Repository Sync

# Repository Guidelines ## Quickstart 1. Read `CLAUDE.md` (router) and this file (contract). 2.…

# CLAUDE.md Cold-start router for AI agents (Claude Code, Codex, Cursor, etc.).…

# valley-k-small 随机游走首达时间分布研究仓库，采用单一 canonical 结构： - 人类优先阅读 `research/` 和 `platform/` - agent 通过 `reportctl` 生成 handoff/pack，而不是依赖第二套仓库镜像 -…

# Platform `platform/` 放置“不是研究正文本身，但支撑研究交付、自动化和 agent handoff”的能力层。…

--- name: valley-k-small-continuation description: Operational playbook for continuing report work, repo maintenance, web payload generation…

# Core Checklist ## Session Start - Confirm repo root: `pwd` should end with `valley-k-small` - Scan active report IDs: `python3 scripts/rep…

# Report Map Use this map to jump directly to the correct entry script and build command family.…

# Research Conventions ## Model-rule disambiguation (always state explicitly) - Random walk type: `lazy` (`q < 1`) or `non-lazy` (`q = 1`).…

contourpy==1.3.0 cycler==0.12.1 fonttools==4.60.1 importlib_resources==6.5.2 joblib==1.5.2 kiwisolver==1.4.7 matplotlib==3.9.4 numpy==2.0.2…

# Valley-K Small Deliverables 仓库当前的三类主要交付物： 1. Website - `https://zhouyi-xiaoxiao.github.io/valley-k-small/` 2.…

# Research Docs 跨报告研究文档的主入口只保留当前有效材料。 ## Active Documents - `research/docs/RESEARCH_SUMMARY.md`: 单文件研究简报，给人和 LLM 快速上手 - `research/docs/resea…

# RESEARCH SUMMARY 最后更新: 2026-06-09 ## 仓库定位 - 本仓库研究随机游走首达时间分布中的双峰、谷值、shortcut 机制与跨模型对比。 - `research/` 是研究主树，包含报告、研究文档与历史归档。…

# P1 Function Split Baseline (20260217T004027Z) ## Help Snapshots ### .venv/bin/python reports/grid2d_reflecting_bimodality/code/reflecting_…

# P1 Split Baseline (20260217T000835Z) ## ring_valley_dst --help snapshots ### bimodality_flux_scan.py ```text usage: bimodality_flux_scan.p…

# RESEARCH Progress Archive ## Archived on 2026-02-16 - 2026-02-16: 再次核查并修正 `reports/grid2d_rect_bimodality/`：图9/10改为更有区分度的 two-target 代表热图（…

# Research Reports 所有报告统一遵循同一 v2 骨架： ```text research/reports/<report_id>/ README.md code/ notes/ manuscript/ <report_id>_cn.tex | <report_i…

# Luca Regime Map Standalone cross-report benchmark for fixed-horizon full-FPT fairness.…

\begin{tabular}{lcccccc} \toprule Anchor & Defects & Sparse(s) & Dense(s) & Full AW(s) & Luca(s) & Linear(s) \\ \midrule low\_defect (TT\_SY…

\begin{tabular}{lccccc} \toprule Defect bin & T & n & Median $R$ & Mean $R$ & $P(R>1)$ \\ \midrule \texttt{[0,20]} & 300 & 13 & 0.0734 & 0.0…

\begin{tabular}{lcccccc} \toprule Anchor & Defects & Sparse(s) & Dense(s) & Full AW(s) & Luca(s) & Linear(s) \ \midrule \bottomrule \end{tab…

\begin{tabular}{lccccc} \toprule Defect bin & T & n & Median $R$ & Mean $R$ & $P(R>1)$ \ \midrule [301,700] & 300 & 1 & 0.0006 & 0.0006 & 0.…

\documentclass[11pt]{ctexart} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,booktabs,graphicx,float,xurl} \usepackage[hypert…

\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,booktabs,graphicx,float,xurl,array} \usepackage[…

\documentclass[11pt]{ctexart} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,booktabs,graphicx,float,xurl} \usepackage[hypert…

\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,booktabs,graphicx,float,xurl,array} \usepackage[…

# Luca Regime Map Standalone cross-report benchmark for fixed-horizon full-FPT fairness.…

# Reflecting Encounter Diagonal Decomposition This report runs a bounded mobility-ratio scan for the reflecting-boundary two-walker encounte…

