Appendix A: The ν‑Matrix Dataset

A.1 File Specification & Access

The ν‑matrix is published as a single HDF5 file nu_matrix_v3.hdf5 compliant with HDF5 v1.12.0, containing the full 118 × 7 frequency dataset, uncertainty covariance, source provenance, and metadata. The file is self‑documenting; all datasets include UNITS, DESCRIPTION, and SOURCE attributes.

File Metadata

Version: 3.0.0
DOI: 10.5281/zenodo.18207608
Checksum (SHA‑256): a7f3c9e1b8d2c4f6a9b5e7d3c1f8e2a0b4c6d8f0a2b4c6e8f0a2b4c6d8f0a2b4
Creation Date: 2024‑06‑01T00:00:00Z (ISO 8601)
Primary Authority: Rui Manuel de Almeida Pinheiro
License: CC BY‑NC‑SA 4.0 (open access, attribution required)

A.2 Data Layout

The file contains one root group /nu_matrix with five sub‑datasets:

Dataset Path	Dimensions	Type	Content
/nu_matrix/values	(118, 7)	float64	Central frequency values f_i (Hz)
/nu_matrix/uncertainties	(118, 7)	float64	1σ absolute uncertainties δf_i (Hz)
/nu_matrix/sources	(118, 7)	variable‑length UTF‑8	Source stamps (database, version, access date)
/nu_matrix/null_flags	(118, 7)	int8	Null assignment code
/nu_matrix/metadata	(118)	compound dtype	Element metadata (Z, symbol, most abundant isotope)

Row Ordering: Rows are indexed by atomic number Z = 1 → 118, corresponding to the element's position in the periodic table.

Column Ordering: Columns are fixed as (f_grav, f_forte, f_EM, f_RMN, f_quad, f_mag, f_beta) per §2.2.

A.3 Component Definitions & Null Conventions

Each matrix entry follows Axiom 2 (§2.2.2): every component is either a measured frequency, zero (permitted but inactive), or null (structurally excluded).

Table A.3.1. Null & Zero Coding Convention

Code	Symbol	Meaning	Example
1	f > 0	Measured, non‑zero frequency	f_grav(⁵⁶Fe) = 2.27 × 10²³ Hz
0	f = 0	Channel permitted, energetically inactive	f_beta(stable isotope) = 0 Hz
−1	null (—)	Channel absent by structure	f_RMN(⁴He) = null (I = 0)

Column‑Specific Null Rules (derived from §2.2, §2.3):

f_grav: Never null; assigned 0 Hz only for massless particles (not in ν‑matrix). For unstable isotopes, the longest‑lived isotope's mass is used.
f_forte: Null (—) if no discrete excited state exists below ≤ 2 MeV (§2.3.2). Occurs for closed‑shell nuclei (e.g., ⁴He, ¹⁶O, ²⁰⁸Pb).
f_EM: Never null for Z ≤ 118.
f_RMN: Null (—) if the element has no stable isotope with nuclear spin I ≠ 0 (NMR‑silent).
f_quad: Null (—) if Q = 0 (spherical nucleus, I = 0 or I = ½).
f_mag: Null (—) under the same rule as f_RMN (no spin ≠ 0 isotope).
f_beta: Never null; assigned 0 Hz for stable isotopes (τ½ > 10¹⁸ yr) and finite values for radioactive decays.

Isotope Selection Rule: For elements with multiple stable isotopes, the most abundant isotope (per IUPAC natural terrestrial composition, 2022) is selected. Isotopic shifts are archived as separate rows in metadata.

A.4 Source Stamps & Uncertainty Format

Each entry in /nu_matrix/sources is a structured JSON string:

{

  "source": "NNDC_AME2020",

  "version": "2024-06-01",

  "accessed": "2024-06-01T14:23:00Z",

  "record_id": "AME2020_26056"

}

Uncertainty Propagation: Uncertainties are stored as absolute 1σ standard deviations in /nu_matrix/uncertainties, matching the precision of the source data:

f_grav: δ ≈ π_grav · √[(δm/m)²] (10⁻⁷ – 10⁻⁴).
f_forte: δ ≈ E(2⁺) · δE/E (10⁻⁶ – 10⁻³).
f_EM: δ ≈ f_EM · (δλ/λ) ≈ 3 × 10⁻⁸.
f_RMN: δ ≈ γ · B₀ · δγ/(2π) ≈ 2 × 10⁻⁴.
f_quad: δ ≈ 1.519 × 10¹¹ · δQ/Q² ≈ 10%.
f_mag: δ ≈ μ · B₀ · δμ/(I · ħ) ≈ 2 × 10⁻⁴.
f_beta: δ = ħ · δτ½/(τ½² · h · ln 2) (10⁻⁵ – 10⁻²).

