Lineum Extension — Vortex–Particle Coupling

Document ID: lineum-extension-vortex-particle-coupling
Version: 1.0.0
Status: Draft
Relates to: lineum-core.md §6.2
Compatibility: core ≥1.0.0,<2.0 ; Eq=4 ; κ static ; 2D periodic
Date: 2025-08-19

1. Abstract

We formalize how phase vortices in Lineum bind into long-lived, particle-like structures (“linons”). Empirically, co-rotating vortex triads (↺↺↺ / ↻↻↻) form near-equilateral configurations stabilized by a symmetric φ basin; their interferential pattern in arg ψ persists for ≥20 steps. This extension provides operational detection rules, output schemas, and a validation plan for independent replication under the canonical 2D, periodic-BC regime of the core paper.

2. Motivation

The core documents robust linon dynamics but leaves micro-mechanics of binding to interpretation. This supplement isolates the vortex-binding rules: rotation sense, geometry, φ-mediated stability, and reproducible criteria. The goal is a falsifiable protocol that other groups can run on their Lineum outputs without modifying the canonical Equation (1).

3. Scope & Assumptions (canonical)

  • Dimensionality: 2D discrete grid, periodic BCs.
  • κ map: static spatial tuner sampled per step (no time evolution).
  • Inputs: snapshots or timeseries of ψ (phase) and φ (value, ∇φ), plus run metadata.
  • Out of scope: dynamic-κ variants, 3D, non-periodic boundaries (treat as separate experiments).

4. Definitions & Notation

Term Meaning
Vortex (↺/↻) Phase singularity from winding of arg ψ (counter-/clockwise).
Triad Unordered set of three vortices.
Near-equilateral Triangle with edge-ratio tolerance ≤ 10% (configurable).
Linon Localized |ψ|² excitation (per core).
Binding basin Local low-|∇φ| pocket (often with a φ extremum) at/near the triad centroid.
Symbolic record e.g., ↺⟨u ⊙ u ⊙ d⟩_△ for a co-rotating triad in triangular topology with φ-bridges (⊙).

ASCII fallback: use CCW<C,C,D>_TRI for ↺⟨u,u,d⟩_△ and BRIDGE for ⊙ when Unicode is unavailable.

5. Expected Data Inputs

  • phi_grid_summary.csv — φ values and (optionally) ∇φ per grid cell / per frame
  • psi_phase.(png|npy) — phase field or images sufficient for vortex detection
  • true_trajectories.csv (optional) — tracked linon centers over time
  • Run metadata: grid size, seeds, κ-map description, noise amplitude, step count

6. Detection Algorithm (operational)

Parameters (defaults):

VORTEX_MIN_SEP = 3           # cells; de-duplicate near-overlapping cores
EQUILATERAL_TOL = 0.10       # 10% edge-ratio tolerance
STABILITY_STEPS = 20         # min consecutive frames
MAX_CENTROID_DRIFT = 2       # cells across STABILITY_STEPS
QUIET_BASIN_Q = 0.20         # centroid |∇φ| ≤ 20th percentile of local neighborhood
PHI_BRIDGE_TOL = 1           # ±1 cell around pair midpoint

Steps:

  1. Vortex identification — compute winding of arg ψ; label each core as ↺ or ↻.
  2. Triad candidates — enumerate 3-tuples with same rotation (↺↺↺ or ↻↻↻); keep near-equilateral by EQUILATERAL_TOL.
  3. Centroid basin check — at triad centroid, require quiet φ pocket: local |∇φ| ≤ QUIET_BASIN_Q quantile; Laplacian indicates extremum (minimum/maximum) consistent with capture.
  4. Interference criterion — between cores, detect persistent striping / nodal pattern in arg ψ (e.g., Fourier anisotropy vs. randomized null).
  5. Temporal stability — track the three cores (or φ proxies) across frames; require ≥ STABILITY_STEPS with centroid drift ≤ MAX_CENTROID_DRIFT.
  6. φ-bridges (⊙) — for each pair, test for φ extremum near pair midpoint (± PHI_BRIDGE_TOL cells).
  7. Emit record — if all pass, write structured record (see §7), including symbolic form, e.g. ↺⟨u ⊙ u ⊙ d⟩_△.

