Lineum Extension — Zeta–RNB Resonance

Document ID: lineum-extension-zeta-rnb-resonance
Version: 1.0.0
Status: Draft
Relates to: lineum-core.md §5.4 (RNB), §5.6 (spectral stability)
Compatibility: core ≥1.0.0,<2.0 ; Eq=4 ; κ static ; 2D periodic
Date: 2025-08-19

1. Abstract

We investigate a putative resonance between Resonant Return Points (RNB) detected in Lineum and the imaginary parts of the nontrivial zeros of the Riemann zeta function on the critical line. Using a normalized axis for comparison, preliminary analyses show a strong distributional similarity (e.g., Pearson correlation ≈ 0.9842 in one canonical family of runs), despite the fact that no zeta-related mathematics is encoded in the model. This extension formalizes the datasets, metrics, and controls required to reproduce or refute the observation under the canonical 2D, periodic-BC regime.

2. Motivation

RNBs are repeatedly visited coordinates that arise from the system’s own dynamics; they were previously nicknamed “deja‑vu points” but have been standardized as Resonant Return Points (RNB). If RNB distributions echo the spacing of zeta zeros, that would suggest a surprising numerical structure emergent from purely local update rules, without explicit number theory in the code base.

3. Scope & Assumptions (canonical)

  • Dimensionality: 2D discrete grid, periodic BCs.
  • κ: static spatial map (no time evolution) in the canonical scope of this extension.
  • Signals: RNB positions measured along a normalized axis; reference set of zeta zeros’ imaginary parts {t_n} on Re(s)=1/2, normalized to [0,1].
  • Evidence to date: initial strong matches were observed on specific runs including spec7_true; some experiments used a κ trajectory (e.g., island_to_constant), which is non‑canonical. We separate canonical from non‑canonical evidence in reporting.

4. Data Requirements

  • RNB dataset: per‑run CSV with normalized positions (e.g., rnb_positions.csv), including run ID, frame bounds, normalization method.
  • Zeta zeros: list of the first N imaginary parts {t_n} of zeros on Re(s)=1/2, normalized to [0,1].
  • Metadata: grid size, Δt, seeds, κ map description (and κ trajectory if used), noise level, detection parameters.
  • Optional: occupancy maps around RNBs, echo/closure flags, spectrum logs for cross‑checks.

5. Definitions

  • RNB (Resonant Return Point): a coordinate (or small neighborhood) that is revisited by distinct linons after prior decay at the same site, beyond a minimal delay window; RNBs are behavioral (not merely structural fossils).
  • Normalized axis: affine map of the comparison coordinate to [0,1]; the same mapping must be used for both RNBs and {t_n}.
  • Distributional match: similarity of empirical CDFs, histograms, or kernel density estimates under the chosen normalization.

6. Methods

6.1 Preprocessing

  1. Build the RNB set for a run: detect decays, define an echo window, record revisits within ε of prior decay locations; deduplicate to unique sites.
  2. Normalize coordinates to [0,1] along the chosen axis (report axis and mapping).
  3. Load the first N zeta zeros {t_n} and normalize to [0,1].

6.2 Comparison Metrics

  • Pearson correlation between binned densities of RNBs and normalized {t_n}.
  • Euclidean distance between normalized histogram vectors.
  • KS statistic between empirical CDFs.
  • Peak‑alignment error: absolute differences between leading modes of the two distributions.

6.3 Controls

  • Null shuffles (position): randomize RNB positions or bootstrap with replacement; correlations should collapse toward chance.
  • Null surrogates (spacing): compare to Poisson or Wigner surrogates with matched counts.
  • Sensitivity: sweep bin counts, bandwidths, and N (e.g., 25, 49, 75) to test stability.
  • Cross‑runs: replicate across seeds, κ maps (constant vs island), and grid sizes.

6.4 Reporting

  • Always report normalization, binning/bandwidth, N, and confidence intervals from bootstrap.
  • Separate canonical (κ static) from non‑canonical (κ trajectory) results.

7. Expected Results (illustrative)

  • High Pearson correlation and low Euclidean distance for canonical runs showing RNB structure.
  • Robustness of the match across reasonable binning/bandwidth choices.
  • Nulls reduce correlation and increase distances toward baseline.
  • Some high‑index deviations (phase offset) are plausible and should be discussed.

8. Limitations & Caveats

  • Normalization bias: different axis choices can alter apparent similarity; pre‑register mapping.
  • Finite‑sample effects: small N and sparse RNBs inflate variance; aggregate across runs.
  • Non‑canonical confounders: κ trajectories can restructure spectra; report separately.
  • Multiple comparisons: control for tuning of N, binning, and bandwidth (e.g., hold‑out or pre‑registration).

9. Reproducibility Checklist

  • Publish RNB CSVs, zeros list, code for normalization and metrics.
  • Share seeds, κ config, Δt, and all detection parameters (ε, τ windows).
  • Provide null/surrogate scripts and cross‑run aggregation notebooks.
  • Include plots of histograms, CDFs, and peak alignments with CIs.

10. Appendix — Minimal Pseudocode

# inputs: rnb_positions[], zeta_zeros[]
x = normalize_to_unit_interval(rnb_positions)
t = normalize_to_unit_interval(zeta_zeros)
h_x = histogram(x, bins=B, density=True)
h_t = histogram(t, bins=B, density=True)
pearson = corr(h_x, h_t)
edist = l2_norm(h_x - h_t)
ks = ks_statistic(ecdf(x), ecdf(t))

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) — datasets, metrics (Pearson, Euclidean, KS), null/surrogate controls, canonical vs non‑canonical reporting, reproducibility checklist.

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