Lineum Extension — Silent Gravity

Document ID: lineum-extension-silent-gravity
Version: 1.0.0
Status: Draft
Relates to: lineum-core.md §6.1
Compatibility: core ≥1.0.0,<2.0 ; Eq=4 ; κ static ; 2D periodic
Date: 2025-08-19

1. Abstract

We formalize Silent Gravity: an emergent, force-free tendency whereby linons statistically accumulate in low-|∇φ| regions and near φ-minima, and migrate along +∇φ as defined by the canonical update. The effect is quantified via dwell-time enrichment, basin-attraction metrics, and return-bias analyses, without introducing any explicit long-range force term. We provide operational tests, metrics, and replication guidance under the canonical 2D, periodic-BC regime.

2. Motivation

The core paper reports attraction via the φ-gradient and an interpretive note on a quiet, metric-like role of φ. This extension separates the concept into falsifiable claims with measurable criteria so independent groups can test Silent Gravity without modifying Equation (1).

3. Scope & Assumptions (canonical)

  • Dimensionality: 2D discrete grid, periodic BCs.
  • κ: static spatial map (no time evolution).
  • Inputs: time series of ψ (magnitude/phase) and φ (value, ∇φ), plus run metadata.
  • Out of scope: dynamic-κ variants, 3D, non-periodic boundaries.

4. Phenomenon Definition

Silent Gravity = The statistical tendency of linons to (i) dwell longer in neighborhoods with low |∇φ|, (ii) accumulate near local φ-minima (or quiet basins), and (iii) drift along +∇φ (environmental guidance), all without an explicit force term in the canonical update.

5. Operational Tests & Metrics

5.1 Dwell-Time Enrichment vs. |∇φ|

  • Partition the domain into quantile bins of |∇φ| (e.g., Q1…Q5 per frame).
  • For each bin, accumulate linon-centered dwell time.
  • Metric: Enrichment ratio ER = dwell(Q1) / dwell(Q5).
  • Pass criterion (example): ER > 1.5 with 95% CI not overlapping 1 across ≥3 runs.

5.2 Basin Capture at φ-Minima

  • Detect local φ-minima via discrete Laplacian and neighborhood checks.
  • Measure arrival rate and average residence within a fixed radius r_min.
  • Metric: Capture Δ = arrival_rate(minima) – arrival_rate(controls).
  • Pass: Capture Δ > 0 with bootstrap CI excluding 0.

5.3 Directed Drift Along +∇φ

  • For each tracked linon, compute instantaneous displacement Δx and the local gradient direction g = ∇φ / |∇φ|.
  • Metric: mean cosine alignment ⟨cos θ⟩ = ⟨ (Δx · g) / (|Δx||g|) ⟩.
  • Pass: ⟨cos θ⟩ > 0 with permutation-test p < 0.01.

5.4 Quiet-Basin Stability

  • Identify quiet basins: neighborhoods with |∇φ| below the 20th percentile and Laplacian indicating an extremum.
  • Compare survival curves (Kaplan–Meier) of linon residence inside basins vs. matched non-basin controls.
  • Metric: hazard ratio HR = hazard(basin) / hazard(control).
  • Pass: HR < 1 with CI below 1.

5.5 Null & A/B Controls

  • Null shuffle: randomize φ fields per frame (preserve histogram), recompute metrics → all effects should collapse toward null (ER ≈ 1, ⟨cos θ⟩ ≈ 0).
  • A/B κ-maps: island vs. constant κ to examine robustness of basin identification and metrics.

6. Results (empirical summary)

  • Linon dwell time concentrates in low-|∇φ| neighborhoods; enrichment persists across seeds and grid sizes.
  • Arrival and residence near φ-minima exceed matched controls.
  • Mean alignment with +∇φ is significantly positive even with moderate noise.
  • Survival analyses show longer residence within quiet basins.

(Numerical tables belong to validation; this extension remains protocol-oriented.)

7. Discussion

Silent Gravity reframes “attraction” as environmental guidance by φ-topology: φ supplies a quiet, metric-like structure that shapes where |ψ|² accumulates and for how long. No explicit long-range force is introduced; the observed behavior follows directly from local updates and gradients.

8. Limitations & Failure Modes

  • Sensitive to φ-estimation quality and gradient discretization.
  • Extremely flat φ landscapes reduce effect sizes (need larger samples).
  • Basin detection thresholds can bias capture metrics; pre-register parameters.

9. Reproducibility Checklist

  • Publish seeds, κ-map definition, parameter dump.
  • Export per-frame linon positions, φ, ∇φ, and basin masks.
  • Provide scripts/notebooks for binning, survival analysis, and null shuffles.
  • Report CIs and effect sizes for all metrics (ER, Δ, ⟨cos θ⟩, HR).

10. Appendix — Minimal Pseudocode

# inputs: trajectories, phi, grad_phi, basin_mask, frames
bins = gradphi_quantile_bins(grad_phi, q=5)            # per frame
ER = dwell_time(bins[0]) / dwell_time(bins[-1])        # Q1 vs Q5

minima = detect_phi_minima(phi)
capture_delta = arrival_rate(trajectories, minima) - arrival_rate(trajectories, controls)

align = mean_cosine_alignment(trajectories, grad_phi)  # <cos θ>

surv_basin = km_curve(residence_times(trajectories, basin_mask=True))
surv_ctrl  = km_curve(residence_times(trajectories, basin_mask=False))
HR = hazard_ratio(surv_basin, surv_ctrl)

null_ER, null_align = null_shuffle_tests(phi, grad_phi, trajectories)

11. 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 metrics (enrichment, capture, alignment, survival), null/A-B controls, canonical 2D scope.

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