nested-recd

Correction (0.2.3). Versions ≤ 0.2.2 computed H_joint and the pairwise entropies inside Syn with np.unique on a list of tuples and no axis=0. NumPy flattens that input, so Syn, excess3 and binary phi3 were not the published definition when N ≥ 3. Surp and Res_pair were not affected. For N = 2, Syn is 0 either way. 0.2.3 counts joint rows. legacy_pooled_counting=True reproduces the old numbers and warns. See CHANGELOG.md. This version: 10.5281/zenodo.22970079. The 0.2.2 deposit remains 10.5281/zenodo.21937204.

Nested ordinal RECD — pure-NumPy implementation of nested ordinal conjunction levels (Φ₁, Φ₂, Φ₃), the continuous excess³ Level-3 proxy (T-IV; not the strong residual), optional Res_pair (=\mathrm{KL}(P|P^{(2)})), Gibbs-α weights, and λ-weighted Discrete Extramental Clock (RECD) accumulation.

This is the canonical nested-clock software for the Systemic Tau / RECD stack (foundations paper, excess³ methods preprint, cardiac CCTP/SDDB pilot, Academy Learning Tau).

Léxico (no negociable): strong Level-3 := Res_pair. excess3 := exportable proxy. Report w with any Res_pair figure. Abundance (excess3) ≠ aggregation share (f_3).

Install

pip install nested-recd

From source (development):

pip install -e ".[dev]"

excess³ contract (methods)

Canonical specification: Padilla-Villanueva (2026), excess³: A Pre-Specified Continuous Proxy…
DOI: 10.5281/zenodo.21385937 · repo: github.com/johelpadilla/excess3

Rule	Software
excess3 = 0.6·Syn + 0.4·Surp	ALPHA_SYN, ALPHA_SURP (fixed a priori)
Continuous primary	out["excess3"] / compute_phi3_excess
Φ₃ secondary	binary ticks of excess3 vs theta3
Null on full contrast	surrogate_pvalue_delta_excess3 (phase-shuffle / permute on |Δ|)
Proxy ≠ full PID	documented; no complete Williams–Beer atom claimed
Parallel nesting	Φ₁, Φ₂, excess³ computed without hard Boolean ascent

Defaults: m=3, delay 1, window 13, theta3=0.10 (software); cardio-like reference DEFAULT_THETA3_CARDIO=0.08.

Quick start

import numpy as np
from nested_recd import compute_recd_from_conjunctions, ALPHA_SYN, ALPHA_SURP

rng = np.random.default_rng(0)
X = rng.normal(size=(800, 3)).cumsum(axis=0)

out = compute_recd_from_conjunctions(X, m=3, d=4, theta3=0.10)

print("mean excess³ (primary):", float(np.nanmean(out["excess3"])))
print("Φ3 active fraction:", float(np.nanmean(out["phi3"] > 0)))
print("weights:", out["params"]["alpha_syn"], out["params"]["alpha_surp"])
print("final T_recd:", float(out["T_recd"][-1]))

Strong residual Res_pair (optional)

out = compute_recd_from_conjunctions(X, compute_res=True)
# or, on a symbol window:
from nested_recd import compute_res_pair_window
# Always report window length with Res_pair (Nota 4: w=13 can bias chaos).

Continuous excess³ only

from nested_recd import generate_multivariate_symbols, compute_phi3_excess

S = generate_multivariate_symbols(X, m=3)
phi3, excess3 = compute_phi3_excess(S, window=13, theta=0.10, stride=1)

Surrogate p-value on |Δ excess3|

from nested_recd import surrogate_pvalue_delta_excess3

res = surrogate_pvalue_delta_excess3(X, split=400, n_surr=99, method="phase", seed=0)
print(res["delta_obs"], res["p_value"])

Optional τ_s for λ regime weights

out = compute_recd_from_conjunctions(X, tau_s=tau_s)
# or
out = compute_recd_from_conjunctions(X, lam_override=0.5)

Note: delta_recd / T_recd use binary Φ₃ in the α-weighted clock (legacy Discrete Extramental Clock). For exportable / scalable Level-3 reporting, use continuous excess³ and nulls on |Δ excess3|. For strong Level-3 claims, report res_pair (enable compute_res=True) and the window w. Do not identify the two.