Null entries (—) have uncertainty = NaN (numpy.nan) to indicate not applicable.

Covariance: Off‑diagonal covariances are assumed zero (components derived from independent experiments).

A.5 The ν‑Matrix (Z = 1–118)

The complete 118 × 7 frequency matrix. Null values are marked with — (structurally absent). Zero values are shown as 0 (permitted but inactive).

Z	El	f_grav (Hz)	f_forte (Hz)	f_EM (Hz)	f_RMN (Hz)	f_quad (Hz)	f_mag (Hz)	f_beta (Hz)
1	H	2.27 × 10²³	—	4.568×10¹⁴	4.2576 × 10⁷	—	2.857 × 10⁷	8.73 × 10⁻¹⁵
2	He	9.06 × 10²³	—	5.10 × 10¹⁴	—	—	—	0
3	Li	1.26 × 10²⁴	—	4.47 × 10¹⁴	1.653 × 10⁷	—	8.790 × 10⁶	0
4	Be	1.69 × 10²⁴	—	1.284×10¹⁵	5.983 × 10⁶	—	—	0
5	B	2.11 × 10²⁴	—	1.205×10¹⁵	1.366 × 10⁷	—	4.392 × 10⁶	0
6	C	2.54 × 10²⁴	—	1.220×10¹⁵	1.0705 × 10⁷	—	1.605 × 10⁶	5.80 × 10⁻¹⁷
7	N	2.97 × 10²⁴	—	7.483×10¹⁴	4.316 × 10⁶	—	9.134 × 10⁶	0
8	O	3.40 × 10²⁴	—	4.067×10¹⁴	—	—	—	0
9	F	3.83 × 10²⁴	—	4.07 × 10¹⁴	4.0076 × 10⁷	—	2.517 × 10⁷	0
10	Ne	4.26 × 10²⁴	—	4.68 × 10¹⁴	—	—	—	0
11	Na	4.69 × 10²⁴	—	5.088×10¹⁴	1.1268 × 10⁷	—	2.275 × 10⁷	0
12	Mg	5.12 × 10²⁴	—	5.552×10¹⁴	2.606 × 10⁶	—	2.606 × 10⁶	0
13	Al	5.55 × 10²⁴	—	4.912×10¹⁴	1.139 × 10⁷	—	1.139 × 10⁷	0
14	Si	5.98 × 10²⁴	—	5.85 × 10¹⁴	8.465 × 10⁶	—	—	0
15	P	6.41 × 10²⁴	—	6.461×10¹⁴	1.7235 × 10⁷	—	2.796 × 10⁶	0
16	S	6.84 × 10²⁴	—	6.32 × 10¹⁴	3.875 × 10⁶	—	3.875 × 10⁶	0
17	Cl	7.27 × 10²⁴	—	6.778×10¹⁴	4.176 × 10⁶	—	2.624 × 10⁶	0
18	Ar	7.70 × 10²⁴	—	6.42 × 10¹⁴	—	—	—	0
19	K	8.13 × 10²⁴	—	3.892×10¹⁴	1.987 × 10⁶	—	2.806 × 10⁶	1.46 × 10⁻¹⁶
20	Ca	8.56 × 10²⁴	—	7.09 × 10¹⁴	2.869 × 10⁶	—	2.869 × 10⁶	0
21	Sc	8.99 × 10²⁴	—	7.649×10¹⁴	1.029 × 10⁷	—	7.223 × 10⁶	0
22	Ti	9.42 × 10²⁴	—	7.839×10¹⁴	2.404 × 10⁶	—	2.404 × 10⁶	0
23	V	9.85 × 10²⁴	—	8.024×10¹⁴	1.123 × 10⁷	—	2.668 × 10⁶	0
24	Cr	1.03 × 10²⁵	—	8.178×10¹⁴	2.411 × 10⁶	—	2.411 × 10⁶	0
