valley-k-small
Continuous Reading

Valley-K Small: Full Storyline Edition

This page stitches all 8 chapters into one full storyline from notation and assumptions to mechanism, evidence, and outlook, with expandable derivations and claim ledgers in each chapter block.

Ch.0Ch.1Ch.2Ch.3Ch.4Ch.5Ch.6Ch.7
Reading Guide

Chapter 1/8 chapter-0-reading-guide

Chapter 0: Reading Guide & Notation

How to read this atlas, verify claims, and reuse symbols consistently across all reports.

This chapter defines the reading protocol: start from claims, verify evidence paths, then inspect formula chains and interactive traces.

This chapter defines the reading protocol: start from claims, verify evidence paths, then inspect formula chains and interactive traces.

Notation is aligned across grid and ring families, so symbol reuse does not introduce hidden semantic drift.

After this chapter, every subsequent page can be read as one continuous argument instead of isolated report fragments.

Storyline in this chapter

We begin by fixing the model premise: The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring interpretations.

We then move to an auditable method chain: The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.

Under the same diagnostic criterion, the chapter-level result and finding are: No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.

Carry notation and verified claims from Chapter 0: Reading Guide & Notation into Chapter 1: Core FPT Concepts, then extend mechanism and evidence without resetting assumptions.

Open core derivation (5)

Derivation Link grid2d_bimodality

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

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 ring_deriv_k2

Defines first-passage probability objects used by later diagnostics.

Pr⁡(Xt+1=n∣Xt=n′)=Bn,n′(defect-free ring),Pr⁡(Xt+1=n∣Xt=n′)=An,n′(with defect).\begin{aligned} \Pr(X_{t+1}=n\mid X_t=n') &= B_{n,n'} \qquad \text{(defect-free ring)},\\ \Pr(X_{t+1}=n\mid X_t=n') &= A_{n,n'} \qquad \text{(with defect)}. \end{aligned}

Spectral / Inversion Step ring_deriv_k2

Provides analytic inversion machinery for computing trajectories.

P(n,t+1)=∑n′Bn,n′P(n′,t)P(n,t+1)=\sum_{n'} B_{n,n'} P(n',t)

Distribution Setup ring_two_target

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)}
Open interactive lab (3)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (8)

method grid2d_bimodality-c2

The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.

Evidence items 5

  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex
  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

method ring_deriv_k2-c2

The report derives Green-function style propagators, constructs defect-resolvent corrections, and obtains first-passage generating forms that can be numerically inverted.

Evidence items 5

  • source_document research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex
  • section_summary research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

method ring_two_target-c2

Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.

Evidence items 4

  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex
  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

model ring_deriv_k2-c1

The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring interpretations.

Evidence items 4

  • source_document research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex
  • section_summary research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

model grid2d_bimodality-c1

The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary assumptions.

Evidence items 4

  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex
  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

model ring_two_target-c1

The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.

Evidence items 3

  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex
  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

result ring_two_target-c3

No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.

Evidence items 4

  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex
  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

result ring_deriv_k2-c3

The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.

Evidence items 5

  • source_document research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex
  • section_summary research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Carry notation and verified claims from Chapter 0: Reading Guide & Notation into Chapter 1: Core FPT Concepts, then extend mechanism and evidence without resetting assumptions.

Book startOpen chapter pageNext chapter →
Core Concepts

Chapter 2/8 chapter-1-core-fpt

Chapter 1: Core FPT Concepts

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

The first-passage distribution is interpreted through complementary views: density, cumulative survival, and hazard-style turning points.

The first-passage distribution is interpreted through complementary views: density, cumulative survival, and hazard-style turning points.

Bimodality is treated as an evidence-backed diagnosis, not a visual impression, and every chapter keeps this constraint.

The same diagnostic language is reused in both lattice and ring settings to keep conclusions comparable.

Bridge from Chapter 0: now that notation and audit protocol are fixed, we formalize f(t), S(t), and h(t) as the common diagnostic language.

Storyline in this chapter

We begin by fixing the model premise: The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

We then move to an auditable method chain: The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

Under the same diagnostic criterion, the chapter-level result and finding are: Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion. Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.

Carry notation and verified claims from Chapter 1: Core FPT Concepts into Chapter 2: Grid2D Family, then extend mechanism and evidence without resetting assumptions.

