valley-k-small
grid2d_blackboard_bimodality

Grid2D Blackboard Endpoint Case

This blackboard-style Grid2D case studies reflecting boundaries with start and target anchored at wall endpoints. The Z/S endpoint configurations are used to test whether a visually delayed hump is a true second peak or a window-edge artifact, with diagnostics linked to channel decomposition and path geometry.

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

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

Model

The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.

Method

The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.

Result

For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.

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-family

← 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

For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.

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_blackboard_bimodality-c1

The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.

Evidence trail
  • source_document research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    " y " ， g y>0 ； g x>0 。 ： p stay 。

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    ， transient， Q transient ， r ， p t+1 =Qp t, f(t)=r^->p p t 。 exact recursion ，AW MC 。

  • math_block research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    Model formula context in Grid2D Blackboard Endpoint Case: The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall

Linked reports Grid2D Rectangle BimodalityGrid2D Reflecting-Boundary BimodalityGrid2D Bimodality BaselineGrid2D Two-Target Double-Peak

METHOD

method grid2d_blackboard_bimodality-c2

The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.

Evidence trail
  • source_document research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    数据与复现: The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    参数表: The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.

  • math_block research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    Method formula context in Grid2D Blackboard Endpoint Case: The pipeline runs blackboard case builders, screenshot-style scans, and

  • dataset /data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json

    Z_scan [probability]: absorption -> delta_core, delta_open [probability]

Linked reports Grid2D Reflecting-Boundary BimodalityTwo-Target Lazy Ring MechanicsGrid2D Bimodality BaselineGrid2D Two-Target Double-Peak

RESULT

result grid2d_blackboard_bimodality-c3

For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.

Evidence trail
  • source_document research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    t_p2 late-window （ ）。

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    ： S=(30,45) ， T=(30,15) ， " + " 。 ： t_p1 102 ，valley ratio =1 ， + ；

  • math_block research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    Result formula context in Grid2D Blackboard Endpoint Case: For the scanned endpoint cases, dominant behavior is single-peak plus long tail

  • dataset /data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json

    Z_scan [probability]: absorption -> delta_core, delta_open [probability]

Linked reports Grid2D Reflecting-Boundary BimodalityFinal Multitimescale FPT and Encounter ReportCross-Model Luca Regime MapGrid2D Rectangle Bimodality

FINDING

finding grid2d_blackboard_bimodality-c4

Endpoint wall configurations remain primarily unimodal with a long tail under reflecting constraints.

Evidence trail
  • source_document research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    Endpoint wall configurations remain primarily unimodal with a long tail under reflecting constraints.

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    ： S=(30,45) ， T=(30,15) ， " + " 。 ： t_p1 102 ，valley ratio =1 ， + ；

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    ： x=28 to 29 x=31 to 32 ， y=15..45 。 bias： x=29..31 ； y=15..45 core =0.95 ， y=10..14,46..50 open =0.6 。

  • dataset /data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json

    Z_scan [probability]: absorption -> delta_core, delta_open [probability]

Linked reports Grid2D Reflecting-Boundary BimodalityFinal Multitimescale FPT and Encounter ReportGrid2D Rectangle Bimodality

finding grid2d_blackboard_bimodality-c5

The delayed hump in late windows is diagnostic-window edge behavior rather than a stable second structural peak.

Evidence trail
  • source_document research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    The delayed hump in late windows is diagnostic-window edge behavior rather than a stable second structural peak.

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    t_p2 late-window （ ）。

  • section_summary research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    ： S=(30,45) ， T=(30,15) ， " + " 。 ： t_p1 102 ，valley ratio =1 ， + ；

  • math_block research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

    Finding formula context in Grid2D Blackboard Endpoint Case: The delayed hump in late windows is diagnostic-window edge behavior rather

  • dataset /data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json

    Z_scan [probability]: absorption -> delta_core, delta_open [probability]

Linked reports Two-Target Lazy Ring MechanicsGrid2D Two-Target Double-PeakRing Derivation BackboneLazy Ring Jump-Over Mechanism (K2 vs K4)

Interactive Dataset

Plot controls
window=1

absorption delta_core, delta_open [probability]

Loading plot data…

Provenance: research/reports/grid2d_blackboard_bimodality/artifacts/data/Z_scan.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.

gy=−0.2, δopen=0.6g_y=-0.2,\ \delta_{\text{open}}=0.6

数值扫描与尝试（基于配置 Z）

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.

qu=α(u)qq_u=\alpha(u)q

模型定义与坐标约定

Derivation Link

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

pstay(u)=1−qup_{\text{stay}}(u)=1-q_u

模型定义与坐标约定

Distribution Setup

Defines first-passage probability objects used by later diagnostics.

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

FPT 定义与精确递推

Mathematical Principles

Showing 6 / 8

数值扫描与尝试（基于配置 Z） EN

gy=−0.2, δopen=0.6g_y=-0.2,\ \delta_{\text{open}}=0.6
Formula source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_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_blackboard_bimodality/manuscript/grid2d_blackboard_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_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

模型定义与坐标约定 EN

qu=α(u)qq_u=\alpha(u)q
Formula source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

模型定义与坐标约定 EN

pstay(u)=1−qup_{\text{stay}}(u)=1-q_u
Formula source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

FPT 定义与精确递推 EN

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

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

Show remaining formulas

FPT 定义与精确递推 EN

f(t)=r⊤Qt−1p0,t≥1,f(t)=r^\top Q^{t-1}p_0,\qquad t\ge 1,
Formula source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

FPT 定义与精确递推 EN

pt+1=Qpt, f(t)=r⊤ptp_{t+1}=Qp_t,\ f(t)=r^\top p_t
Formula source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

Narrative Sections

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

FPT 定义与精确递推

， transient， Q transient ， r ， p t+1 =Qp t, f(t)=r^->p p t 。 exact recursion ，AW MC 。

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

参数表

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

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

图形阅读指南

： 、 ； 、 door、 sticky ； local bias； (g x,g y) 。 。

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

数值扫描与尝试（基于配置 Z）

（ ）。Z0--Z4 t_p2 late-window （ ）。 tab:scan Z0--Z4 + ， ， " " 。 sticky ； （Z5/Z6） ， 。

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

数据与复现

数据与复现: this section connects Grid2D Blackboard Bimodality assumptions to measurable observables, verification commands, and downstream chapter claims.

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

模型定义与坐标约定

" y " ， g y>0 ； g x>0 。 ： p stay 。

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

Show more sections

结论与关键图示

： S=(30,45) ， T=(30,15) ， " + " 。 ： t_p1 102 ，valley ratio =1 ， + ；

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

统计指标

t_p2 late-window （ ）。

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

配置 S：小 N 竖向矩形（墙端点起止）

： 18×30 （ x=1..18, y=1..30 ； N=30 ）。 ： x=7 to 8 x=10 to 11 ， y=9..22 。

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

配置 Z：走廊端点起止（墙段内端点）

： x=28 to 29 x=31 to 32 ， y=15..45 。 bias： x=29..31 ； y=15..45 core =0.95 ， y=10..14,46..50 open =0.6 。

Source

research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report grid2d_blackboard_bimodality --lang en
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

Download Assets

Showing 20 / 40

Figure Gallery

Showing 12 / 40

Show remaining figures