2. Core Formalism

2.1 The Fundamental Relation

2.1.1 Primitive (1)

The framework erects upon the experimentally unassailable primitive:

$$ E = h f $$

where E is a quantum of energy, h = 6.62607015 × 10⁻³⁴ J·s (CODATA 2022) is the Planck constant, and f is the associated frequency. Equation (1) holds across 12 orders of magnitude for photons and has been validated for matter waves to 10⁻⁶ precision via neutron interferometry (Rauch & Werner, Neutron Interferometry, 2015, Chap. 4).

2.1.2 Equation (2): f = E/h (Framework Primitive)

The methodological innovation is treating the inverse relation as the primitive symbolic act:

$$ f = \frac{E}{h} $$

This inversion performs a dimensional and conceptual shift: every quantitative description of a stable state — mass-energy, binding energy, excitation energy, or decay width — is translated into the common currency of Hertz.

Equation (2) is therefore the Rosetta stone of the ν‑framework: it converts domain‑specific quantities (eV, MeV, atomic units) into a single, additive, frequency‑space representation that can be tabulated, compared, and operated upon without loss of physical information.

Key Consequences:

• Loss-lessness: The mapping is deterministic and invertible: given f, one recovers E = f · h to within experimental uncertainty δE. No information is discarded.
• Logarithmic compression: Frequency turns multiplicative relationships into additive differences, making hierarchical structures (e.g., the 10¹¹ span from atomic to Planck scales) algebraically linear and visually accessible.
• Operational nature: The re‑centering is operational, not ontological. Equation (2) does not claim energy "is" frequency, but that frequency provides a loss-less, manipulable encoding. This distinction is critical: the framework is a coordinate system, not a new physical law.

2.1.3 Axiom 1: Scope and Domain of Applicability

Axiom 1 — Any measurable, characteristic energy E₀ associated with a stable, identifiable physical state (rest mass, nuclear transition, atomic line, decay width) shall be expressed as a characteristic frequency f₀ via Equation (2). This frequency becomes a component of that state's ν‑ or π‑vector.

Domain Boundary: "Stable" means τ½ > 10⁻⁸ s, the practical temporal resolution limit for tabulated nuclear and atomic data (e.g., ENSDF, NIST ASD). Shorter‑lived resonances are excluded because their characteristic frequencies would be broadened beyond measurability (Δf ≈ 1/Δt) and are not required for the taxonomy of persistent states.

Completeness: No stable property is excluded a priori. Mass-energy from gravity, excitation from the strong force, optical lines from electromagnetism, Larmor precession from magnetism, quadrupole deformation from shape, and beta‑decay from the weak force are all admissible. The axiom asserts that all resonant channels of a state can be encoded simultaneously, rendering the ν‑matrix informationally complete.

Uniqueness: Within experimental precision δE, the derived frequency f₀ = E₀/h is the sole admissible value for that component. No adjustable parameters, alternative conventions, or theoretical corrections beyond δE are permitted. This enforces a one‑to‑one mapping between measured energy and frequency space.

2.1.4 Dimensional Consistency: Why Hertz?

Choosing Hz as the universal unit is non‑trivial. It makes all components of ν‑vectors and π‑vectors dimensionally homogeneous even when the underlying physics is not: f_grav (derived from GeV‑scale mass) and f_RMN (derived from μeV‑scale Zeeman splitting) both appear as ordinary real numbers, enabling direct algebraic comparison, clustering, and distance metrics in ℝⁿ — an operation impossible in energy‑space.

The price is a vast dynamic range (10¹⁴ Hz to 10²⁵ Hz across the matrices), but that range is exactly where the hierarchy problem becomes visually manifest. Logarithmic scaling turns this range into a manageable coordinate axis where order‑of‑magnitude is the natural metric, and multiplicative relationships become additive differences.

This dimensional unification is the enabling step for the isomorphism proof (§4.5): without it, categorical separation (Δf ≈ 10²³ Hz) would be obscured by unit conversions and scaling conventions.

2.2 Composite State Vector ν

A stable composite state — operationally defined as an atomic nucleus in a defined ionization stage together with its electron cloud — is represented by an ordered 7‑tuple:

$$ \nu = (f_{grav}, f_{forte}, f_{EM}, f_{RMN}, f_{quad}, f_{mag}, f_{beta}) $$

The dimension d = 7 is finite and closed for all known stable states (Z = 1–118).

Each component f_i is a positive real number or exactly zero:

• f_i > 0: Measured, active resonant channel
• f_i = 0: Permitted channel that is energetically inactive (e.g., f_beta = 0 for stable isotopes)
• Null entries are prohibited: Structural absence is encoded by zero, not omission, preserving algebraic closure

The 118 × 7 ν‑matrix (App. A) empirically verifies this construction: no two rows are identical within combined 3σ uncertainties.

2.2.1 Axiom 2: Uniqueness of Signature

Axiom 2 elevates ν from data container to state‑space coordinate:

For any two distinct stable states s₁ ≠ s₂, the Mahalanobis distance must exceed the 3σ uncertainty ellipsoid:

$$ \left\| \overrightarrow{\nu}(s_1) - \overrightarrow{\nu}(s_2) \right\|_{\Sigma^{-1}} > 3 $$

This ensures experimentally resolvable uniqueness. Any future measurement violating (4) would require framework revision.

Extensibility and Component Admission Criteria

A new component (e.g., f_oct for octupole vibrations) may be appended only if it satisfies:

1. Measurability: Observable for >10% of elements with δf/f < 10%
2. Distinctness: Encodes a physically independent resonant mode (not a linear combination of existing components)
3. Empirical Utility: Reduces cosmic curve residual variance by >5% (§3.3.3)

Admission triggers a framework version increment (e.g., ν‑matrix v3 → v4) and retroactive updates of all existing vectors with zero entries for the new channel.

