valley-k-small
grid2d_reflecting_bimodality

Grid2D Reflecting-Boundary Bimodality

This reflecting-boundary Grid2D report tests whether bimodality survives when periodic shortcuts are removed and all walls reflect. Using representative detour, pore, and transport-track cases, it shows that multi-channel timing structure can persist under strict boundary confinement.

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

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

Model

The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).

Method

Case families are evaluated under common PMF/hazard diagnostics and compared by channel decomposition and timing windows.

Result

Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.

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 2: Grid2D Family

Also appears in chapter-2-grid2d-familychapter-6-repro-validation

← 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.

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

Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.

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_reflecting_bimodality-c1

The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).

Evidence trail
  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    " y " ， g y>0 。 ： p stay 。 (1) g x>0 p left p right ， g x>0 （ g x<0 ）

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    （first-passage time） PMF f(t)= ，CDF F(t)= (t) ， ， f(t) “ t ” ， f(t) 。 ， F(t) “ ”， S(t) “ ”

  • math_block research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Model formula context in Grid2D Reflecting-Boundary Bimodality: The model keeps a reflecting 2D lattice with absorbing target and

Linked reports Grid2D Two-Target Double-PeakGrid2D Bimodality BaselineGrid2D Blackboard Endpoint CaseGrid2D Rectangle Bimodality

METHOD

method grid2d_reflecting_bimodality-c2

Case families are evaluated under common PMF/hazard diagnostics and compared by channel decomposition and timing windows.

Evidence trail
  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Case families are evaluated under common PMF/hazard diagnostics and compared by channel decomposition and timing windows.

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    （first-passage time） PMF f(t)= ，CDF F(t)= (t) ， ， f(t) “ t ” ， f(t) 。 ， F(t) “ ”， S(t) “ ”

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Method formula context in Grid2D Reflecting-Boundary Bimodality: Case families are evaluated under common PMF/hazard diagnostics and

  • dataset /data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json

    cases_reflecting_summary [probability]: gx -> q [probability]

Linked reports Grid2D Blackboard Endpoint CaseGrid2D Bimodality BaselineGrid2D Rectangle BimodalityGrid2D Two-Target Double-Peak

RESULT

result grid2d_reflecting_bimodality-c3

Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.

Evidence trail
  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    ： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。 vs ：R1/R7/C3 " + " / ；R6 door ；NB4/NB5 + ；S1/S2 sticky + ， " " fast/slow

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    双峰指标汇总: Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.

  • math_block research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Result formula context in Grid2D Reflecting-Boundary Bimodality: Several reflecting cases preserve clear early/late channel separation

  • dataset /data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json

    cases_reflecting_summary [probability]: gx -> q [probability]

Linked reports Grid2D Blackboard Endpoint CaseLazy Ring Flux BaselineFinal Multitimescale FPT and Encounter Report

FINDING

finding grid2d_reflecting_bimodality-c4

Bimodality can persist under fully reflecting boundaries when competing pathways remain topologically distinct.

Evidence trail
  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Bimodality can persist under fully reflecting boundaries when competing pathways remain topologically distinct.

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    ： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。 vs ：R1/R7/C3 " + " / ；R6 door ；NB4/NB5 + ；S1/S2 sticky + ， " " fast/slow

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    双峰指标汇总: Bimodality can persist under fully reflecting boundaries when competing pathways remain topologically distinct.

  • math_block research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Finding formula context in Grid2D Reflecting-Boundary Bimodality: Bimodality can persist under fully reflecting boundaries when competing

  • dataset /data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json

    cases_reflecting_summary [probability]: gx -> q [probability]

Linked reports Grid2D Blackboard Endpoint CaseLazy Ring Jump-Over Mechanism (K2 vs K4)Ring Valley Regime MapGrid2D Bimodality Baseline

finding grid2d_reflecting_bimodality-c5

Pore/track structures modify delay channels through accessibility, not only through drift magnitude.

Evidence trail
  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Pore/track structures modify delay channels through accessibility, not only through drift magnitude.