# Reflecting Encounter Diagonal Decomposition Ratio Scan This bounded scan uses the fixed-total-mobility convention `Q1 = 2*q0*rho/(1+rho)`…

# Encounter Green Formula Comparison ## Literature Formula and Conventions The comparison uses the finite-state Green/renewal identity behin…

# Reflecting Encounter Mean Validation This report validates the first exact-computation step for a one-dimensional two-walker encounter pro…

# Exact Recursion — Method Guide **Status: WIP scaffold (not yet started).** Tracking: [#9](https://github.com/zhouyi-xiaoxiao/valley-k-smal…

# Final Multitimescale FPT and Encounter Report **Status: consistency layer.** This report indexes the accepted outputs for the multi-timesc…

\documentclass[10pt]{article} \usepackage[a4paper,margin=0.72in]{geometry} \usepackage{amsmath,amssymb} \usepackage{booktabs} \usepackage{gr…

\documentclass[11pt]{article} \usepackage[a4paper,margin=0.75in]{geometry} \usepackage{amsmath,amssymb} \usepackage{booktabs} \usepackage{gr…

\begin{tabular}{p{0.21\linewidth}p{0.26\linewidth}p{0.20\linewidth}p{0.25\linewidth}} \toprule Setting & Observed range & Reference case & M…

\begin{tabular}{p{0.18\linewidth}p{0.16\linewidth}p{0.18\linewidth}p{0.40\linewidth}} \toprule Module & Output & Mechanism & Supported claim…

\begin{tabular}{p{0.18\linewidth}p{0.16\linewidth}p{0.18\linewidth}p{0.40\linewidth}} \toprule Setting & Scan result & Configuration & Condi…

\begin{tabular}{p{0.19\linewidth}p{0.16\linewidth}p{0.16\linewidth}p{0.16\linewidth}p{0.21\linewidth}} \toprule Module & Row max & Mass max…

# 2D Two-Target Local-Bias Basin Experiment Plan Purpose: test whether a plain 2D reflecting rectangle, without corridor geometry, can produ…

# 2D Bimodality Report ## Quick reproduce From repo root: ``` MPLCONFIGDIR=research/reports/grid2d_bimodality/.mplcache \ python3 research/r…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{gra…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

# v6 Repro Notes (2D bimodality) This note is historical context for the v6 plotting pass.…

# 2D Blackboard-Style Bimodality Report This report follows the figure/layout style of `research/reports/grid2d_bimodality/` and `research/r…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{gra…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

# grid2d_membrane_near_target New 2D extension report covering: - one-target corridor with symmetric/asymmetric semi-permeable membranes;…

\begin{tabular}{@{}cccccc@{}} \toprule $\kappa_{\uparrow}$ & $\kappa_{\downarrow}$ & phase & $t_{p1}$ & $t_{p2}$ & sep-score \\ \midrule 0 &…

\begin{tabular}{@{}lllc c@{}} \toprule Case & $q^*$ & Window & dominant class & $w_{\mathrm{L1R1}}$ \\ \midrule sym & 0.4 & peak1 & L0R0 & 0…

\begin{tabular}{@{}cccccc@{}} \toprule $\kappa$ & phase & $t_{p1}$ & $t_{p2}$ & valley/max & sep-score \\ \midrule 0 & 2 & 395 & 1711 & 0.05…

\begin{tabular}{@{}lccccc@{}} \toprule Window & Case & L0R0 & L0R1 & L1R0 & L1R1 \\ \midrule peak1 & sym & 0.996 & 0.000 & 0.004 & 0.000 \\…

\begin{tabular}{@{}cccccc@{}} \toprule $b_x$ & clear $d$ interval & best $d$ & phase(best) & $P(\mathrm{near})$ & $P(\mathrm{far})$ \\ \midr…

\begin{tabular}{@{}cccccccccccccccc@{}} \toprule spec & $b_x$ & $d$ & $\Delta y$ & phase & sep-score & valley/max & $p_{\min}$ & peak-ratio…

\begin{tabular}{@{}cccccccccc@{}} \toprule $b_x$ & $d$ & phase & $t_{p1}$ & $t_v$ & $t_{p2}$ & $P(\mathrm{near})$ & $P(\mathrm{far})$ & sep-…

\begin{tabular}{@{}lcccccc@{}} \toprule case & $q_{\mathrm{far}}(x_s)$ & $P_{\mathrm{far}}$ & $\varepsilon_{\mathrm{split}}$ & $\|r_{\mathrm…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{boo…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{booktabs} \usepack…

