valley-k-small
Cross-Model Synthesis

Chapter 5: Cross-Model Synthesis

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

Read time 12 min Reports 4 Interactive panels 4

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Chapter Guide

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.

Narrative Walkthrough

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.

Concept Cards

First-passage distribution

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

Reports 26

Survival and hazard

Links between f(t), S(t), and hazard-style diagnostics.

Reports 14

Hazard interpretation

Peak/valley interpretation using hazard dynamics.

Reports 12

AW inversion

Discrete Cauchy / FFT inversion from generating functions.

Reports 11

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

Reports 13

Theory Chain

Distribution Setup cross_luca_regime_map

Cross-Model Luca Regime Map · Distribution Setup

Defines first-passage probability objects used by later diagnostics.

f(t)=Pr⁡[T=t],S(t)=Pr⁡[T>t],h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=\Pr[T>t],\quad h(t)=\frac{f(t)}{S(t-1)}

Distribution Setup grid2d_two_target_double_peak

Grid2D Two-Target Double-Peak · Distribution Setup

Defines first-passage probability objects used by later diagnostics.

f(t)=Pr⁡[T=t],S(t)=Pr⁡[T>t],h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=\Pr[T>t],\quad h(t)=\frac{f(t)}{S(t-1)}

Distribution Setup ring_two_target

Two-Target Lazy Ring Mechanics · Distribution Setup

Defines first-passage probability objects used by later diagnostics.

f(t)=Pr⁡[T=t],S(t)=Pr⁡[T>t],h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=\Pr[T>t],\quad h(t)=\frac{f(t)}{S(t-1)}

Distribution Setup ring_valley_dst

Destination-Scan Valley Control · Distribution Setup

Defines first-passage probability objects used by later diagnostics.

f(t)=Pr⁡[T=t],S(t)=Pr⁡[T>t],h(t)=f(t)S(t−1)f(t)=\Pr[T=t],\quad S(t)=\Pr[T>t],\quad h(t)=\frac{f(t)}{S(t-1)}

Interactive Evidence Panel

Cross-Model Luca Regime Map · runtime_raw

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

Interactive Dataset

Plot controls
window=1

t_max defect_pairs, local_bias_sites, sparse_seconds

Loading plot data…

Provenance: research/reports/cross_luca_regime_map/artifacts/data/runtime_raw.csv

Grid2D Two-Target Double-Peak · method_comparison_c1 [probability]

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

Interactive Dataset

Plot controls
window=1

mfpt_truncation_scan_t_max mfpt_truncation_scan_mass_any [probability]

Loading plot data…

Provenance: research/reports/grid2d_two_target_double_peak/artifacts/data/method_comparison_c1.json

Two-Target Lazy Ring Mechanics · small_scan_metrics [probability]

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

Interactive Dataset

Plot controls
window=1

beta q [probability]

Loading plot data…

Provenance: research/reports/ring_two_target/artifacts/data/small_scan_metrics.csv

Destination-Scan Valley Control · scan [probability]

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

Interactive Dataset

Plot controls
window=1

steps mass, remaining [probability]

Loading plot data…

Provenance: research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/scan.csv

Evidence Trail

This chapter is presented as one coherent story. The underlying report artifacts are preserved as auditable evidence nodes.

Open evidence-node index
  • Cross-Model Luca Regime Map (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.)
  • Grid2D Two-Target Double-Peak (Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.)
  • Two-Target Lazy Ring Mechanics (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.)
  • Destination-Scan Valley Control (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.)

Claim Ledger

finding cross_luca_regime_map-c4 cross_luca_regime_map

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

Open evidence links
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

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

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

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Finding formula context in Cross-Model Luca Regime Map: Pooled timing medians indicate sparse exact dominates most scanned regimes in

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

finding ring_two_target-c4 ring_two_target

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

Open evidence links
  • source_document research/reports/ring_two_target/artifacts/tables/case_configs.tex

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

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

    Research report ring_two_target.

  • math_block research/reports/ring_two_target/artifacts/tables/case_configs.tex

    Finding formula context in Two-Target Lazy Ring Mechanics: Two-target geometry introduces competing fast and delayed channels, making

  • dataset /data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json

    small_scan_metrics [probability]: beta -> q [probability]

finding ring_valley_dst-c5 ring_valley_dst

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

Open evidence links
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

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

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

    Research report ring_valley_dst.

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

    Finding formula context in Destination-Scan Valley Control: Scanning dst alone can substantially change second-peak height ratio and

  • dataset /data/v1/reports/ring_valley_dst/series/scan-probability.json

    scan [probability]: steps -> mass, remaining [probability]

result cross_luca_regime_map-c3 cross_luca_regime_map

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.

Open evidence links
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    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

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

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Result formula context in Cross-Model Luca Regime Map: Across the scanned workload, sparse exact remains the dominant full-FPT baseline

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

finding ring_valley_dst-c4 ring_valley_dst

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

Open evidence links
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

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

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

    Research report ring_valley_dst.

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

    Finding formula context in Destination-Scan Valley Control: Deterministic flux/master-equation results and Monte Carlo class decomposition

  • dataset /data/v1/reports/ring_valley_dst/series/scan-probability.json

    scan [probability]: steps -> mass, remaining [probability]

finding cross_luca_regime_map-c5 cross_luca_regime_map

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

Open evidence links
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

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

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

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Finding formula context in Cross-Model Luca Regime Map: The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

method cross_luca_regime_map-c2 cross_luca_regime_map

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

Open evidence links
  • source_document research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

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

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

    Research report cross_luca_regime_map.

  • math_block research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex

    Method formula context in Cross-Model Luca Regime Map: Runs use warm-up plus repeated timed executions, defect-pair routing rules, and

  • dataset /data/v1/reports/cross_luca_regime_map/series/runtime_raw.json

    runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds

model ring_valley_dst-c1 ring_valley_dst

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.

Open evidence links
  • source_document research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex

    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.

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

    Research report ring_valley_dst.

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

    Model formula context in Destination-Scan Valley Control: The model uses a K=6 ring with one directed shortcut src->dst and an absorbing

Chapter Summary

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

Open chapter glossary links
  • AW Inversion: Discrete Cauchy/FFT-based inversion from generating functions to time-domain FPT quantities.
  • Beta Scan: Parameter sweep over shortcut strength β to identify phase shifts and regime boundaries.
  • Bimodality Criterion: Operational criterion to separate true two-peak structure from noisy shoulders.
  • Claim Ledger: Structured mapping from statement to evidence paths and cross-report links.
  • Equal4 Baseline: Four-way equalized baseline used to compare shortcut effects under symmetric local movement.
  • First-Passage Time (FPT): Random time needed for the trajectory to hit an absorbing target for the first time.
  • Hazard Rate: Conditional probability of first passage at step t given survival up to t.
  • Renormalize Shortcut Mode: Base transition weights are rescaled after shortcut injection to preserve normalization constraints.
  • Selfloop Shortcut Mode: Shortcut probability mass is taken from self-loop probability without renormalizing other moves.
  • Survival Function: Probability that first passage has not happened by step t.
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