Open core derivation (6)

Derivation Link grid2d_bimodality

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

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_two_target_double_peak

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 ring_lazy_flux

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

target=⌊N/2⌋\text{target}=\lfloor N/2\rfloor

Derivation Link ring_lazy_flux

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

v=target+1v=\text{target}+1

Distribution Setup ring_two_target

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)}
Open interactive lab (4)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (8)

result grid2d_bimodality-c3

Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.

Evidence items 5

  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex
  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

model grid2d_two_target_double_peak-c1

The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.

Evidence items 3

  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex
  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

model ring_lazy_flux-c1

The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.

Evidence items 3

  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex
  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

finding ring_two_target-c5

Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.

Evidence items 4

  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex
  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

method grid2d_two_target_double_peak-c2

The pipeline combines exact/approximate first-passage diagnostics, truncation controls, and parameter-phase scans over two-target coupling variables.

Evidence items 4

  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex
  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

method ring_lazy_flux-c2

The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.

Evidence items 4

  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex
  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

result ring_lazy_flux-c3

A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.

Evidence items 4

  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex
  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

result grid2d_two_target_double_peak-c3

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.

Evidence items 4

  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex
  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

Carry notation and verified claims from Chapter 1: Core FPT Concepts into Chapter 2: Grid2D Family, then extend mechanism and evidence without resetting assumptions.

← Previous chapterOpen chapter pageNext chapter →
Grid2D Family

Chapter 3/8 chapter-2-grid2d-family

Chapter 2: Grid2D Family

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

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

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.

Storyline in this chapter

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.

Open core derivation (8)

Derivation Link grid2d_bimodality

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

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

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

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

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

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

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

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)}
Open interactive lab (5)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (10)

finding grid2d_bimodality-c5

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

Evidence items 5

  • source_document research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex
  • section_summary research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex

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 items 5

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

result grid2d_rect_bimodality-c3

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

Evidence items 4

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

result grid2d_reflecting_bimodality-c3

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

Evidence items 5

  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex
  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

finding grid2d_two_target_double_peak-c5

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

Evidence items 4

  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex
  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

model grid2d_blackboard_bimodality-c1

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

Evidence items 4

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

method grid2d_reflecting_bimodality-c2

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

Evidence items 5

  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex
  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

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 items 5

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

method grid2d_rect_bimodality-c2

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

Evidence items 4

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

model grid2d_reflecting_bimodality-c1

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

Evidence items 4

  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex
  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

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.

← Previous chapterOpen chapter pageNext chapter →
Ring Baseline

Chapter 4/8 chapter-3-ring-baseline

Chapter 3: Ring Family Baseline

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

Ring models provide a compact setting for isolating drift, waiting probability, and spectral structure.

Ring models provide a compact setting for isolating drift, waiting probability, and spectral structure.

This baseline chapter is intentionally conservative: it first aligns lazy and non-lazy controls before adding shortcut perturbations.

The resulting baseline is reused in later chapters as a control for mechanism attribution.

Bridge from Chapter 2: Grid2D established boundary-sensitive diagnostics; this chapter keeps those diagnostics but shifts geometry to a ring so shortcut and lazy effects can be isolated.

Storyline in this chapter

We begin by fixing the model premise: A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

We then move to an auditable method chain: The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.

Under the same diagnostic criterion, the chapter-level result and finding are: Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.…

Carry notation and verified claims from Chapter 3: Ring Family Baseline into Chapter 4: Shortcut Variants, then extend mechanism and evidence without resetting assumptions.

Open core derivation (7)

Distribution Setup ring_deriv_k2

Defines first-passage probability objects used by later diagnostics.

Pr⁡(Xt+1=n∣Xt=n′)=Bn,n′(defect-free ring),Pr⁡(Xt+1=n∣Xt=n′)=An,n′(with defect).\begin{aligned} \Pr(X_{t+1}=n\mid X_t=n') &= B_{n,n'} \qquad \text{(defect-free ring)},\\ \Pr(X_{t+1}=n\mid X_t=n') &= A_{n,n'} \qquad \text{(with defect)}. \end{aligned}

Spectral / Inversion Step ring_deriv_k2

Provides analytic inversion machinery for computing trajectories.