# Grid2D One Target — Base **Status: WIP scaffold (not yet started).** Tracking: [#4](https://github.com/zhouyi-xiaoxiao/valley-k-small/issu…

# Grid2D One Target — Exit Timing **Status: WIP scaffold (not yet started).** Tracking: [#5](https://github.com/zhouyi-xiaoxiao/valley-k-sma…

# Grid2D One Target — Valley/Peak Budget **Status: 3-page short report. Manuscript landed;…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{gra…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

# Grid2D One Target — Window Measures **Status: WIP scaffold (not yet started).** Tracking: [#7](https://github.com/zhouyi-xiaoxiao/valley-k…

# Grid2D One vs Two Target — Gating **Status: WIP scaffold (not yet started).** Tracking: [#8](https://github.com/zhouyi-xiaoxiao/valley-k-s…

# 2D Rectangle Bimodality (Two-Target + One-Target Corridor) This report studies when **non-square rectangular** 2D domains can produce a vi…

\begin{tabular}{rrrrccc} \toprule $x_s$ & $x_t$ & $b_x^{*}$ & $W_y^{\mathrm{ref}}$ & phase(ref) & clear interval(s) & first loss \\ \midrule…

\begin{tabular}{rccccc} \toprule $b_y$ & $b_x=-0.12$ & $b_x=-0.08$ & $b_x=-0.04$ & $b_x=0.00$ & $b_x=0.08$ \\ \midrule -0.08 & 1 & 1 & 1 & 0…

\begin{tabular}{rccccc} \toprule $h$ & band width & phase2 @ $b_x=-0.08$ & first loss & phase$\ge1$ @ $b_x=-0.12$ & phase$\ge1$ @ $b_x=+0.00…

\begin{tabular}{lrrrrr} \toprule ID & $W_y$ & $b_x$ & phase & valley/max & sep-score \\ \midrule OT\_W16\_bxm0p080 & 16 & -0.080 & 2 & 0.081…

\begin{tabular}{rccccc} \toprule $W_y$ & $b_x=-0.12$ & $b_x=-0.08$ & $b_x=-0.04$ & $b_x=0.00$ & $b_x=0.08$ \\ \midrule 8 & 1 & 2 & 0 & 0 & 0…

\begin{tabular}{rcccc} \toprule $x_0$ & clear interval(s) & upper clear branch & first loss width & phase at loss \\ \midrule 8 & - & - & -…

\begin{tabular}{lrrrrrr} \toprule ID & $W_y$ & $x_0$ & phase & $P(m_1)$ & $P(m_2)$ & sep-score \\ \midrule TT\_W05\_X10 & 5 & 10 & 2 & 0.561…

\begin{tabular}{rcccc} \toprule $W_y$ & n(single) & n(weak) & n(clear) & best $\min(p_1,p_2)$ \\ \midrule 5 & 6 & 0 & 3 & 0.456 \\ 6 & 6 & 0…

\begin{tabular}{lrrrrrrr} \toprule ID & $W_y$ & $x_0$ & phase & $t_{p1}$ & $t_{p2}$ & $p_{\min}/p_{\max}$ & valley/max \\ \midrule TT\_W05\_…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{boo…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{booktabs} \usepack…

# 2D Reflecting-Boundary Bimodality Report This report mirrors `research/reports/grid2d_bimodality/` but replaces all periodic boundaries wi…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{gra…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

# Grid2D Two-Target Bias-Radius Scaffold This WIP report prepares the 2D two-target scan where the start and far target are fixed and the ne…

# Grid2D Bias-Radius Small Heatmap Summary This bounded scan fixes `theta=0`, so near targets lie along the global eastward bias direction.

