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

边界条件的物理含义与保留价值

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

Source

research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py run --report grid2d_bimodality -- python3 code/bimodality_2d_pipeline.py
  • python3 scripts/reportctl.py build --report grid2d_bimodality --lang cn
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

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