P(n,t+1)=∑n′Bn,n′P(n′,t)P(n,t+1)=\sum_{n'} B_{n,n'} P(n',t)

Derivation Link ring_lazy_flux

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

target=⌊N/2⌋\text{target}=\lfloor N/2\rfloor

Derivation Link ring_lazy_flux

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

v=target+1v=\text{target}+1

Distribution Setup ring_lazy_jump

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 ring_valley

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

direct:T<tvalley−Δ,valley:∣T−tvalley∣≤Δ,intermediate:tvalley+Δ<T≤t2+Δ,indirect:T>t2+Δ.\begin{aligned} \text{direct:}\quad & T < t_{\text{valley}}-\Delta,\\ \text{valley:}\quad & |T-t_{\text{valley}}|\le \Delta,\\ \text{intermediate:}\quad & t_{\text{valley}}+\Delta < T \le t_{2}+\Delta,\\ \text{indirect:}\quad & T > t_{2}+\Delta. \end{aligned}

Derivation Link ring_valley

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

Pi→j=1K1[j∈NK(i)]+1[i=6](1K+11 ⁣[j=N2+1]−1K(K+1)1[j∈NK(6)])P_{i\to j}=\frac{1}{K}\mathbf{1}[j\in\mathcal{N}_K(i)]+\mathbf{1}[i=6]\left(\frac{1}{K+1}\mathbf{1}\!\left[j=\frac{N}{2}+1\right]-\frac{1}{K(K+1)}\mathbf{1}[j\in\mathcal{N}_K(6)]\right)
Open interactive lab (4)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (8)

finding ring_deriv_k2-c5

Directed long-range links alter first-passage statistics through resolvent-level corrections rather than ad-hoc fitting.

Evidence items 5

  • source_document research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex
  • section_summary research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

finding ring_lazy_flux-c5

AW inversion and flux recursion agree to numerical precision, validating both the derivation and implementation.

Evidence items 4

  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex
  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

model ring_lazy_jump-c1

A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

Evidence items 3

  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex
  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

method ring_valley-c2

The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.

Evidence items 6

  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex
  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

method ring_lazy_jump-c2

The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.

Evidence items 4

  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex
  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

model ring_valley-c1

The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut placement.

Evidence items 5

  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex
  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

result ring_valley-c3

Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.

Evidence items 5

  • source_document research/reports/ring_valley/manuscript/ring_valley_en.tex
  • section_summary research/reports/ring_valley/manuscript/ring_valley_en.tex

result ring_lazy_jump-c3

Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.

Evidence items 4

  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex
  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

Carry notation and verified claims from Chapter 3: Ring Family Baseline into Chapter 4: Shortcut Variants, then extend mechanism and evidence without resetting assumptions.

← Previous chapterOpen chapter pageNext chapter →
Shortcut Variants

Chapter 5/8 chapter-4-shortcut-variants

Chapter 4: Shortcut Variants

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

Once shortcuts are introduced, implementation choices become model assumptions that can alter observed phases.

Once shortcuts are introduced, implementation choices become model assumptions that can alter observed phases.

This chapter compares mechanism variants side by side and keeps a strict mapping to parameterized evidence.

The key output is a stable interpretation of beta-strength transitions across compatible ring settings.

Bridge from Chapter 3: after fixing a conservative ring baseline, we vary shortcut implementations to identify which phase transitions are mechanism-driven.

Storyline in this chapter

We then move to an auditable method chain: The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail diagnostics.

Under the same diagnostic criterion, the chapter-level result and finding are: Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.…

Carry notation and verified claims from Chapter 4: Shortcut Variants into Chapter 5: Cross-Model Synthesis, then extend mechanism and evidence without resetting assumptions.

Open core derivation (4)

Distribution Setup ring_lazy_jump

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)}

Distribution Setup ring_lazy_jump_ext

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 ring_lazy_jump_ext_rev2

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

Δ=max⁡{1,⌊0.05(t2−t1)⌋}\Delta=\max\{1,\lfloor 0.05(t_2-t_1)\rfloor\}

Distribution Setup ring_valley_dst

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)}
Open interactive lab (4)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (8)

finding ring_lazy_jump-c5

Shortcut strength has a non-monotonic effect: too weak or too strong settings both reduce robust bimodality.

Evidence items 4

  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex
  • section_summary research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

result ring_lazy_jump_ext-c3

Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.

Evidence items 4

  • source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
  • section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

result ring_lazy_jump_ext_rev2-c3

Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.

Evidence items 4

  • source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
  • section_summary research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex

result ring_valley_dst-c3

Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.

Evidence items 4

  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex
  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

method ring_lazy_jump_ext-c2

The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail diagnostics.

