valley-k-small
Grid2D Family

Chapter 2: Grid2D Family

From periodic baseline to reflecting and two-target variants, showing which structures preserve or break bimodality.

Read time 13 min Reports 5 Interactive panels 5

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Chapter Guide

Grid2D reports are organized as one progression: baseline, geometric constraints, reflecting boundaries, and two-target interactions.

The chapter highlights which mechanisms remain stable and which are sensitive to boundary or geometry changes.

Every subsection links back to the same claim ledger so conclusions can be checked without rereading all PDFs.

Bridge from Chapter 1: with shared FPT diagnostics in place, we test which Grid2D geometric constraints preserve or break bimodality.

Narrative Walkthrough

We begin by fixing the model premise: The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.

We then move to an auditable method chain: Case families are evaluated under common PMF/hazard diagnostics and compared by channel decomposition and timing windows.

Under the same diagnostic criterion, the chapter-level result and finding are: 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.…

Bridge note: keep the same hazard/survival diagnostics from Grid2D, then switch geometry to ring so shortcut and lazy parameters can be isolated without boundary-shape confounders.

Concept Cards

First-passage distribution

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

Reports 26

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

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

Reports 13

AW inversion

Discrete Cauchy / FFT inversion from generating functions.

Reports 11

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}

Derivation Link grid2d_blackboard_bimodality

Grid2D Blackboard Endpoint Case · 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

Derivation Link grid2d_blackboard_bimodality

Grid2D Blackboard Endpoint Case · 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_rect_bimodality

Grid2D Rectangle Bimodality · 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 grid2d_reflecting_bimodality

Grid2D Reflecting-Boundary Bimodality · 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}

Derivation Link grid2d_reflecting_bimodality

Grid2D Reflecting-Boundary Bimodality · Derivation Link

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

Δ=δpstay\Delta=\delta p_{\text{stay}}

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)}

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 Blackboard Endpoint Case · Z_scan [probability]

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

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

Grid2D Rectangle Bimodality · ot_scan_bias2d

Toggle series and tune smoothing to see how parameter shifts reweight fast versus delayed pathways.

Interactive Dataset

Plot controls
window=1

bx_values by_values

Loading plot data…

Provenance: research/reports/grid2d_rect_bimodality/artifacts/data/ot_scan_bias2d.json

Grid2D Reflecting-Boundary Bimodality · cases_reflecting_summary [probability]

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

Interactive Dataset

Plot controls
window=1

gx q [probability]

Loading plot data…

Provenance: research/reports/grid2d_reflecting_bimodality/code/config/cases_reflecting_summary.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

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 Blackboard Endpoint Case (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.)
  • Grid2D Rectangle Bimodality (Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.)
  • Grid2D Reflecting-Boundary Bimodality (Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.)
  • 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.)

Claim Ledger

finding grid2d_bimodality-c5 grid2d_bimodality

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

Open evidence links
  • 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]

result grid2d_blackboard_bimodality-c3 grid2d_blackboard_bimodality

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.

Open evidence links
  • 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]

result grid2d_rect_bimodality-c3 grid2d_rect_bimodality

Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.

Open evidence links
  • source_document research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.

  • section_summary research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Research report grid2d_rect_bimodality.

  • math_block research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Result formula context in Grid2D Rectangle Bimodality: Aspect ratio and endpoint arrangement shift the balance between direct and detour

  • dataset /data/v1/reports/grid2d_rect_bimodality/series/ot_scan_bias2d.json

    ot_scan_bias2d: bx_values -> by_values

result grid2d_reflecting_bimodality-c3 grid2d_reflecting_bimodality

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

Open evidence links
  • 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]

finding grid2d_two_target_double_peak-c5 grid2d_two_target_double_peak

Two-target competition creates a controlled mechanism for multi-timescale first-passage behavior.

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

    Two-target competition creates a controlled mechanism for multi-timescale first-passage behavior.

  • 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

    Finding formula context in Grid2D Two-Target Double-Peak: Two-target competition creates a controlled mechanism for multi-timescale

  • 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]

model grid2d_blackboard_bimodality-c1 grid2d_blackboard_bimodality

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

Open evidence links
  • 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

method grid2d_reflecting_bimodality-c2 grid2d_reflecting_bimodality

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

Open evidence links
  • 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]

method grid2d_blackboard_bimodality-c2 grid2d_blackboard_bimodality

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

Open evidence links
  • 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]

method grid2d_rect_bimodality-c2 grid2d_rect_bimodality

The workflow scans geometry and bias parameters while keeping first-passage diagnostics fixed, then compares pathway composition across rectangular configurations.

Open evidence links
  • source_document research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    The workflow scans geometry and bias parameters while keeping first-passage diagnostics fixed, then compares pathway composition across rectangular configurations.

  • section_summary research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Research report grid2d_rect_bimodality.

  • math_block research/reports/grid2d_rect_bimodality/artifacts/tables/ot_anchor_selection.tex

    Method formula context in Grid2D Rectangle Bimodality: The workflow scans geometry and bias parameters while keeping first-passage

  • dataset /data/v1/reports/grid2d_rect_bimodality/series/ot_scan_bias2d.json

    ot_scan_bias2d: bx_values -> by_values

model grid2d_reflecting_bimodality-c1 grid2d_reflecting_bimodality

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

Open evidence links
  • 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

Chapter Summary

From periodic baseline to reflecting and two-target variants, showing which structures preserve or break bimodality.

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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