valley-k-small
ring_deriv_k2

Ring Derivation Backbone

This derivation report provides the analytical backbone for ring models with one directed long-range link, including defect-free propagators, defect corrections, and first-passage generating functions. It serves as the shared mathematical base used by later shortcut and valley studies.

Book PositionReading PathConnected ReportsNarrative Arc PositionVerifiable ClaimsInteractive FiguresMathematical LogicFormula LibraryNarrative NotesAppendix & Assets

Updated: 9 Jun 2026, 21:57:42 UTC

Model

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.

Method

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

Result

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

Book Position

This report is part of the chapterized mainline. Use chapter links to keep continuity instead of reading reports in isolation.

Primary chapter Chapter 0: Reading Guide & Notation

Also appears in chapter-0-reading-guidechapter-3-ring-baseline

Next chapter →

Reading Path

  1. Scan key findings first to decide whether this report is relevant.
  2. Use the interactive panel to test parameter and shape sensitivity.
  3. Then read the mathematical chain and formula library for derivation details.

Key Findings

Connected Reports

This report sits inside a shared chain. Use links below to move upstream/downstream and across model families.

Narrative Arc Position

This report appears in one or more global arcs. Use these checkpoints to keep reading continuity across pages.

Grid and ring mechanisms converge into cross-report synthesis.

Global storyline that connects all report families from mechanism to synthesis.

Checkpoints: 27 Claims: 130

Open arc sequence
  1. Lazy Ring Jump-Over Mechanism (K2 vs K4)
  2. Lazy Ring Shortcut Beta Scan
  3. Lazy Ring Shortcut Figure-1 Revision
  4. Lazy Ring Flux Baseline
  5. Ring Valley Regime Map
  6. Destination-Scan Valley Control
  7. Ring Derivation Backbone
  8. Two-Target Lazy Ring Mechanics
  9. 1D Ring Two-Walker Encounter With Shortcut
  10. Grid2D Bimodality Baseline
  11. Grid2D Reflecting-Boundary Bimodality
  12. Grid2D Blackboard Endpoint Case
  13. Grid2D Two-Target Double-Peak
  14. 2D Two-Walker Encounter With Shortcut
  15. Grid2D Rectangle Bimodality
  16. Grid2D Membrane Near Target
  17. Grid2D One Target — Base
  18. Grid2D One Target — Exit Timing
  19. Grid2D One Target — Valley/Peak Budget
  20. Grid2D One Target — Window Measures
  21. Grid2D One vs Two Target — Gating
  22. Grid2D Two-Target Bias-Radius Scaffold
  23. Cross-Model Luca Regime Map
  24. Final Multitimescale FPT and Encounter Report
  25. Exact Recursion — Method Guide
  26. Reflecting Encounter Diagonal Decomposition
  27. Reflecting Encounter Mean Validation

Verifiable Claims

Claims below are tied to explicit evidence paths so each statement can be audited.

Report Objective

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

Verification Steps
  1. Read the key claims and their evidence references first.
  2. Verify at least one equation card and one dataset panel against source paths.
  3. Cross-check this report with upstream/downstream linked reports.

MODEL

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

    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

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

    We consider a discrete-time random walk on a 1D ring (cycle) of size N with periodic boundary conditions (PBC). Transition matrices and convention.

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

    We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.

  • math_block research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Model formula context in Ring Derivation Backbone: The setting is a finite ring random walk with periodic indexing and one directed

Linked reports Lazy Ring Jump-Over Mechanism (K2 vs K4)Lazy Ring Flux BaselineLazy Ring Shortcut Beta ScanFinal Multitimescale FPT and Encounter Report

METHOD

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

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

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

    The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n

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

    We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.

  • math_block research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Method formula context in Ring Derivation Backbone: The report derives Green-function style propagators, constructs defect-resolvent

  • dataset /data/v1/reports/ring_deriv_k2/series/note_k2-probability.json

    note_k2 [tabular probability]: N -> q, p [probability]

Linked reports Grid2D Bimodality BaselineLazy Ring Flux BaselineTwo-Target Lazy Ring MechanicsDestination-Scan Valley Control

RESULT

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

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

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

    We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.

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

    The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n

  • math_block research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Result formula context in Ring Derivation Backbone: The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and

  • dataset /data/v1/reports/ring_deriv_k2/series/note_k2-probability.json

    note_k2 [tabular probability]: N -> q, p [probability]

Linked reports Lazy Ring Flux BaselineLazy Ring Jump-Over Mechanism (K2 vs K4)Lazy Ring Shortcut Beta ScanLazy Ring Shortcut Figure-1 Revision

FINDING

finding ring_deriv_k2-c4

Defect-free and defect-corrected propagators can be written in a unified analytic framework.

Evidence trail
  • source_document research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Defect-free and defect-corrected propagators can be written in a unified analytic framework.

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

    The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n

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

    We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.

  • math_block research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Finding formula context in Ring Derivation Backbone: Defect-free and defect-corrected propagators can be written in a unified analytic

  • dataset /data/v1/reports/ring_deriv_k2/series/note_k2-probability.json

    note_k2 [tabular probability]: N -> q, p [probability]

Linked reports Grid2D Bimodality BaselineCross-Model Luca Regime Map

finding ring_deriv_k2-c5

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

Evidence trail
  • source_document research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

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

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

    We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.