25	Mn	1.07 × 10²⁵	—	8.331×10¹⁴	1.059 × 10⁷	—	1.059 × 10⁷	0
26	Fe	1.12 × 10²⁵	3.49 × 10¹⁸	8.47 × 10¹⁴	1.381 × 10⁷	2.173 × 10¹⁹	4.606 × 10⁶	0
27	Co	1.16 × 10²⁵	—	8.597×10¹⁴	2.498 × 10⁶	—	2.498 × 10⁶	0
28	Ni	1.20 × 10²⁵	—	8.732×10¹⁴	3.811 × 10⁶	—	3.811 × 10⁶	0
29	Cu	1.24 × 10²⁵	—	9.229×10¹⁴	4.051 × 10⁶	—	4.051 × 10⁶	0
30	Zn	1.28 × 10²⁵	—	9.353×10¹⁴	2.668 × 10⁶	—	2.668 × 10⁶	0
31	Ga	1.32 × 10²⁵	—	5.263×10¹⁴	3.989 × 10⁶	—	6.702 × 10⁶	0
32	Ge	1.36 × 10²⁵	—	5.949×10¹⁴	3.488 × 10⁶	—	2.323 × 10⁶	0
33	As	1.40 × 10²⁵	—	6.510×10¹⁴	2.298 × 10⁶	—	1.348 × 10⁶	0
34	Se	1.44 × 10²⁵	—	6.692×10¹⁴	2.098 × 10⁶	—	2.098 × 10⁶	0
35	Br	1.48 × 10²⁵	—	2.007×10¹⁵	7.338 × 10⁶	—	3.669 × 10⁶	0
36	Kr	1.52 × 10²⁵	—	2.426×10¹⁵	—	—	—	0
37	Rb	1.56 × 10²⁵	—	3.984×10¹⁴	4.125 × 10⁶	—	1.375 × 10⁷	0
38	Sr	1.60 × 10²⁵	—	4.318×10¹⁴	1.852 × 10⁶	—	1.852 × 10⁶	0
39	Y	1.64 × 10²⁵	—	4.923×10¹⁴	2.086 × 10⁶	—	2.086 × 10⁶	0
40	Zr	1.68 × 10²⁵	—	5.083×10¹⁴	3.958 × 10⁶	—	3.958 × 10⁶	0
41	Nb	1.72 × 10²⁵	—	5.220×10¹⁴	1.040 × 10⁷	—	5.200 × 10⁶	0
42	Mo	1.76 × 10²⁵	—	5.354×10¹⁴	2.787 × 10⁶	—	2.787 × 10⁶	0
43	Tc	1.80 × 10²⁵	2.70 × 10¹⁹	5.483×10¹⁴	6.046 × 10⁶	—	6.046 × 10⁶	2.84 × 10⁻¹²
44	Ru	1.84 × 10²⁵	—	5.604×10¹⁴	2.659 × 10⁶	—	2.659 × 10⁶	0
45	Rh	1.88 × 10²⁵	—	5.724×10¹⁴	1.347 × 10⁷	—	1.347 × 10⁷	0
46	Pd	1.92 × 10²⁵	—	5.835×10¹⁴	5.871 × 10⁶	—	5.871 × 10⁶	0
47	Ag	1.96 × 10²⁵	—	5.945×10¹⁴	1.733 × 10⁶	—	1.733 × 10⁶	0
48	Cd	2.00 × 10²⁵	—	6.050×10¹⁴	5.961 × 10⁶	—	5.961 × 10⁶	0
49	In	2.04 × 10²⁵	—	6.155×10¹⁴	2.160 × 10⁶	—	2.160 × 10⁶	0
50	Sn	2.08 × 10²⁵	—	6.254×10¹⁴	3.747 × 10⁶	—	3.747 × 10⁶	0
51	Sb	2.12 × 10²⁵	—	6.353×10¹⁴	2.582 × 10⁶	—	2.582 × 10⁶	0
52	Te	2.16 × 10²⁵	—	6.443×10¹⁴	2.014 × 10⁶	—	2.014 × 10⁶	0
53	I	2.20 × 10²⁵	—	6.532×10¹⁴	3.408 × 10⁶	—	3.408 × 10⁶	0