7. Output Schema (per confirmed triad)

CSV columns (suggested):

run_id, frame_start, frame_end, rotation (CCW/CW),
x1,y1, x2,y2, x3,y3,  # vortex core coords
a,b,c, equilateral_tol_pass,
centroid_x,centroid_y, phi_centroid, gradphi_centroid, laplace_phi_centroid,
bridge_12, bridge_23, bridge_31,        # boolean flags
interference_score, stability_steps, centroid_drift_max,
symbol, notes

8. Validation Plan

A/B κ-maps: island vs constant to contrast triad incidence, basin symmetry, bridge rate.
Noise sweep (ξ): plot survival curves vs. noise amplitude.
Tolerance sweep: 5–15 % equilateral tolerance; evaluate precision/recall vs. manual labels.
Shuffled null: per-frame randomization of vortex positions; estimate false-positive rate (FPR).
Cross-runs: replicate on spec6_true, spec7_true; pool metrics with CIs.

Primary metrics:

  • Incidence: triads / 1000 frames (by rotation class).
  • Stability: mean (±SD) confirmed frames; Kaplan–Meier survival if censoring.
  • Basin contrast: centroid |∇φ| quantiles vs. neighborhood / background.
  • Interference score: Fourier energy ratio in oriented bands vs. isotropic null.
  • Bridge rate: fraction of edges with detected φ-bridge.

9. Results (empirical summary)

  • Co-rotating triads occur significantly more often as stable configurations than mixed-rotation triples.
  • Passing triads consistently show a quiet φ basin at the centroid and robust interferential structure in arg ψ.
  • The ≥20-step threshold filters turbulence without suppressing genuine bindings.

(Numerical tables belong to the validation report; this extension stays model-procedural.)

10. Discussion

Binding emerges without explicit forces: φ provides the quiet, metric-like background shaping where |ψ|² accumulates; arg ψ interference encodes phase-locking between cores. Rotation sense (↺/↻) acts as a particle/antiparticle tag, while binding topology (triangle vs. chain) encodes species. We do not fix a universal taxonomy here; the symbolic layer is a pragmatic recording language.

11. Limitations & Failure Modes

  • 2D canonical scope; 3D or non-periodic boundaries may alter vortex statistics.
  • Vortex detection quality depends on phase unwrapping and image SNR.
  • Near-equilateral constraint is pragmatic; other motifs (chains, multi-rings) need dedicated criteria.
  • False positives can rise when φ gradients are globally shallow; require null controls.

12. Reproducibility Checklist

  • Publish seeds, κ-map, parameter dump.
  • Include raw phase fields (or reproducible FFT pipeline), φ grids, and code to recompute vortices.
  • Provide overlay figures: arg ψ with vortex markers; φ heatmap with centroid and bridges; drift tracks.
  • Export full triad CSVs and a README with parameter values matching §6 defaults or stating deviations.

13. Appendix A — Minimal Pseudocode

# inputs: vortices = [(x,y,rot), ...], phi_grid, phase_field, frames
V = deduplicate_close_vortices(vortices, min_sep=VORTEX_MIN_SEP)
triads = [T for T in combinations_same_rotation(V, k=3) if near_equilateral(T, tol=EQUILATERAL_TOL)]

for T in triads:
    c = centroid(T)
    if quiet_phi_basin(phi_grid, c, q=QUIET_BASIN_Q) and persistent_interference(phase_field, T):
        if stable_over_time(T, frames, min_steps=STABILITY_STEPS, max_centroid_drift=MAX_CENTROID_DRIFT):
            bridges = detect_phi_bridges(phi_grid, T, tol=PHI_BRIDGE_TOL)  # (b12, b23, b31)
            emit_record(T, c, bridges, symbol="↺⟨u ⊙ u ⊙ d⟩_△")

14. Appendix B — Symbol Key

Symbol Meaning
↺ / ↻ CCW/CW rotation (particle/antiparticle tag)
⊙ φ-bridge (local extremum between a pair)
△ triangular binding topology
u, d visual/role tags for cores within triad (analogy labels; non-essential to detection)

15. Versioning & Changelog

Policy. Semantic Versioning applies to this document; compatibility with the core is pinned in the header.
1.0.0 — 2025-08-19 (initial)

  • Operational criteria for co-rotating vortex triads, output schema, and validation plan (canonical 2D, periodic BCs).
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