Lineum Extension — Return Echo

Document ID: lineum-extension-return-echo
Version: 1.0.0
Status: Draft
Relates to: lineum-core.md §5.4
Compatibility: core ≥1.0.0,<2.0 ; Eq=4 ; κ static ; 2D periodic
Date: 2025-08-19

1. Abstract

We formalize Return Echo: a behavioral bias where future linon trajectories revisit the ε-neighborhood of prior decay locations after a delay, distinct from Structural Closure. Closure denotes a static φ remnant after decay; echo denotes later arrivals steered by local ∇φ shaping toward the old site. This document provides operational detection, metrics, controls, and validation guidance under the canonical 2D, periodic-BC scope of the core paper.

2. Motivation

The core interprets closure as a field memory independent of active ψ, and notes a separate echo phenomenon in which new linons return to prior decay coordinates. A dedicated extension is required to (i) disambiguate echo from closure, (ii) define falsifiable tests and controls, and (iii) standardize outputs for replication across runs.

3. Scope & Assumptions (canonical)

Dimensionality: 2D discrete grid, periodic BCs.
κ map: static spatial tuner (no temporal variation).
Inputs: time series of linon detections/decays, φ and ∇φ fields (or summaries), run metadata.
Out of scope: dynamic-κ variants, 3D, non-periodic boundaries.

4. Definitions

Decay event: time t0 and location L where a tracked linon drops below persistence criteria.
Echo window: a time interval [t0 + τ_min, t0 + τ_max] during which revisits are checked.
ε-neighborhood: set of coordinates within distance ε from L.
Revisit: detection of a new (distinct) linon whose center enters the ε-neighborhood in the echo window.
Matched control sites: K locations per decay matched on φ and |∇φ| quantiles but unrelated to any decay; used to estimate baseline revisit rates.
Null-shuffle: randomized per-frame positions of candidate targets or time-shuffled trajectories to estimate chance revisits.

5. Operational Detection

5.1 Parameters (defaults)

EPSILON = 2                 # cells (radius)
TAU_MIN = 5                 # steps after decay (avoid immediate, trivial proximity)
TAU_MAX = 500               # steps after decay
K_CONTROLS = 10             # matched control sites per decay (φ, |∇φ| ±10% quantiles)
MAX_ID_GAP = 1              # frames tolerated between detections in tracking
PHI_MATCH_TOL = 0.10        # matching tolerance for φ and |∇φ| quantiles

Tune TAU_MAX to your run length; report chosen values in metadata.

5.2 Procedure

Detect and log all decay events (L, t0).
For each decay, construct K matched control sites (same frame t0, similar φ, |∇φ| quantiles, not overlapping any decay).
Scan frames t ∈ [t0 + τ_min, t0 + τ_max] for revisits: new linon centers entering B(L, ε).
Compute echo metrics (below) and repeat for all decays; aggregate across runs.

5.3 Metrics

Echo Rate (ER): ER = P(revisit | decay) / P(revisit | control); report 95% CIs (bootstrap).
Delay Distribution: histogram / KDE of (t - t0) for revisits.
Offset Distribution: distances from L at revisit time, to assess ε sensitivity.
Approach Alignment: mean ⟨cos θ⟩ of approach vector vs local ∇φ near entry into B(L, ε).
Occupancy Surplus: revisit density map vs. control density map around L.

5.4 Controls

Null-shuffle: randomize candidate positions or time indices; ER should → 1, alignment → 0.
A/B κ-maps: island vs constant κ; echo should persist if driven by φ topology rather than κ edges alone.
ε/τ Sensitivity: sweep ε ∈ {1,2,3,4} and τ_max ∈ {250,500,1000}; report ER stability.

5.5 Disambiguation from Closure

Closure check: flag whether a static φ remnant (persistent imprint) exists at L post-decay (e.g., low-|∇φ| pocket or Laplacian extremum persisting ≥ M steps).
Independence test: compute ER separately for decays with and without detected φ remnants. An echo effect that remains >1 in both strata supports a behavioral interpretation.
Co-occurrence report: report fraction of echoes that occur at sites with closure vs. without closure.

6. Expected Results (summary)

ER > 1 across seeds and κ-maps, with a unimodal delay distribution.
Positive approach alignment (⟨cos θ⟩ > 0) indicating guidance by local ∇φ near re-entry.
Occupancy surplus localized around prior decay sites; null-shuffles collapse ER → 1.

(Full numerical tables belong in validation reports.)

7. Limitations & Failure Modes

Echo detection is sensitive to tracking quality (ID switches).
Very flat φ landscapes reduce alignment signal; larger samples may be needed.
Poor matching of control sites can bias ER; pre-register matching rules.

8. Reproducibility Checklist

Publish seeds, κ-map, parameter dump.
Export per-decay logs, control-site selection, revisit events, and φ/∇φ fields at relevant frames.
Provide scripts for ER bootstrap, alignment calculations, and null-shuffles.
Report ε/τ sensitivity and stratified ER (with/without closure).

9. Appendix — Minimal Pseudocode

# inputs: decays[(L, t0)], detections[t], phi[t], gradphi[t]
for (L, t0) in decays:
    controls = match_controls(L, t0, phi[t0], gradphi[t0], k=K_CONTROLS, tol=PHI_MATCH_TOL)
    echo_hits = 0; control_hits = 0

    for t in range(t0 + TAU_MIN, min(t0 + TAU_MAX, T)):
        # echo test
        if exists_new_linon_in_ball(detections[t], center=L, eps=EPSILON):
            echo_hits += 1; break

    for C in controls:
        for t in range(t0 + TAU_MIN, min(t0 + TAU_MAX, T)):
            if exists_new_linon_in_ball(detections[t], center=C, eps=EPSILON):
                control_hits += 1; break

    record_ER_sample(echo_hit=(echo_hits>0), control_hit=(control_hits>0))

ER = bootstrap_ratio(p_echo, p_control)

10. 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 definition, ER metric, controls (null, A/B κ), alignment and occupancy analyses; canonical 2D scope.