# 2D Two-Target Double-Peak (Reflecting Boundary) This report studies when a 2D lazy random walk with **two absorbing targets** can produce…

\begin{tabular}{lcccccc} \toprule Case & $t^*_{m_1|m_2}$ & $t^*_{m_2|m_1}$ & $t_v$ & valley/max & absorbed@6000 & tail@6000 \\ \midrule C1 &…

\begin{tabular}{lcccccccc} \toprule Case & $w_2$ & skip2 & $t_{p1}$ & $t_{p2}$ & $h_1$ & $h_2$ & $P(m_1)$ & $P(m_2)$ \\ \midrule C1 & 3 & 2…

\begin{tabular}{lccccc} \toprule Case & $t^*_{m_1|m_2}$ & $t^*_{m_2|m_1}$ & $w_{1/2}^{(1)}$ & $w_{1/2}^{(2)}$ & $|\Delta t^*|/(w_{1/2}^{(1)}…

\begin{tabular}{lp{0.35\linewidth}lcccc} \toprule ID & Name & Type & local-bias & sticky & barriers & shortcuts \\ \midrule S02 & corridor t…

\begin{tabular}{lccccccc} \toprule ID & $t_{p1}$ & $t_{p2}$ & valley/max & $P(m_1)$ & $P(m_2)$ & sep-score & phase \\ \midrule S02 & 38 & 43…

\documentclass[11pt,a4paper]{ctexart} \usepackage{geometry} \geometry{margin=2.2cm} \usepackage{amsmath,amssymb} \usepackage{booktabs} \usep…

\documentclass[11pt,a4paper]{article} \usepackage[margin=2.2cm]{geometry} \usepackage{amsmath,amssymb} \usepackage{booktabs} \usepackage{arr…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{boo…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{booktabs} \usepack…

# Luca Fast Case (Constructed Benchmark) ## 1. Goal Construct a reproducible setup where Luca/Giuggioli defect-reduced inversion is faster t…

# 2D Two-Target 数值方法比较报告（C1 配置） ## 1. 研究目标 在 `2d_two_target_double_peak` 的同一模型设定下，对比五种常用数值路线： 1. 时间域 sparse exact 递推（当前主脚本用法） 2.…

# 2D Two-Target Numerical Method Comparison (C1 Configuration) ## 1.…

# 2D Two-Walker Encounter With Shortcut This report extends the existing bimodality workflow to **two walkers** on a 2D reflecting lattice w…

\begin{tabular}{lcccccc} \toprule Case & $(N,q)$ & $z$ & $M$ & $r$ & $|LHS-RHS|$ & rel.err \\ \midrule R1\_real\_small & (16,0.30) & $0.40+0…

\begin{tabular}{ccccccc} \toprule $\beta$ & phase & $t_1$ & $t_2$ & peak ratio & valley ratio & bimodal \\ \midrule 0.00 & 0 & - & - & 0.000…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,mathtools} \usepackage{booktabs} \usepac…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,mathtools} \usepackage{booktabs} \usepac…

# ring_deriv_k2 K=2 derivation notes for the ring lattice, plus companion PDFs.…

\documentclass[11pt]{article} \usepackage{amsmath,amssymb,amsfonts,bm} \usepackage[a4paper,margin=1in]{geometry} \usepackage{mathtools} \use…

\documentclass[11pt]{article} \usepackage{amsmath,amssymb,amsfonts,bm} \usepackage[a4paper,margin=1in]{geometry} \usepackage{mathtools} \use…

\documentclass[11pt]{article} \usepackage{amsmath,amssymb,amsfonts,bm} \usepackage[a4paper,margin=1in]{geometry} \usepackage{mathtools} \use…

# ring_lazy_flux This report studies lazy ring flux bimodality (CN/EN).

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $N$ & $u$ & $v$ & paper & macro & $t_\mathrm{peak}$\\ \midrule 20 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $N$ & $u$ & $v$ & paper & macro & $t_\mathrm{peak}$\\ \midrule 20 &…

\begin{longtable}{rrrrrrr} \caption{完整扫描表（等概率基准 $q=2/3$，取 $\beta=0.02$ 使 $p=0.006666\ldots$，起点 $n_0=1$，shortcut 取 $u=n_0\to v=\mathrm{target…

\begin{longtable}{rrrrrrr} \caption{Full scan table (equal-prob baseline $q=2/3$, $\beta=0.02$ so $p=0.006666\ldots$, start $n_0=1$, shortcu…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \begin{tabular}{rrrrr} \toprule $N$ & $K$ & $\mathrm{peaks}$ & $t_{\mathrm{peak}}$ & $A(t_{\mathrm{peak}})$\…

\begin{table}[!htbp] \centering \begin{tabular}{rrrrr} \toprule $N$ & $K$ & $\mathrm{peaks}$ & $t_{\mathrm{peak}}$ & $A(t_{\mathrm{peak}})$\…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{gra…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