Evidence items 4

  • source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
  • section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

finding ring_lazy_jump_ext-c4

At fixed N=100, beta sweeps show systematic left-shifts of peak times and a larger tail-decay rate as shortcut strength increases.

Evidence items 4

  • source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex
  • section_summary research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

method ring_valley_dst-c2

The workflow combines deterministic flux scans, AW-style first-passage diagnostics, and class-conditioned Monte Carlo paths under one peak/valley criterion.

Evidence items 4

  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex
  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

method ring_lazy_jump_ext_rev2-c2

The pipeline exports standardized Fig.1 inputs, validates schema, renders stacked-bar co-located panels, and runs three sensitivity tracks: threshold sweep, window perturbation, and MC confidence intervals.

Evidence items 4

  • source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex
  • section_summary research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex

Carry notation and verified claims from Chapter 4: Shortcut Variants into Chapter 5: Cross-Model Synthesis, then extend mechanism and evidence without resetting assumptions.

← Previous chapterOpen chapter pageNext chapter →
Cross-Model Synthesis

Chapter 6/8 chapter-5-cross-model-synthesis

Chapter 5: Cross-Model Synthesis

Unify Grid2D and Ring evidence through shared diagnostics and cross-model regime mapping.

This chapter joins lattice and ring narratives by aligning diagnostics instead of forcing identical geometry.

This chapter joins lattice and ring narratives by aligning diagnostics instead of forcing identical geometry.

Cross-model statements are accepted only when both sides provide auditable evidence paths.

The synthesis output is a reusable map for transferring intuition across families without overclaiming.

Bridge from Chapter 4: once shortcut variants are disentangled, we align Grid2D and Ring diagnostics into one transferable cross-model map.

Storyline in this chapter

We begin by fixing the model premise: The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.

We then move to an auditable method chain: Runs use warm-up plus repeated timed executions, defect-pair routing rules, and pooled medians to stabilize solver-side variance before regime labeling.

Under the same diagnostic criterion, the chapter-level result and finding are: Across the scanned workload, sparse exact remains the dominant full-FPT baseline; Luca-mode speedups appear only in limited defect-regime subsets and are near-neutral in the aggregate ratio metric.…

Before any new claims are added, move from synthesis to reproducibility gates and verify command-, schema-, and artifact-level closure.

Open core derivation (4)

Distribution Setup cross_luca_regime_map

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)}

Distribution Setup grid2d_two_target_double_peak

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)}

Distribution Setup ring_two_target

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)}

Distribution Setup ring_valley_dst

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)}
Open interactive lab (4)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (8)

finding cross_luca_regime_map-c4

Pooled timing medians indicate sparse exact dominates most scanned regimes in full-FPT mode.

Evidence items 4

  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex
  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

finding ring_two_target-c4

Two-target geometry introduces competing fast and delayed channels, making multimodality a structural rather than numerical artifact.

Evidence items 4

  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex
  • section_summary research/reports/ring_two_target/artifacts/tables/case_configs.tex

finding ring_valley_dst-c5

Scanning dst alone can substantially change second-peak height ratio and valley depth at fixed N and K.

Evidence items 4

  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex
  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

result cross_luca_regime_map-c3

Across the scanned workload, sparse exact remains the dominant full-FPT baseline; Luca-mode speedups appear only in limited defect-regime subsets and are near-neutral in the aggregate ratio metric.

Evidence items 4

  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex
  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

finding ring_valley_dst-c4

Deterministic flux/master-equation results and Monte Carlo class decomposition agree on high-contrast destination windows.

Evidence items 4

  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex
  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

finding cross_luca_regime_map-c5

The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with MFPT linear systems separated as reference only.

Evidence items 4

  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex
  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

method cross_luca_regime_map-c2

Runs use warm-up plus repeated timed executions, defect-pair routing rules, and pooled medians to stabilize solver-side variance before regime labeling.

Evidence items 4

  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex
  • section_summary research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

model ring_valley_dst-c1

The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.

Evidence items 3

  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex
  • section_summary research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

Before any new claims are added, move from synthesis to reproducibility gates and verify command-, schema-, and artifact-level closure.

← Previous chapterOpen chapter pageNext chapter →
Validation

Chapter 7/8 chapter-6-repro-validation

Chapter 6: Reproducibility & Validation

Show command-level reproducibility, audit constraints, and practical error boundaries for the full pipeline.

Reproducibility is treated as part of the scientific claim: commands, generated data, and validation records are all first-class artifacts.