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    ： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。 vs ：R1/R7/C3 " + " / ；R6 door ；NB4/NB5 + ；S1/S2 sticky + ， " " fast/slow

  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

    Finding formula context in Grid2D Reflecting-Boundary Bimodality: Pore/track structures modify delay channels through accessibility, not

  • dataset /data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json

    cases_reflecting_summary [probability]: gx -> q [probability]

Linked reports Grid2D Rectangle BimodalityCross-Model Luca Regime MapFinal Multitimescale FPT and Encounter ReportGrid2D Bimodality Baseline

Interactive Dataset

Plot controls
window=1

aw_inversion_seconds_median aw_estimated_total_seconds, aw_eval_per_z_seconds [metric]

Loading plot data…

Provenance: research/reports/grid2d_reflecting_bimodality/artifacts/data/aw_exact_speed.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.

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

Derivation Link

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

τ=inf⁡{t≥1: Xt=(xt,yt)},\tau=\inf\{t\ge 1:\,X_t=(x_t,y_t)\},

FPT 基础：定义、直观与三种计算方法

Distribution Setup

Defines first-passage probability objects used by later diagnostics.

F(t)=Pr⁡(τ≤t)F(t)=\Pr(\tau\le t)

FPT 基础：定义、直观与三种计算方法

Distribution Setup

Defines first-passage probability objects used by later diagnostics.

S(t)=Pr⁡(τ>t)=1−F(t).S(t)=\Pr(\tau>t)=1-F(t).

FPT 基础：定义、直观与三种计算方法

Mathematical Principles

Showing 6 / 14

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_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

FPT 基础：定义、直观与三种计算方法 EN

τ=inf⁡{t≥1: Xt=(xt,yt)},\tau=\inf\{t\ge 1:\,X_t=(x_t,y_t)\},
Formula source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

FPT 基础：定义、直观与三种计算方法 EN

F(t)=Pr⁡(τ≤t)F(t)=\Pr(\tau\le t)
Formula source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

FPT 基础：定义、直观与三种计算方法 EN

S(t)=Pr⁡(τ>t)=1−F(t).S(t)=\Pr(\tau>t)=1-F(t).
Formula source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

Show remaining formulas

FPT 基础：定义、直观与三种计算方法 EN

pt+1=Qpt,f(t)=r⊤pt,t≥1,p_{t+1}=Qp_t,\qquad f(t)=r^\top p_t,\quad t\ge 1,
Formula source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

FPT 基础：定义、直观与三种计算方法 EN

f(t)≈r−t1m∑k=0m−1f~(zk)e−2πikt/m,f(t)\approx r^{-t}\frac{1}{m}\sum_{k=0}^{m-1}\tilde f(z_k)e^{-2\pi i k t/m},
Formula source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

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_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

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

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

解析推导：从无缺陷 propagator 到时域 FPT EN

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

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

Narrative Sections

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

FPT 基础：定义、直观与三种计算方法

（first-passage time） PMF f(t)= ，CDF F(t)= (t) ， ， f(t) “ t ” ， f(t) 。 ， F(t) “ ”， S(t) “ ”

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

参数表

参数表: this section connects Grid2D Reflecting Bimodality assumptions to measurable observables, verification commands, and downstream chapter claims.

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

双峰指标汇总

双峰指标汇总: this section connects Grid2D Reflecting Bimodality assumptions to measurable observables, verification commands, and downstream chapter claims.

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

图形阅读指南

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

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

模型定义与坐标约定

" y " ， g y>0 。 ： p stay 。 (1) g x>0 p left p right ， g x>0 （ g x<0 ）

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

Show more sections

直观机制图解：两通道混合 + 时间尺度分离

： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

经典构造与机制

（R1、R7、C3、R6、NB4、NB5、S1、S2） 。 R1： vs U " " ： fast， U slow。

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

结论与关键图示

： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。 vs ：R1/R7/C3 " + " / ；R6 door ；NB4/NB5 + ；S1/S2 sticky + ， " " fast/slow

Source

research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_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_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py run --report grid2d_reflecting_bimodality -- python3 code/reflecting_bimodality_pipeline.py
  • python3 scripts/reportctl.py build --report grid2d_reflecting_bimodality --lang cn
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

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