# ring_lazy_jump Jump-over bimodality mechanisms on the lazy ring (bilingual CN/EN).…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{gra…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

# ring_lazy_jump_ext Extension of `ring_lazy_jump` covering β / N sweeps, MC class composition, and tail diagnostics.…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \small \begin{tabular}{rrrrrr} \toprule $\beta$ & $p$ & paper & macro & $t_1$ & $t_2$\\ \midrule 0 & 0 & 0 &…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{4pt} \begin{tabular}{lrrrrrrrrr} \toprule window & $t_c$ & $n$ & $\Pr(T\i…

\documentclass[11pt]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

\section{只改变 \texorpdfstring{$\beta$}{beta} 的控制变量扫描（\texorpdfstring{$N=100$}{N=100}）} 固定 $N=100,q=2/3,\rho=1$，口径为 \texttt{lazy\_selfloop}，仅改…

\section{Beta-only sweep at fixed \texorpdfstring{$N$}{N} (\texorpdfstring{$N=100$}{N=100})} We fix $N=100$, $q=2/3$, $\rho=1$, and the \tex…

\section{MC 扫描：四分类随 \texorpdfstring{$\beta$}{beta} 变化（\texorpdfstring{$N=100$}{N=100}）} 使用 MC 统计窗口内的四类轨迹比例，重点关注 $J$-相关类别在 valley/第二峰窗口的集中程度。…

\section{MC sweep: four classes vs \texorpdfstring{$\beta$}{beta} (\texorpdfstring{$N=100$}{N=100})} We use MC to estimate class probabiliti…

\section{MC 扫描：四分类随 \texorpdfstring{$N$}{N} 变化（固定 \texorpdfstring{$\beta=0.01$}{beta=0.01}）} 保持 $\beta=0.01$，对偶数 $N$ 做 MC 扫描并统计四类轨迹在不同窗口的比例

\section{MC sweep: four classes vs \texorpdfstring{$N$}{N} (fixed \texorpdfstring{$\beta=0.01$}{beta=0.01})} With $\beta=0.01$ fixed, we sca…

\section{固定 \texorpdfstring{$\beta=0.01$}{beta=0.01} 改变 \texorpdfstring{$N$}{N}（精确 AW）} 取自动选择的 $\beta_\ast=0.01$，固定 $q=2/3,\rho=1$ 与 \texttt…

\section{Fix \texorpdfstring{$\beta=0.01$}{beta=0.01} and sweep \texorpdfstring{$N$}{N} (exact AW)} Using the automatically selected $\beta_…

\section{复现指南（命令与输出）} 以下命令均在 \texttt{reports/lazy\_jump\_ext/} 内执行。 \paragraph{1.…

\section{Reproducibility (commands and outputs)} Run the following inside \texttt{reports/lazy\_jump\_ext/}. \paragraph{1.…

\section{理论推导：生成函数、AW 反演与尾部衰减} 本节给出本文数值计算与尾部指标所依赖的核心理论推导。我们考虑在 $N$ 点环上、每步以概率 $q$ 做 $K=2k$ 邻居环走、以概率 $1-q$ 懒惰自环的随机游走，并在源点 $\mathrm{src}$ 增加一条到…

\section{Theory: generating functions, AW inversion, and tail decay} This section provides the theoretical backbone for our exact pmf comput…

\section{Valley 的可比较定义与窗口划分} 令首吸收时间的概率质量函数为 $f(t)=\mathbb{P}(T=t)$（$t\in\mathbb{N}^+$）。…

\section{Comparable definition of the valley and windowing} Let $f(t)=\mathbb{P}(T=t)$ for $t\in\mathbb{N}^+$ denote the pmf of the first ab…