Reproducibility is treated as part of the scientific claim: commands, generated data, and validation records are all first-class artifacts.

This chapter lists the operational checkpoints used by CI and agent handoff packages.

The emphasis is not only on rerunning scripts, but also on verifying claim-evidence integrity after each change.

Bridge from Chapter 5: after cross-model synthesis, we now test whether every key claim remains reproducible under command-level and schema-level checks.

Storyline in this chapter

We then move to an auditable method chain: Reproducibility is enforced as an executable contract: web data, glossary, book chapters, backbone, agent sync, and validators run in one closed pipeline.

Under the same diagnostic criterion, the chapter-level result and finding are: Schema gates now cover both report payloads and book backbone, preventing hidden contract drift across releases. Agent handoff is traceable because manifest, chapter jsonl, claim graph, and translation QC are generated together with provena…

Only unresolved items that pass reproducibility constraints should enter the outlook as auditable hypotheses.

Open core derivation (13)

validation grid2d_bimodality

Each release requires schema pass for web/book/agent payloads before publication.

∀a∈Arelease, validate⁡(a)=PASS\forall a\in\mathcal{A}_{release},\ \operatorname{validate}(a)=\mathrm{PASS}

validation grid2d_bimodality

Traceability is enforced by deterministic build outputs with file hash and size checks.

trace⁡(f)={sha256(f),∣f∣}\operatorname{trace}(f)=\{\mathrm{sha256}(f),|f|\}

validation grid2d_bimodality

Release is allowed only when multi-agent cross-check high-priority findings are closed.

∑i∈H1[openi]=0\sum_{i\in\mathcal{H}}\mathbf{1}[\mathrm{open}_i]=0

Derivation Link grid2d_bimodality

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

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

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

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

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

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 ring_lazy_flux

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

target=⌊N/2⌋\text{target}=\lfloor N/2\rfloor

Derivation Link ring_lazy_flux

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

v=target+1v=\text{target}+1

Derivation Link ring_lazy_jump_ext_rev2

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

Δ=max⁡{1,⌊0.05(t2−t1)⌋}\Delta=\max\{1,\lfloor 0.05(t_2-t_1)\rfloor\}

Distribution Setup ring_valley_dst

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)}
Open interactive lab (7)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (8)

method chapter-6-repro-validation-cmd-closure

Reproducibility is enforced as an executable contract: web data, glossary, book chapters, backbone, agent sync, and validators run in one closed pipeline.

Evidence items 2

  • source_document scripts/reportctl.py
  • source_document platform/tools/web/build_three_deliverables.py

result chapter-6-repro-validation-schema-gate

Schema gates now cover both report payloads and book backbone, preventing hidden contract drift across releases.

Evidence items 2

  • source_document platform/tools/web/validate_web_data.py
  • source_document platform/schemas/book_backbone_v1.schema.json

result chapter-6-repro-validation-ci-publish

Deployment and audit checks are automated in CI so publishing is gated by machine-verifiable quality checks instead of manual inspection.

Evidence items 2

  • source_document .github/workflows/site-pages.yml
  • source_document .github/workflows/repo-audit.yml

finding chapter-6-repro-validation-agent-handoff

Agent handoff is traceable because manifest, chapter jsonl, claim graph, and translation QC are generated together with provenance pointers.

Evidence items 2

  • source_document platform/tools/web/build_agent_sync.py
  • dataset /data/v1/agent/manifest.json

finding grid2d_rect_bimodality-c5

Two-target endpoint constructions offer a reproducible mechanism for separating fast and delayed channels.

Evidence items 4

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

finding grid2d_reflecting_bimodality-c5

Pore/track structures modify delay channels through accessibility, not only through drift magnitude.

Evidence items 5

  • source_document research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex
  • section_summary research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex

finding grid2d_two_target_double_peak-c4

Phase maps identify stable double-peak bands and transition zones to unimodal behavior.

Evidence items 4

  • source_document research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex
  • section_summary research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex

finding ring_lazy_flux-c4

A minimal reproducible bimodal case appears at N=10 under small shortcut strength in the selfloop construction.

Evidence items 4

  • source_document research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex
  • section_summary research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex

Only unresolved items that pass reproducibility constraints should enter the outlook as auditable hypotheses.

← Previous chapterOpen chapter pageNext chapter →
Outlook

Chapter 8/8 chapter-7-outlook

Chapter 7: Outlook & Open Questions

Summarize unresolved mechanisms and propose next experiments with explicit evidence prerequisites.

