valley-k-small
final_multitimescale_fpt_encounter

Final Multitimescale FPT and Encounter Report

shows the actual spatial configuration for each selected case. The left panel in each row is the physical reflecting one-dimensional lattice: the two walker starts are marked, the reflecting walls are drawn at sites (1) and (N), and the dominant early/late encounter sites are shown directly on the lattice.

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

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

Model

Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).

Method

For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.

Result

For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.

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 6: Reproducibility & Validation

Also appears in chapter-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.

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 current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.

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 final_multitimescale_fpt_encounter-c1

Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).

Evidence trail
  • source_document research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    shows the actual spatial configuration for each selected case. The left panel in each row is the physical reflecting one-dimensional lattice: the two walker starts are marked

  • math_block research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Model formula context in Final Multitimescale FPT and Encounter Report: Walker (i) has mobility (q i (0,1]).

Linked reports Lazy Ring Flux BaselineGrid2D Bimodality BaselineLazy Ring Shortcut Beta ScanGrid2D Two-Target Bias-Radius Scaffold

METHOD

method final_multitimescale_fpt_encounter-c2

For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.

Evidence trail
  • source_document research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Stage 2 introduced smoothing persistence, parity-pair filtering, and negative controls. Stage 2b added mechanism-focused checks: boundary-null controls, target-shift reranking

  • math_block research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Method formula context in Final Multitimescale FPT and Encounter Report: For the current finite one-dimensional reflecting lattice with

  • dataset /data/v1/reports/final_multitimescale_fpt_encounter/series/grid2d_local_bias_basin_scan-probability.json

    grid2d_local_bias_basin_scan [probability]: early_near_fraction -> global_bias, local_bias [probability]

Linked reports Cross-Model Luca Regime MapGrid2D Bimodality BaselineGrid2D Rectangle BimodalityGrid2D Two-Target Double-Peak

RESULT

result final_multitimescale_fpt_encounter-c3

[misc] Result statement: For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.

Evidence trail
  • source_document research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    [misc] Result statement: For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Stage 2 introduced smoothing persistence, parity-pair filtering, and negative controls. Stage 2b added mechanism-focused checks: boundary-null controls, target-shift reranking

  • math_block research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Result formula context in Final Multitimescale FPT and Encounter Report: [misc] Result statement: For the current finite one-dimensional

  • dataset /data/v1/reports/final_multitimescale_fpt_encounter/series/grid2d_local_bias_basin_scan-probability.json

    grid2d_local_bias_basin_scan [probability]: early_near_fraction -> global_bias, local_bias [probability]

Linked reports Grid2D Rectangle BimodalityGrid2D Two-Target Double-PeakCross-Model Luca Regime MapGrid2D Bimodality Baseline

FINDING

finding final_multitimescale_fpt_encounter-c4

For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.

Evidence trail
  • source_document research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.

  • section_summary research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    shows the actual spatial configuration for each selected case. The left panel in each row is the physical reflecting one-dimensional lattice: the two walker starts are marked

  • math_block research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

    Finding formula context in Final Multitimescale FPT and Encounter Report: For the current finite one-dimensional reflecting lattice with

  • dataset /data/v1/reports/final_multitimescale_fpt_encounter/series/grid2d_local_bias_basin_scan-probability.json

    grid2d_local_bias_basin_scan [probability]: early_near_fraction -> global_bias, local_bias [probability]

Linked reports Ring Valley Regime MapCross-Model Luca Regime MapGrid2D Bimodality BaselineGrid2D Membrane Near Target

Interactive Dataset

Plot controls
window=1

early_near_fraction global_bias, local_bias [probability]

Loading plot data…

Provenance: research/reports/final_multitimescale_fpt_encounter/artifacts/data/grid2d_local_bias_basin_scan.csv

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.

N=21,(a,b)=(6,2),q1=0.02,q2=0.90N=21,\qquad (a,b)=(6,2),\qquad q_1=0.02,\qquad q_2=0.90

Conclusion

Derivation Link

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

N=31,(a,b)=(2,4),q1=0.70,q2=0.20N=31,\qquad (a,b)=(2,4),\qquad q_1=0.70,\qquad q_2=0.20

Conclusion

Model Constraint

States the structural constraints and parameter ranges of the model.

Pi(1,1)=1−qi/2,Pi(1,2)=qi/2,Pi(N,N)=1−qi/2,Pi(N,N−1)=qi/2.P_i(1,1)=1-q_i/2,\quad P_i(1,2)=q_i/2,\qquad P_i(N,N)=1-q_i/2,\quad P_i(N,N-1)=q_i/2.

Model and Exact Absorbing-Chain Formulation

Model Constraint

States the structural constraints and parameter ranges of the model.

P=P1⊗P2.P=P_1\otimes P_2.

Model and Exact Absorbing-Chain Formulation

Model Constraint

States the structural constraints and parameter ranges of the model.

D={(k,k):1≤k≤N}.D=\{(k,k):1\leq k\leq N\}.