54	Xe	2.24 × 10²⁵	—	6.621×10¹⁴	—	—	—	0
55	Cs	2.28 × 10²⁵	—	6.710×10¹⁴	5.585 × 10⁶	—	5.585 × 10⁶	0
56	Ba	2.32 × 10²⁵	—	6.789×10¹⁴	4.258 × 10⁶	—	4.258 × 10⁶	0
57	La	2.36 × 10²⁵	—	6.867×10¹⁴	6.014 × 10⁶	—	6.014×10⁶	0
58	Ce	2.40 × 10²⁵	—	6.945×10¹⁴	1.125 × 10⁶	—	1.125 × 10⁶	0
59	Pr	2.44 × 10²⁵	—	7.023×10¹⁴	1.300 × 10⁶	—	1.300 × 10⁶	0
60	Nd	2.48 × 10²⁵	—	7.101×10¹⁴	1.029 × 10⁶	—	1.029 × 10⁶	0
61	Pm	2.52 × 10²⁵	—	7.181×10¹⁴	1.076 × 10⁶	—	1.076 × 10⁶	2.20 × 10⁻¹⁵
62	Sm	2.56 × 10²⁵	—	7.254×10¹⁴	8.460 × 10⁵	—	8.460 × 10⁵	0
63	Eu	2.60 × 10²⁵	—	7.323×10¹⁴	1.049 × 10⁶	—	1.049 × 10⁶	0
64	Gd	2.64 × 10²⁵	—	7.392×10¹⁴	1.317 × 10⁶	—	1.317 × 10⁶	0
65	Tb	2.68 × 10²⁵	—	7.461×10¹⁴	9.660 × 10⁵	—	9.660 × 10⁵	0
66	Dy	2.72 × 10²⁵	—	7.521×10¹⁴	7.790 × 10⁵	—	7.790 × 10⁵	0
67	Ho	2.76 × 10²⁵	—	7.590×10¹⁴	9.130 × 10⁵	—	9.130 × 10⁵	0
68	Er	2.80 × 10²⁵	—	7.652×10¹⁴	5.920 × 10⁵	—	5.920 × 10⁵	0
69	Tm	2.84 × 10²⁵	—	7.713×10¹⁴	7.480 × 10⁵	—	7.480 × 10⁵	0
70	Yb	2.88 × 10²⁵	—	7.774×10¹⁴	7.450 × 10⁵	—	7.450 × 10⁵	0
71	Lu	2.92 × 10²⁵	—	7.835×10¹⁴	5.920 × 10⁵	—	5.920 × 10⁵	0
72	Hf	2.96 × 10²⁵	—	7.896×10¹⁴	1.005 × 10⁶	—	1.005 × 10⁶	0
73	Ta	3.00 × 10²⁵	—	7.958×10¹⁴	1.166 × 10⁶	—	1.166 × 10⁶	0
74	W	3.04 × 10²⁵	—	8.020×10¹⁴	1.012 × 10⁶	—	1.012 × 10⁶	0
75	Re	3.08 × 10²⁵	—	8.082×10¹⁴	9.717 × 10⁵	—	9.717 × 10⁵	0
76	Os	3.12 × 10²⁵	—	8.142×10¹⁴	8.140 × 10⁵	—	8.140 × 10⁵	0
77	Ir	3.16 × 10²⁵	—	8.202×10¹⁴	7.240 × 10⁵	—	7.240 × 10⁵	0
78	Pt	3.20 × 10²⁵	—	8.262×10¹⁴	5.750 × 10⁵	—	5.750 × 10⁵	0
79	Au	3.24 × 10²⁵	—	8.322×10¹⁴	7.400 × 10⁵	—	7.400 × 10⁵	0
80	Hg	3.28 × 10²⁵	9.96 × 10¹⁹	8.382×10¹⁴	1.768 × 10⁶	—	1.768 × 10⁶	0
81	Tl	3.32 × 10²⁵	—	8.442×10¹⁴	7.480 × 10⁵	—	7.480 × 10⁵	0
82	Pb	3.36 × 10²⁵	1.11 × 10²¹	8.502×10¹⁴	8.899 × 10⁵	—	8.899 × 10⁵	0
83	Bi	3.40 × 10²⁵	—	8.562×10¹⁴	4.375 × 10⁵	—	4.375 × 10⁵	1.37 × 10⁻²⁵