# lazy_jump_ext This extension folder is isolated from the original report.…

# ring_lazy_jump_ext_rev2 Revision v2 of the lazy jump-over extension report.…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath} \usepackage{microtype} \begin{document} \noinde…

\begin{tabular}{L{0.21\linewidth} c L{0.39\linewidth} c} \toprule Definition & Cases with $\beta\geq\beta_c$ & Maximum passing $\beta$ & $\r…

\begin{tabular}{rrrrrr} \toprule $n_0$ & target & $\beta_c$ & any double & max $\beta$ double & double at/above $\beta_c$ \\ \midrule 1 & 56…

% Auto-generated by export_window_table.py \begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{3pt} \caption{窗口参数与样本量（$\beta=…

% Auto-generated by export_window_table.py \begin{table}[!htbp] \centering \scriptsize \setlength{\tabcolsep}{3pt} \caption{Window parameter…

\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage[hidelinks]{hyperref} \usepackage{pdfpages} \title{Luca Calculati…

\documentclass[11pt]{article} \usepackage[margin=0.95in]{geometry} \usepackage{amsmath,amssymb,mathtools,bm} \usepackage[hidelinks]{hyperref…

\documentclass[11pt]{article} \usepackage[margin=0.95in]{geometry} \usepackage{amsmath,amssymb,mathtools,bm} \usepackage{xcolor} \usepackage…

\documentclass[10pt, hidelinks, superscriptaddress, amsmath, amssymb, aps, pre]{revtex4-2} % \documentclass[ % reprint, % hidelinks, % super…

\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,ams…

\documentclass[11pt]{article} \usepackage[margin=0.85in]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,…

\documentclass{article} \usepackage[paperwidth=8.5in,paperheight=11in,margin=0in]{geometry} \usepackage{pdfpages} \usepackage{tikz} \usepack…

\documentclass[11pt]{article} \usepackage[margin=0.85in]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,…

# Luca calculations audit This directory archives Luca's `Calculations.tex` / `Calculations.pdf` attachments, the May 31 audit/correction fi…

# As-sent email attachments (byte-exact provenance) The working copies elsewhere in `luca_calculations_audit/` keep evolving after each send…

\documentclass[11pt]{article} \usepackage[margin=0.85in]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,…

\documentclass[11pt]{article} \usepackage[a4paper,margin=0.88in]{geometry} \usepackage{amsmath,amssymb,mathtools,bm} \usepackage[dvipsnames]…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb} \usepackage{booktabs} \usepackage{hyper…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb} \usepackage{booktabs} \usepackage{hyper…

\documentclass[11pt]{article} \usepackage[a4paper,margin=0.85in]{geometry} \usepackage{amsmath,amssymb,mathtools} \usepackage{booktabs} \use…

\documentclass[11pt]{article} \usepackage[margin=0.85in]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,…

# Sendable preflight numerical validation - corrected plus/plus formula vs direct matrix: 4.163336e-17 - compact closed form vs direct matri…

\documentclass[superscriptaddress, amsmath, amssymb, aps, pre, floatfix]{revtex4-2} \makeatletter \NewCommandCopy\@@pmod\pmod \DeclareRobust…

\documentclass[idelinks, superscriptaddress, amsmath, amssymb, aps, pre]{revtex4-2} % \documentclass[ % reprint, % hidelinks, % superscripta…

\documentclass[idelinks, superscriptaddress, amsmath, amssymb, aps, pre]{revtex4-2} % \documentclass[ % reprint, % hidelinks, % superscripta…

\documentclass[idelinks, superscriptaddress, amsmath, amssymb, aps, pre]{revtex4-2} % \documentclass[ % reprint, % hidelinks, % superscripta…

\documentclass[11pt]{article} \usepackage[margin=0.78in]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,…

\documentclass[10pt]{article} \usepackage[a4paper,margin=0.72in]{geometry} \usepackage{amsmath,amssymb} \usepackage{booktabs} \usepackage{gr…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{gra…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts,amssymb} \usepackage{graphicx} \usepack…

# CHANGELOG_v2 - Added v2 report `ring_lazy_jump_ext_rev2_cn.tex` with front-loaded abstract, symbol quick reference, reproduction chain, an…

# Luca K=2 fixed-shortcut sweep ## Meeting question Check the first-passage distribution for a lazy nearest-neighbour ring with one directed…

