valley-k-small
Book Mainline

Valley-K Small Research Book

Read from Chapter 0 to Chapter 7 as one continuous theory-and-evidence narrative. Each chapter is linked to interactive figures and auditable claims.

Start continuous reading Open Chapter 0

8

Book chapters

27

Mapped reports

PASS

Translation gate

Backbone System

This system extracts a canonical logic spine so we can iterate details without losing narrative continuity.

Act I: Foundations

Unify notation, FPT objects, and baseline assumptions before model-specific branches.

Chapters: chapter-0-reading-guide, chapter-1-core-fpt

Act II: Family Mechanisms

Expand Grid2D and Ring families with derivation-backed interaction checkpoints.

Chapters: chapter-2-grid2d-family, chapter-3-ring-baseline, chapter-4-shortcut-variants

Act III: Cross-Model Synthesis

Merge claims into a shared explanatory frame and validate reproducibility pathways.

Chapters: chapter-5-cross-model-synthesis, chapter-6-repro-validation

Act IV: Outlook

Stabilize conclusions, isolate unresolved gaps, and define next-step research questions.

Chapters: chapter-7-outlook

Claim Coverage Policy

Book chapters prioritize non-redundant continuity claims from report-level content_map records. Excluded report claims remain available in report pages and content_map for deep audit. Chapter-native synthesis claims are tracked separately.

Global claims 130 In chapters 54 Chapter-native synthesis claims 10 Excluded from mainline 76

Open excluded claim IDs

Excluded claims are still available in report pages and content_map for deep audit.

Claim ID
cross_luca_regime_map-c1
encounter_reflecting_diagonal_decomp-c1
encounter_reflecting_diagonal_decomp-c2
encounter_reflecting_diagonal_decomp-c3
encounter_reflecting_diagonal_decomp-c4
encounter_reflecting_mean_validation-c1
encounter_reflecting_mean_validation-c2
encounter_reflecting_mean_validation-c3
encounter_reflecting_mean_validation-c4
encounter_reflecting_mean_validation-c5
exact_recursion_method_guide-c1
exact_recursion_method_guide-c2
exact_recursion_method_guide-c3
exact_recursion_method_guide-c4
final_multitimescale_fpt_encounter-c1
final_multitimescale_fpt_encounter-c2
final_multitimescale_fpt_encounter-c3
final_multitimescale_fpt_encounter-c4
grid2d_bimodality-c4
grid2d_blackboard_bimodality-c4
grid2d_blackboard_bimodality-c5
grid2d_membrane_near_target-c1
grid2d_membrane_near_target-c2
grid2d_membrane_near_target-c3
grid2d_membrane_near_target-c4
grid2d_membrane_near_target-c5
grid2d_one_target_base-c1
grid2d_one_target_base-c2
grid2d_one_target_base-c3
grid2d_one_target_base-c4
grid2d_one_target_base-c5
grid2d_one_target_exit_timing-c1
grid2d_one_target_exit_timing-c2
grid2d_one_target_exit_timing-c3
grid2d_one_target_exit_timing-c4
grid2d_one_target_valley_peak_budget-c1
grid2d_one_target_valley_peak_budget-c2
grid2d_one_target_valley_peak_budget-c3
grid2d_one_target_valley_peak_budget-c4
grid2d_one_target_window_measures-c1
grid2d_one_target_window_measures-c2
grid2d_one_target_window_measures-c3
grid2d_one_target_window_measures-c4
grid2d_one_target_window_measures-c5
grid2d_one_two_target_gating-c1
grid2d_one_two_target_gating-c2
grid2d_one_two_target_gating-c3
grid2d_one_two_target_gating-c4
grid2d_one_two_target_gating-c5
grid2d_rect_bimodality-c1
grid2d_rect_bimodality-c4
grid2d_reflecting_bimodality-c4
grid2d_two_target_bias_radius-c1
grid2d_two_target_bias_radius-c2
grid2d_two_target_bias_radius-c3
grid2d_two_target_bias_radius-c4
grid2d_two_target_bias_radius-c5
grid2d_two_walker_encounter_shortcut-c1
grid2d_two_walker_encounter_shortcut-c2
grid2d_two_walker_encounter_shortcut-c3
grid2d_two_walker_encounter_shortcut-c4
grid2d_two_walker_encounter_shortcut-c5
ring_deriv_k2-c4
ring_lazy_jump-c4
ring_lazy_jump_ext-c1
ring_lazy_jump_ext-c5
ring_lazy_jump_ext_rev2-c1
ring_lazy_jump_ext_rev2-c4
ring_lazy_jump_ext_rev2-c5
ring_two_walker_encounter_shortcut-c1
ring_two_walker_encounter_shortcut-c2
ring_two_walker_encounter_shortcut-c3
ring_two_walker_encounter_shortcut-c4
ring_two_walker_encounter_shortcut-c5
ring_valley-c4
ring_valley-c5

Table of Contents

Chapter 1

Chapter 1: Core FPT Concepts

Build the common vocabulary of f(t), survival, hazard, and practical bimodality diagnostics.

Evidence nodes 4 Claims 8

Chapter 2

Chapter 2: Grid2D Family

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

Evidence nodes 5 Claims 10

Chapter 3

Chapter 3: Ring Family Baseline

Establish lazy and non-lazy ring baselines, then align inversion logic and reference behavior before shortcut variants.

Evidence nodes 4 Claims 8

Chapter 4

Chapter 4: Shortcut Variants

Compare selfloop/renormalize/equal4 mechanisms and beta scans to explain when shortcut strength flips phase behavior.

Evidence nodes 4 Claims 8

Glossary Lock Table

Terms are double-written in EN/CN and reused across all chapters to keep notation stable.

AW Inversion

Discrete Cauchy/FFT-based inversion from generating functions to time-domain FPT quantities.

ft≈FFT(F(zk))f_t \approx FFT(F(z_k))

Beta Scan

Parameter sweep over shortcut strength β to identify phase shifts and regime boundaries.

β∈[0,1]β ∈ [0,1]

Bimodality Criterion

Operational criterion to separate true two-peak structure from noisy shoulders.

peak−valley−peakconsistencypeak-valley-peak consistency

Claim Ledger

Structured mapping from statement to evidence paths and cross-report links.

claim−>evidence−>linkedreportsclaim -> evidence -> linked reports

Equal4 Baseline

Four-way equalized baseline used to compare shortcut effects under symmetric local movement.

p(±1)=p(±2)p(±1)=p(±2)

First-Passage Time (FPT)

Random time needed for the trajectory to hit an absorbing target for the first time.

f(t)f(t)

Hazard Rate

Conditional probability of first passage at step t given survival up to t.

h(t)=f(t)/S(t−1)h(t)=f(t)/S(t-1)

Renormalize Shortcut Mode

Base transition weights are rescaled after shortcut injection to preserve normalization constraints.

pi′=c⋅pip_i' = c · p_i