valley-k-small
Shortcut Variants

Chapter 4: Shortcut Variants

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

Read time 12 min Reports 4 Interactive panels 4

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Chapter Guide

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.

Narrative Walkthrough

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.

Concept Cards

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

Reports 13

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

Theory Chain

Distribution Setup ring_lazy_jump

Lazy Ring Jump-Over Mechanism (K2 vs K4) · 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_lazy_jump_ext

Lazy Ring Shortcut Beta Scan · 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)}

Derivation Link ring_lazy_jump_ext_rev2

Lazy Ring Shortcut Figure-1 Revision · Derivation Link

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

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

Lazy Ring Jump-Over Mechanism (K2 vs K4) · scan_N_K4_beta002 [probability]

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

Interactive Dataset

Plot controls
window=1

N q [probability]

Loading plot data…

Provenance: research/reports/ring_lazy_jump/artifacts/data/scan_N_K4_beta002.csv

Lazy Ring Shortcut Beta Scan · scan_N_K4_beta002 [probability]

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

Interactive Dataset

Plot controls
window=1

N q [probability]

Loading plot data…

Provenance: research/reports/ring_lazy_jump_ext/artifacts/data/scan_N_K4_beta002.csv

Lazy Ring Shortcut Figure-1 Revision · luca_k2_fixed_shortcut_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_lazy_jump_ext_rev2/artifacts/data/luca_k2_fixed_shortcut_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
  • Lazy Ring Jump-Over Mechanism (K2 vs K4) (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.)
  • Lazy Ring Shortcut Beta Scan (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.)
  • Lazy Ring Shortcut Figure-1 Revision (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.)
  • 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 ring_lazy_jump-c5 ring_lazy_jump

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

Open evidence links
  • source_document research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

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

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

    Research report ring_lazy_jump.

  • math_block research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex

    Finding formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Shortcut strength has a non-monotonic effect: too weak or too strong

  • dataset /data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

result ring_lazy_jump_ext-c3 ring_lazy_jump_ext

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.

Open evidence links
  • source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

    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.

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

    Research report ring_lazy_jump_ext.

  • math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

    Result formula context in Lazy Ring Shortcut Beta Scan: Increasing beta advances both peaks and steepens tail decay

  • dataset /data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

result ring_lazy_jump_ext_rev2-c3 ring_lazy_jump_ext_rev2

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.

Open evidence links
  • source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex

    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

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

    Research report ring_lazy_jump_ext_rev2.

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

    Result formula context in Lazy Ring Shortcut Figure-1 Revision: Across the three sensitivity tracks, the qualitative mechanism

  • dataset /data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json

    luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]

result ring_valley_dst-c3 ring_valley_dst

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.

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

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

  • 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

    Result formula context in Destination-Scan Valley Control: Destination scanning reveals structured dst windows where the second peak is

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

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

method ring_lazy_jump_ext-c2 ring_lazy_jump_ext

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.

Open evidence links
  • source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

    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

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

    Research report ring_lazy_jump_ext.

  • math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

    Method formula context in Lazy Ring Shortcut Beta Scan: The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N

  • dataset /data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

finding ring_lazy_jump_ext-c4 ring_lazy_jump_ext

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

Open evidence links
  • source_document research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

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

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

    Research report ring_lazy_jump_ext.

  • math_block research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex

    Finding formula context in Lazy Ring Shortcut Beta Scan: At fixed N=100, beta sweeps show systematic left-shifts of peak times and a

  • dataset /data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json

    scan_N_K4_beta002 [probability]: N -> q [probability]

method ring_valley_dst-c2 ring_valley_dst

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

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

  • 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

    Method formula context in Destination-Scan Valley Control: The workflow combines deterministic flux scans, AW-style first-passage

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

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

method ring_lazy_jump_ext_rev2-c2 ring_lazy_jump_ext_rev2

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.

Open evidence links
  • source_document research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex

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

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

    Research report ring_lazy_jump_ext_rev2.

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

    Method formula context in Lazy Ring Shortcut Figure-1 Revision: The pipeline exports standardized Fig.1 inputs, validates schema, renders

  • dataset /data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json

    luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]

Chapter Summary

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

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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