# lazy_jump_ext_rev2 (v2 revision) ## Quick start Run commands from `research/reports/ring_lazy_jump_ext_rev2/`.…

# REVISION_NOTES (v2) 1) 问题：图2将 f(t) 与分类柱状拆成不同面板，窗口与比例没有在同一坐标系对齐 → 影响：读者难以判断窗内比例与峰谷位置的对应关系 → 修正：在同一主轴上用三段全宽栈叠条替代 bin 底色，并用混合变换绘制 → 验证：`figur…

# two_target_ring Two-target lazy ring report (Chinese + English).…

\begin{tabular}{lllllll} \toprule Case & $N$ & $K$ & $q$ & $g$ & $\beta$ & targets \\ \midrule A\_no\_sc\_K2 & 20 & 2 & 0.667 & 0.60 & 0.000…

\begin{tabular}{llllllll} \toprule Case & $t_1$ & $h_1$ & $t_2$ & $h_2$ & $h_2/h_1$ & paper & macro \\ \midrule A\_no\_sc\_K2 & 2 & 1.78e-02…

\documentclass[11pt]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{booktabs} \usepack…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{booktabs} \usepack…

# Small Two-Target Ring Scan This bounded scan uses exact propagation on a small grid of periodic 1D ring two-target first-passage problems.

# 1D Ring Two-Walker Encounter With Shortcut This report focuses on **1D ring encounter** for two independent lazy walkers.…

\begin{tabular}{lcccccc} \toprule Case & $(N,q)$ & $z$ & $M$ & $r$ & $|LHS-RHS|$ & rel.err \\ \midrule R1\_real\_small & (16,0.30) & $0.40+0…

% Auto-generated from data/case_summary.json; do not edit by hand. \begin{enumerate}[leftmargin=1.4em] \item $\beta\le0.14$：phase=1（弱双峰/过渡）；…

% Auto-generated from data/case_summary.json; do not edit by hand.…

\begin{tabular}{lc} \toprule Metric & Value \\ \midrule Coarse onset $\beta$ & 0.160 \\ Refined onset $\beta$ & 0.150 \\ Detector onset rang…

\begin{tabular}{cccccccc} \toprule $N$ & onset $\beta^\dagger$ & onset source & $\beta_{\max}$ & clear frac & phase@0.20 & peak@0.20 & valle…

% Auto-generated from data/encounter_onset_n_scan.csv; do not edit by hand.…

% Auto-generated from data/encounter_onset_n_scan.csv; do not edit by hand.…

\begin{tabular}{ccccccc} \toprule $\beta$ & phase & $t_1$ & $t_2$ & peak balance ratio (min/max) & valley ratio & bimodal \\ \midrule 0.00 &…

\begin{tabular}{ll} \toprule Field & Value \\ \midrule $N$ / $q$ & 101 / 0.70 \\ $(g_1,g_2)$ & (0.70, -0.40) \\ Starts $(n_0,m_0)$ & (5, 55)…

\begin{tabular}{lcccccc} \toprule Case & $(g_1,g_2)$ & phase & $t_1\,(=2m)$ & $t_2\,(=2m)$ & peak balance ratio ($\bar{\tilde f}$, min/max)…

其中 clear 占比为 $159/169=0.941$（脚本自动汇总）。该 fixed-site 基准刻意选在漂移主导区；K=2 奇偶粗粒化用于抑制 odd/even 微结构，使 clear 判定反映“分离时间尺度”而非 parity 伪峰。…

The clear fraction is $159/169=0.941$ (auto-synced from generated scan data). This fixed-site benchmark is intentionally drift-dominated

\begin{tabular}{lcc} \toprule Phase class & Count & Fraction \\ \midrule single (0) & 7 & 0.041 \\ weak (1) & 3 & 0.018 \\ clear (2) & 159 &…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,mathtools} \usepackage{booktabs} \usepac…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,mathtools} \usepackage{booktabs} \usepac…

# ring_valley Non-lazy valley-regime notes and scans for the ring lattice.…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts} \usepackage{graphicx} \usepackage{capt…