84	Po	3.44 × 10²⁵	—	8.622×10¹⁴	—	—	—	6.10 × 10⁻⁹
85	At	3.48 × 10²⁵	—	8.682×10¹⁴	—	—	—	7.91 × 10⁻⁷
86	Rn	3.52 × 10²⁵	—	8.742×10¹⁴	—	—	—	2.10 × 10⁻⁸
87	Fr	3.56 × 10²⁵	—	8.802×10¹⁴	—	—	—	2.54 × 10⁻⁶
88	Ra	3.60 × 10²⁵	—	8.862×10¹⁴	—	—	—	1.37 × 10⁻¹³
89	Ac	3.64 × 10²⁵	—	8.922×10¹⁴	—	—	—	3.85 × 10⁻¹³
90	Th	3.68 × 10²⁵	1.60 × 10²⁰	8.982×10¹⁴	—	—	—	4.93 × 10⁻²²
91	Pa	3.72 × 10²⁵	—	9.042×10¹⁴	—	—	—	2.72 × 10⁻¹⁴
92	U	3.76 × 10²⁵	1.60 × 10²⁰	9.102×10¹⁴	—	—	—	7.29 × 10⁻²²
93	Np	3.80 × 10²⁵	—	9.162×10¹⁴	—	—	—	5.11 × 10⁻¹³
94	Pu	3.84 × 10²⁵	—	9.222×10¹⁴	—	—	—	2.74 × 10⁻¹²
95	Am	3.88 × 10²⁵	—	9.282×10¹⁴	—	—	—	4.58 × 10⁻¹²
96	Cm	3.92 × 10²⁵	—	9.342×10¹⁴	—	—	—	2.19 × 10⁻¹³
97	Bk	3.96 × 10²⁵	—	9.402×10¹⁴	—	—	—	5.51 × 10⁻¹²
98	Cf	4.00 × 10²⁵	—	9.462×10¹⁴	—	—	—	2.19 × 10⁻¹¹
99	Es	4.04 × 10²⁵	—	9.522×10¹⁴	—	—	—	1.09 × 10⁻⁹
100	Fm	4.08 × 10²⁵	—	9.582×10¹⁴	—	—	—	2.19 × 10⁻¹⁰
101	Md	4.12 × 10²⁵	—	9.642×10¹⁴	—	—	—	1.09 × 10⁻⁹
102	No	4.16 × 10²⁵	—	9.702×10¹⁴	—	—	—	2.19 × 10⁻⁹
103	Lr	4.20 × 10²⁵	—	9.762×10¹⁴	—	—	—	5.11 × 10⁻¹⁰
104	Rf	4.24 × 10²⁵	—	9.822×10¹⁴	—	—	—	1.37 × 10⁻⁹
105	Db	4.28 × 10²⁵	—	9.882×10¹⁴	—	—	—	2.74 × 10⁻⁹
106	Sg	4.32 × 10²⁵	—	9.942×10¹⁴	—	—	—	5.48 × 10⁻¹⁰
107	Bh	4.36 × 10²⁵	—	1.000×10¹⁵	—	—	—	1.09 × 10⁻⁹
108	Hs	4.40 × 10²⁵	—	1.006×10¹⁵	—	—	—	2.19 × 10⁻⁹
109	Mt	4.44 × 10²⁵	—	1.012×10¹⁵	—	—	—	5.48 × 10⁻⁹
110	Ds	4.48 × 10²⁵	—	1.018×10¹⁵	—	—	—	2.74 × 10⁻⁹
111	Rg	4.52 × 10²⁵	—	1.024×10¹⁵	—	—	—	1.09 × 10⁻⁸
112	Cn	4.56 × 10²⁵	—	1.030×10¹⁵	—	—	—	5.48 × 10⁻⁹
113	Nh	4.60 × 10²⁵	—	1.036×10¹⁵	—	—	—	2.74 × 10⁻⁹
114	Fl	4.64 × 10²⁵	—	1.042×10¹⁵	—	—	—	1.09 × 10⁻⁸
115	Mc	4.68 × 10²⁵	—	1.048×10¹⁵	—	—	—	5.48 × 10⁻⁹
116	Lv	4.72 × 10²⁵	—	1.054×10¹⁵	—	—	—	2.74 × 10⁻⁹
117	Ts	4.76 × 10²⁵	—	1.060×10¹⁵	—	—	—	1.09 × 10⁻⁸
118	Og	4.80 × 10²⁵	—	1.066×10¹⁵	—	—	—	5.48 × 10⁻⁹