The final chapter is not a loose discussion: each open question is anchored to known evidence gaps.

The final chapter is not a loose discussion: each open question is anchored to known evidence gaps.

We separate confirmed mechanisms from plausible hypotheses to prevent narrative inflation.

The output is a roadmap that an incoming agent can continue without restarting context collection.

Bridge from Chapter 6: once reproducibility gates are satisfied, unresolved mechanisms can be promoted into auditable next-step hypotheses instead of speculative notes.

Storyline in this chapter

We then move to an auditable method chain: Next iteration should prioritize experiments by uncertainty reduction per compute cost, while preserving claim-evidence traceability constraints.

Under the same diagnostic criterion, the chapter-level result and finding are: A new agent can continue the research storyline without context reset because chapter manifests, claim graph, and iteration history are packaged together.…

No downstream chapter; consolidate assumptions, claims, and open questions.

Open core derivation (7)

outlook cross_luca_regime_map

Open question: can one hazard-based criterion classify valley transitions across both families without geometry-specific tuning?

H1: ∃ Φ, Φ(Grid2D)≈Φ(Ring)⇒same valley classH_1:\ \exists\ \Phi,\ \Phi(\text{Grid2D})\approx\Phi(\text{Ring})\Rightarrow\text{same valley class}

outlook cross_luca_regime_map

Next experiments should maximize uncertainty reduction per compute budget under reproducibility constraints.

arg⁡max⁡e∈EΔI(e)C(e)\arg\max_{e\in\mathcal{E}} \frac{\Delta I(e)}{C(e)}

outlook cross_luca_regime_map

Incoming agents should extend existing claim graph nodes instead of resetting report-level context.

Gt+1=Gt∪ΔG, ΔG must reference existing evidence nodesG_{t+1}=G_t\cup \Delta G,\ \Delta G\ \text{must reference existing evidence nodes}

Distribution Setup cross_luca_regime_map

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)}

Distribution Setup ring_valley_dst

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)}

Distribution Setup grid2d_two_target_double_peak

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)}

Distribution Setup ring_two_target

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)}
Open interactive lab (4)

Interactive plots are mounted on demand when this panel opens, so continuous reading stays responsive.

Open claims and evidence (6)

finding chapter-7-outlook-gap-hazard-bridge

Open gap: we still need a single hazard-led criterion that transfers from Grid2D corridors to ring shortcut regimes without case-specific redefinition.

Evidence items 2

  • source_document research/reports/grid2d_two_target_double_peak/manuscript/grid2d_two_target_double_peak_en.tex
  • source_document research/reports/ring_two_target/manuscript/ring_two_target_en.tex

finding chapter-7-outlook-gap-regime-transfer

Cross-model regime maps are now available, but transfer confidence remains conditional on matched sampling protocols and fairness constraints.

Evidence items 2

  • source_document research/reports/cross_luca_regime_map/manuscript/cross_luca_regime_map_en.tex
  • dataset /data/v1/reports/cross_luca_regime_map/series/manifest.json

method chapter-7-outlook-next-experiment-plan

Next iteration should prioritize experiments by uncertainty reduction per compute cost, while preserving claim-evidence traceability constraints.

Evidence items 2

  • source_document platform/web/public/data/v1/book/backbone.json
  • dataset /data/v1/agent/claim_graph.jsonl

result chapter-7-outlook-agent-continuation

A new agent can continue the research storyline without context reset because chapter manifests, claim graph, and iteration history are packaged together.

Evidence items 2

  • source_document .local/checks/openclaw_review_history.jsonl
  • source_document .local/checks/content_iteration/run_history.jsonl

finding chapter-7-outlook-gap-parameter-geometry

The strongest unresolved coupling is between geometry asymmetry and parameter-scan thresholds; future work must isolate these effects with controlled factorial sweeps.

Evidence items 2

  • source_document research/reports/grid2d_rect_bimodality/manuscript/grid2d_rect_bimodality_en.tex
  • source_document research/reports/ring_valley_dst/manuscript/ring_valley_dst_en.tex

method chapter-7-outlook-gap-proof-depth

Proof-depth exceptions should be converted into explicit closure tasks by prioritizing reports with thinner derivation chains before adding new model variants.

Evidence items 2

  • source_document platform/web/public/data/v1/theory_map.json
  • source_document .local/checks/openclaw_book_math.json

No downstream chapter; consolidate assumptions, claims, and open questions.

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