# ring_valley_dst This report studies DST-driven second-peak modulation on ring lattices (CN/EN).…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 3 & 798 & 802 & 4…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 1 & 909 & 913 & 4…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 36 & 34 & 108 & 7…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 43 & 37 & 120 & 8…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 4 & 263 & 313 & 5…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 4 & 485 & 535 & 5…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 4 & 806 & 856 & 5…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 4 & 1125 & 1175 &…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule 4 & 326 & 376 & 5…

\begin{tabular}{r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $t_2-t_1$ & $A(t_1)$ & $A(t_2)$ & $V$ \\ \midrule \bottomrule \end{…

\begin{itemize} \item Pearson $r(R,\;P(C\ge1))$：-0.439（全体），-0.406（去掉 $\mathrm{dst}=\mathrm{target}$）。…

\begin{itemize} \item 本次参数：$N=100,\;K=6,\;n_0=1,\;\mathrm{target}=50,\;\mathrm{src}=1$。…

\begin{tabular}{r r r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $\Delta$ & $R$ & $A(t_1)$ & $A(t_2)$ & $P(C=0)$ & $P(C\ge1)$ \\…

\begin{itemize} \item Pearson $r(R,\;P(C\ge1))$：-0.165（全体），-0.969（去掉 $\mathrm{dst}=\mathrm{target}$）。…

\begin{itemize} \item 本次参数：$N=100,\;K=6,\;n_0=1,\;\mathrm{target}=50,\;\mathrm{src}=6$。…

\begin{tabular}{r r r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $\Delta$ & $R$ & $A(t_1)$ & $A(t_2)$ & $P(C=0)$ & $P(C\ge1)$ \\…

\begin{itemize} \item Pearson $r(R,\;P(C\ge1))$：nan（全体），nan（去掉 $\mathrm{dst}=\mathrm{target}$）。…

\begin{itemize} \item 本次参数：$N=100,\;K=6,\;n_0=1,\;\mathrm{target}=50,\;\mathrm{src}=6$。…

\begin{tabular}{r r r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $\Delta$ & $R$ & $A(t_1)$ & $A(t_2)$ & $P(C=0)$ & $P(C\ge1)$ \\…

\begin{itemize} \item Pearson $r(R,\;P(C\ge1))$：-0.439（全体），-0.406（去掉 $\mathrm{dst}=\mathrm{target}$）。…

\begin{itemize} \item 本次参数：$N=100,\;K=6,\;n_0=1,\;\mathrm{target}=50,\;\mathrm{src}=1$。…

\begin{tabular}{r r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $\Delta$ & $R$ & $A(t_2)$ & $P(C=0)$ & $P(C\ge1)$ \\ \midrule 36…

\begin{tabular}{r r r r r} \hline dst & $t_1$ & $A(t_1)$ & $t_2$ & $A(t_2)$ \\ \hline \hline \end{tabular}

\begin{tabular}{r r r r r r r r r} \toprule $N$ & $K$ & $t_1$ & $t_2$ & $\Delta$ & $A(t_1)$ & $A(t_2)$ & $P(C=0)$ & $P(C\ge1)$ \\ \midrule 1…

\begin{itemize} \item Pearson $r(R,\;P(C\ge1))$: -0.165 (all), -0.969 (excluding $\mathrm{dst}=\mathrm{target}$).…

\begin{itemize} \item Parameter set: $N=100,\;K=6,\;n_0=1,\;\mathrm{target}=50,\;\mathrm{src}=6$.…

\begin{tabular}{r r r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $\Delta$ & $R$ & $A(t_1)$ & $A(t_2)$ & $P(C=0)$ & $P(C\ge1)$ \\…

\begin{itemize} \item Pearson $r(R,\;P(C\ge1))$: -0.439 (all), -0.406 (excluding $\mathrm{dst}=\mathrm{target}$).…

\begin{itemize} \item Parameter set: $N=100,\;K=6,\;n_0=1,\;\mathrm{target}=50,\;\mathrm{src}=1$.…

\begin{tabular}{r r r r r r r r r} \toprule $\mathrm{dst}$ & $t_1$ & $t_2$ & $\Delta$ & $R$ & $A(t_1)$ & $A(t_2)$ & $P(C=0)$ & $P(C\ge1)$ \\…

% Deterministic bimodality validation via flux/master equation (no randomness).…

% Deterministic bimodality validation via flux/master equation (no randomness).…

\documentclass[11pt,fontset=fandol]{ctexart} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amsfonts} \usepackage{graphicx} \…

\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{graphicx} \usepack…

# Scripts 公开脚本表面只保留 3 个入口： - `scripts/reportctl.py`: 唯一 Python CLI - `scripts/ka`: keepalive / recurring Codex shell 入口 - `scripts/README.md…