What it computes

Symbol	Meaning
Φ₁	Co-occurrence of identical ordinal symbols across variable pairs
Φ₂	Persistence of pairwise ordinal relations over lag d
excess³	Continuous hybrid: 0.6·Syn + 0.4·Surp (exportable Level-3 proxy; T-IV)
Φ₃	Binary ticks of excess³ above theta3 (secondary)
Res_pair	Strong Level-3 residual KL(P ‖ P^(2)) (optional; always report w)
λ	Chaos / reorganization intensity (from τ_s or lam_override)
α(λ)	Design-engine weights (template or Gibbs-α); not a joint observable
ΔRECD	α₁Φ₁ + α₂Φ₂ + α₃Φ₃ (legacy clock; binary Φ₃)
T_recd	Cumulative sum of ΔRECD

Ordinal symbols use Bandt–Pompe patterns (m=3 by default).

Surrogates

from nested_recd import phase_shuffle_independent, random_permutation_independent

X_null = phase_shuffle_independent(X, seed=42)
X_perm = random_permutation_independent(X, seed=42)

API surface

from nested_recd import (
    ALPHA_SYN, ALPHA_SURP,
    DEFAULT_THETA3, DEFAULT_THETA3_CARDIO,
    compute_recd_from_conjunctions,
    compute_phi1, compute_phi2, compute_phi3,
    compute_phi3_excess, compute_excess3_window,
    mean_excess_pre_post, surrogate_pvalue_delta_excess3,
    compute_lambda, alpha_weights, alpha_weights_gibbs, regime_lambda_proxy,
    compute_res_pair, compute_res_pair_window,
    compute_weighted_contributions, high_level3_rate,
    generate_multivariate_symbols,
    phase_shuffle_independent, generate_surrogate_ensemble,
)

Relation to other projects

Project	Role
systemictau	Full Systemic Tau library (τ_s, gate RECD, studio) — optional [nested] extra
systemictau-web	Streamlit app; depends on this package
excess3	Methods + intro ES + primer (specification)
cctp-sddb-systemic-tau	Cardiac VF pilot
This package	Lightweight, installable nested-time / excess³ core

Citation

If you use excess³ / Level-3 from this package, cite the methods preprint (canonical specification):

Padilla-Villanueva, J. (2026). excess³: A Pre-Specified Continuous Proxy for Order-3 Synergistic Surplus (methods). Zenodo. https://doi.org/10.5281/zenodo.21385937

This software (0.2.3):

Padilla-Villanueva, J. (2026). nested-recd 0.2.3: Nested ordinal RECD, excess³ proxy, and Res_pair. Zenodo. https://doi.org/10.5281/zenodo.22970079

The previous release (0.2.2), which pooled symbols inside Syn, remains citable as https://doi.org/10.5281/zenodo.21937204.

Software / related stack:

Padilla-Villanueva, J. (2026). Systemic Tau software archive. Zenodo. https://doi.org/10.5281/zenodo.20576241

CCTP/SDDB pilot:

Padilla-Villanueva, J. (2026). CCTP/SDDB: Systemic Tau and ordinal RECD before spontaneous ventricular fibrillation (v1.0.1). Zenodo. https://doi.org/10.5281/zenodo.21270699

@software{padilla_nested_recd_2026,
  author  = {Padilla-Villanueva, Johel},
  title   = {nested-recd: Nested ordinal RECD levels and continuous excess³},
  year    = {2026},
  url     = {https://github.com/johelpadilla/nested-recd},
  version = {0.2.3},
  doi     = {10.5281/zenodo.22970079}
}

License

MIT © 2026 Johel Padilla-Villanueva
ORCID: 0000-0002-5797-6931