Model and Exact Absorbing-Chain Formulation

Distribution Setup

Defines first-passage probability objects used by later diagnostics.

fk(t)=αQt−1R⋅,k,f(t)=∑kfk(t).f_k(t)=\alpha Q^{t-1}R_{\cdot,k},\qquad f(t)=\sum_k f_k(t).

Model and Exact Absorbing-Chain Formulation

Mathematical Principles

Showing 6 / 10

Conclusion EN

N=21,(a,b)=(6,2),q1=0.02,q2=0.90N=21,\qquad (a,b)=(6,2),\qquad q_1=0.02,\qquad q_2=0.90
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Conclusion EN

N=31,(a,b)=(2,4),q1=0.70,q2=0.20N=31,\qquad (a,b)=(2,4),\qquad q_1=0.70,\qquad q_2=0.20
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Model and Exact Absorbing-Chain Formulation EN

Pi(1,1)=1−qi/2,Pi(1,2)=qi/2,Pi(N,N)=1−qi/2,Pi(N,N−1)=qi/2.P_i(1,1)=1-q_i/2,\quad P_i(1,2)=q_i/2,\qquad P_i(N,N)=1-q_i/2,\quad P_i(N,N-1)=q_i/2.
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Model and Exact Absorbing-Chain Formulation EN

P=P1⊗P2.P=P_1\otimes P_2.
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Model and Exact Absorbing-Chain Formulation EN

D={(k,k):1≤k≤N}.D=\{(k,k):1\leq k\leq N\}.
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Model and Exact Absorbing-Chain Formulation EN

fk(t)=αQt−1R⋅,k,f(t)=∑kfk(t).f_k(t)=\alpha Q^{t-1}R_{\cdot,k},\qquad f(t)=\sum_k f_k(t).
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Show remaining formulas

Operational Double-Peak Rule and Reading EN

F2(n)=f(2n−1)+f(2n),F_2(n)=f(2n-1)+f(2n),
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Formula and Green-Function Validation EN

E[T]=α(I−Q)−11,pk=α(I−Q)−1R⋅,k,\mathbb{E}[T]=\alpha (I-Q)^{-1}{\bf 1},\qquad p_k=\alpha (I-Q)^{-1}R_{\cdot,k},
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Formula and Green-Function Validation EN

τk=α(I−Q)−1(I−Q)−1R⋅,kpk.\tau_k =\frac{\alpha (I-Q)^{-1}(I-Q)^{-1}R_{\cdot,k}}{p_k}.
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Weak Target-Shift Shoulder EN

N=31,(a,b)=(2,4),q1=0.7,q2=0.2.N=31,\qquad (a,b)=(2,4),\qquad q_1=0.7,\qquad q_2=0.2.
Formula source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Narrative Sections

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

Conclusion

For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Formula and Green-Function Validation

The fundamental-matrix identities are [E [T]=alpha (I-Q)e-1 1, p k=alpha (I-Q)e-1 R ·,k,] and The validation shows numerical closure of mass, mean, encounter-position probabilities, and the (p k k) decomposition.

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Mean Encounter Time Versus Mobility Ratio

The mean-ratio pilot shows that an interior maximum of the exact mean encounter time is not universal. Two starts have an interior maximum over the scanned range;

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Model and Exact Absorbing-Chain Formulation

Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Operational Double-Peak Rule and Reading

A raw second bump is not enough. The final rule first forms the parity-pair curve which removes odd/even synchronous-update oscillations.

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Same-Target Long-Tail Cases A--C

Cases A--C are shoulders or long tails, not double peaks. Their early and late encounter-site distributions have the same dominant (k), and the translated interior controls do not show a large boundary-specific amplification.

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Show more sections

Spatial Configuration Views

shows the actual spatial configuration for each selected case. The left panel in each row is the physical reflecting one-dimensional lattice: the two walker starts are marked, the reflecting walls are drawn at sites (1) and (N), and the dominant early/late encounter sites are shown directly on the lattice.

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Stage-2 and Stage-2b: No Robust (F 2) Double Peaks

Stage 2 introduced smoothing persistence, parity-pair filtering, and negative controls. Stage 2b added mechanism-focused checks: boundary-null controls, target-shift reranking, encounter-site windows, grouped (f G(t)) curves, and leading eigenmodes of (Q).

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Weak Target-Shift Shoulder

The clearest retained target-shift case is Its early/late encounter-site distributions shift from top site to, with, maximum positive (k) at, and target-shift score (1.43×1e-3).

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Why the Stage-1 Double Peaks Were Artifacts

Stage 1 ranked raw curve shapes. This promoted short-distance parity combs: raw (f(t)) has repeated alternating local maxima, but the parity-pair curve collapses the pattern to a single meaningful peak.

Source

research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex

Reproducibility Commands

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

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A boundary 1 5 (stage2b/boundary_vs_interior)

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B boundary 1 7 (stage2b/boundary_vs_interior)

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C boundary 1 4 (stage2b/boundary_vs_interior)

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