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

    The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n

  • math_block research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

    Finding formula context in Ring Derivation Backbone: Directed long-range links alter first-passage statistics through resolvent-level

  • dataset /data/v1/reports/ring_deriv_k2/series/note_k2-probability.json

    note_k2 [tabular probability]: N -> q, p [probability]

Linked reports Grid2D Blackboard Endpoint CaseGrid2D Two-Target Double-PeakTwo-Target Lazy Ring Mechanics1D Ring Two-Walker Encounter With Shortcut

Interactive Dataset

Plot controls
window=1

N q, p [probability]

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Provenance: research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Mathematical Logic Chain

From model assumptions to interpretation in a short, ordered chain.

Distribution Setup

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}

Notation and model

Spectral / Inversion Step

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)

Notation and model

Spectral / Inversion Step

Provides analytic inversion machinery for computing trajectories.

∑nBn,n′=∑nAn,n′=1\sum_n B_{n,n'}=\sum_n A_{n,n'}=1

Notation and model

Model Constraint

States the structural constraints and parameter ranges of the model.

Bn,n=1−q,Bn±1,n=q2,Bm,n=0 otherwise.B_{n,n}=1-q,\qquad B_{n\pm 1,n}=\frac{q}{2},\qquad B_{m,n}=0\text{ otherwise}.

Notation and model

Model Constraint

States the structural constraints and parameter ranges of the model.

Au,u=1−q−p,Au±1,u=q2,Av,u=Bv,u+p,An,u=Bn,u otherwise,A_{u,u}=1-q-p,\qquad A_{u\pm 1,u}=\frac q2,\qquad A_{v,u}=B_{v,u}+p,\qquad A_{n,u}=B_{n,u}\ \text{otherwise},

Notation and model

Model Constraint

States the structural constraints and parameter ranges of the model.

X‾≡B‾−A‾.\underline{X}\equiv \underline{B}-\underline{A}.

Notation and model

Mathematical Principles

Showing 6 / 14

Notation and model EN

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}
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

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

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

∑nBn,n′=∑nAn,n′=1\sum_n B_{n,n'}=\sum_n A_{n,n'}=1
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

Bn,n=1−q,Bn±1,n=q2,Bm,n=0 otherwise.B_{n,n}=1-q,\qquad B_{n\pm 1,n}=\frac{q}{2},\qquad B_{m,n}=0\text{ otherwise}.
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

Au,u=1−q−p,Au±1,u=q2,Av,u=Bv,u+p,An,u=Bn,u otherwise,A_{u,u}=1-q-p,\qquad A_{u\pm 1,u}=\frac q2,\qquad A_{v,u}=B_{v,u}+p,\qquad A_{n,u}=B_{n,u}\ \text{otherwise},
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

X‾≡B‾−A‾.\underline{X}\equiv \underline{B}-\underline{A}.
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Show remaining formulas

Notation and model EN

ηu,v≡Xv,u=Bv,u−Av,u,ηv,u≡Xu,v=Bu,v−Au,v.\eta_{u,v}\equiv X_{v,u}=B_{v,u}-A_{v,u},\qquad \eta_{v,u}\equiv X_{u,v}=B_{u,v}-A_{u,v}.
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

ηu,v=−p,ηv,u=0,\boxed{\eta_{u,v}=-p,\qquad \eta_{v,u}=0,}
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

Xu,u=Bu,u−Au,u=p=−ηu,vX_{u,u}=B_{u,u}-A_{u,u}=p=-\eta_{u,v}
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

Qn0(n,t)=Pr⁡(Xt=n∣X0=n0)Q_{n_0}(n,t)=\Pr(X_t=n\mid X_0=n_0)
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

Sn0(n,t)=Pr⁡(Xt=n∣X0=n0)S_{n_0}(n,t)=\Pr(X_t=n\mid X_0=n_0)
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

Q~n0(n,z)=∑t≥0Qn0(n,t)zt\tilde Q_{n_0}(n,z)=\sum_{t\ge 0} Q_{n_0}(n,t)z^t
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

d(a,b)≡min⁡(∣a−b∣,  N−∣a−b∣)∈{0,1,…,⌊N2⌋}.d(a,b)\equiv \min\bigl(|a-b|,\;N-|a-b|\bigr)\in\Bigl\{0,1,\dots,\Bigl\lfloor \frac N2\Bigr\rfloor\Bigr\}.
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Notation and model EN

Tn(x)=cos⁡ ⁣(narccos⁡x),Un(x)=sin⁡ ⁣((n+1)arccos⁡x)sin⁡(arccos⁡x).\T_n(x)=\cos\!\bigl(n\arccos x\bigr),\qquad \U_n(x)=\frac{\sin\!\bigl((n+1)\arccos x\bigr)}{\sin(\arccos x)}.
Formula source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Narrative Sections

Cleaned chapter summaries are shown first; low-value placeholders are hidden.

Notation and model

We consider a discrete-time random walk on a 1D ring (cycle) of size N with periodic boundary conditions (PBC). Transition matrices and convention.

Source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Numerical validation (double precision)

We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent. For each parameter set (N,q,p,u,v,z) we built the column-stochastic transition matrices B (ring) and A (directed shortcut), and computed The ground-truth generating functions are Q

Source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Part 3 --- First-passage time (FPT) generating function

The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n (z) = Q(d(n,n0),z) - z p (Q(d(n,u),z)- Q(d(n,v),z)) Q(d(u,n0),z) 1+z p (Q(0,z)- Q(d(u,v),z)) Q(0,z) - z p (Q(d(n,u),z)- Q(d(n,v),z)) Q(d(u,n)

Source

research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex

Reproducibility Commands

Open command list
  • python3 scripts/reportctl.py build --report ring_deriv_k2 --lang en
  • python3 scripts/reportctl.py translation-qc
  • python3 scripts/reportctl.py web-build --mode changed --skip-npm-ci

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