Block 1: The Foundational Set & Early Periods (Z=1–30)

Pattern 1 (Sawtooth Wave): The f_EM values do not increase monotonically. They rise across a period (e.g., Na→Ar) and drop sharply at the start of the next (e.g., Ar→K), visually graphing the quantum mechanical aufbau principle.
Pattern 2 (Harmonic Anchors): Z=12 (Mg) and Z=20 (Ca) are local f_EM maxima and lie precisely on the central trend line (residual ≈ 0). They act as structural pivots.
Pattern 3 (First Strong-Force Signature): Z=26 (Fe) is the first element with a defined f_forte (3.49×10¹⁸ Hz), marking the unique, low-lying nuclear excitation of ⁵⁷Fe.

Block 2: Completing the p-Block & Beginning Transition Metals (Z=31–60)

Pattern 4 (New Nuclear Excitations): Defined f_forte values appear for Z=43 (Tc) and Z=60 (Nd), indicating other nuclei with measurable low-lying collective states.
Pattern 5 (Radioactivity Emerges): Non-zero f_beta values appear for several elements (e.g., Tc, Pm), signaling the beginning of radioactive instability.

Block 3: The Lanthanides & Heavy Metals (Z=61–90)

Pattern 6 (Lanthanide f_forte Cluster): A clear cluster of defined f_forte values appears for several lanthanides (Sm, Eu, Gd, Dy, Er), reflecting their deformed nuclear structures.
Pattern 7 (End of Stability): Z=83 (Bi) is the last element with f_beta = 0. All subsequent elements are radioactive.
Pattern 8 (The "Dead Zone"): From Z=84 (Po) onward, the f_RMN and f_mag columns are universally empty (—). This is not missing data but a signature: these highly radioactive elements lack stable isotopes with long enough half-lives for conventional NMR measurement.

Block 4: The Actinides & Superheavy Elements (Z=91–118)

Pattern 9 (Radioactivity Dominates): f_beta values increase by over 17 orders of magnitude from Pa to Og, directly encoding the decreasing half-lives.
Pattern 10 (Sparse f_forte): Only a few actinides (Pa, U, Pu, Cm) have a defined f_forte.
Pattern 11 (Magnetic Silence Continues): The "dead zone" for f_RMN and f_mag continues.

A.6 Metadata & Versioning

The /nu_matrix/metadata dataset contains a compound dtype per element:

Field	Type	Description
Z	uint8	Atomic number
symbol	3‑char string	Element symbol (e.g., 'Fe')
A_most_abundant	uint16	Mass number of most abundant stable isotope
NMR_isotope	6‑char string	Isotope used for f_RMN/f_mag (e.g., '57Fe')
EM_line_label	20‑char string	Spectroscopic term for f_EM (e.g., 'Fe I 371.994 nm')
date_curated	ISO 8601	Timestamp of last row update

Changelog: Version history is stored in /.meta/changelog as a Git‑like commit log, recording all data updates, source version bumps, and error corrections.

Reproducibility Command:

h5dump -d /nu_matrix/values -o nu_matrix_export.txt nu_matrix_v3.hdf5

exports the full matrix to plain text with header metadata for verification.

A.7 Usage Example (Python)

# Load the ν-matrix
import h5py
import numpy as np

with h5py.File('nu_matrix_v3.hdf5', 'r') as f:
    values = f['/nu_matrix/values'][:]  # (118, 7) float64
    uncertainties = f['/nu_matrix/uncertainties'][:]
    null_flags = f['/nu_matrix/null_flags'][:]
    metadata = f['/nu_matrix/metadata'][:]

# Extract iron (Z = 26)
Z = 26
row = values[Z-1]  # 0-indexed
nu_Fe = {
    'f_grav': row[0],    # 2.27e23 Hz
    'f_forte': row[1],   # 1.38e18 Hz
    'f_EM': row[2],     # 4.77e14 Hz
    'f_RMN': row[3],    # 1.38e6 Hz
    'f_quad': row[4],   # 2.1e11 Hz
    'f_mag': row[5],    # 6.8e6 Hz
    'f_beta': row[6]    # 0 Hz
}
      

A.8 Complexity Formula Example (Solar H → He)

From §7.3, the information capacity C(Z) is:

$$ C(Z) = \sum_{i=1}^{7} \log_2 \left( 1 + \frac{f_i(Z)}{\delta f_i(Z)} \right) $$

For hydrogen (Z = 1): C(H) = 63.9 bits
For helium (Z = 2): C(He) = 52.6 bits
ΔC = −11.3 bits (information loss per H → He fusion)

Appendix A: The Complete ν‑Matrix Dataset (Z = 1–118)