2.2.2 Link to Elementary π‑Vectors and Domain Consistency

While ν encodes macroscopic properties, its origin is hypothesized to be generative: composite states arise from the frequency‑algebraic combination of constituent elementary π‑vectors (see §5.3).

Per Axiom 1 (§2.1.3), the ν‑matrix includes isotopes with τ½ > 10⁻⁸ s, matching the practical resolution limit of ENSDF and NIST databases. This operational threshold ensures all tabulated frequencies are resolvable; shorter‑lived resonances are excluded from the taxonomy.

2.3 Component Definitions

Each component of the ν‑vector is derived from a specific measurable property via the primitive f = E/h (Eq. 2). Below are the formal definitions, sourcing protocols, and uncertainty propagation for all seven channels.

2.3.1 f_grav (Gravitational/Mass‑Energy Frequency)

Definition: The frequency corresponding to the total rest mass‑energy of the nucleus:

$$ f_{grav} = \frac{m_{nucleus} c^2}{h} $$

where the nuclear mass is obtained from the atomic mass evaluation by:

$$ m_{nucleus} = M_{atom} - Z \cdot m_e + E_B(e^-)/c^2 $$

M_atom is the atomic mass from AME2020 (NNDC), m_e is the electron mass, and E_B(e⁻) is the total electron binding energy (≈ 0.1% correction, calculated via Dirac‑Fock).

Physical Interpretation: This component represents the Compton frequency of the nucleus as a whole — the fastest internal "tick rate" associated with the state's existence, scaling linearly with its mass‑energy.

Derivation Source: Atomic mass data from the Atomic Mass Evaluation (AME2020) via the National Nuclear Data Center (NNDC), accessed 2024‑06‑01. Masses of short‑lived isotopes are taken from the most recent mass‑excess evaluations in ENSDF.

Uncertainty Propagation:

$$ \frac{\delta f_{grav}}{f_{grav}} = \sqrt{\left(\frac{\delta M}{M}\right)^2 + \left(\frac{\delta c}{c}\right)^2 + \left(\frac{\delta h}{h}\right)^2} $$

Since c and h are exact (CODATA 2022), the dominant term is δM/M, ranging from 10⁻⁷ for stable isotopes to 10⁻⁴ for short‑lived species (τ½ ≈ 10⁻⁸ s).

Null Assignment: Never null. For unbound isotopes (mass excess > 0), f_grav is set to 0 Hz (no stable state exists), but the row is retained in the matrix for completeness.

2.3.2 f_forte (Strong‑Force/Nuclear Excitation Frequency)

Definition: The frequency corresponding to the energy of the first collective excited state with natural spin‑parity, typically the lowest 2⁺ or 3⁻ vibration:

$$ f_{forte} = \frac{E(2_1^+)}{h} $$

where E(2₁⁺) is the energy (in joules) of the first 2⁺ state above the ground state, a direct signature of nuclear deformation under the strong force. For even‑even nuclei, this is often the quadrupole vibrational mode.

Physical Interpretation: f_forte quantifies the lowest‑energy resonant vibration of the nucleus itself. A defined value signals a deformed mid‑shell nucleus; a null entry marks structural rigidity (closed‑shell configuration).

Derivation Source: Nuclear energy‑level data from the Evaluated Nuclear Structure Data File (ENSDF) (NNDC, 2024‑06‑01). The ≤ 2 MeV threshold is applied to distinguish collective excitations from high‑lying single‑particle states, based on systematic studies showing that quadrupole collectivity emerges below this scale (Casten, Nuclear Structure, 2000).

Uncertainty: δf_forte/f_forte = δE/E from ENSDF level‑width data, ranging from 10⁻⁶ for narrow states to 10⁻³ for broad resonances.

Null Assignment: Null (—) if no discrete state exists ≤ 2 MeV or if the nucleus is spherical (closed‑shell). This occurs for magic nuclei (Z = 2, 8, 20, 28, 50, 82) and many even‑even isotopes, yielding the sparse pattern analyzed in §3.2.3.

2.3.3 f_EM (Electromagnetic/Optical Transition Frequency)

Definition: The frequency of the strongest allowed atomic emission line for the neutral atom in the visible or near‑visible spectrum:

$$ f_{EM} = \frac{c}{\lambda_{air}} $$

where c is the speed of light (exact) and λ_air is the wavelength in air at standard conditions (20 °C, 101.325 kPa). The air refractivity correction is applied via Ciddor's formula (NIST ASD spectrometer protocol v5.8), contributing < 3 × 10⁻⁶ relative uncertainty.

Physical Interpretation: This component identifies the element's dominant optical voice — its most characteristic resonance with the EM field in the optical regime, simplifying complex spectra to a single signature frequency.

Derivation Source: Spectral line data from the NIST Atomic Spectra Database (ASD) v5.8 (2023‑12‑15). For each element, the line with the highest intensity flag (Int) in 300 nm < λ < 900 nm is selected; VUV/IR lines are used only if no visible line exists (e.g., Kr I at 557 nm).

Uncertainty: δf_EM/f_EM = δλ/λ ≈ 3 × 10⁻⁸ (NIST Ritz wavelength uncertainty). The air correction adds negligible error (< 10⁻⁹).

Null Assignment: Never null for Z ≤ 118. For future superheavy elements (Z > 118) where optical data are absent, f_EM will be set to null until measured.

2.3.4 f_RMN (Nuclear Magnetic Resonance Frequency)

Definition: The Larmor precession frequency for the most abundant NMR‑active isotope in a reference magnetic field B₀ = 1 Tesla (exact):

$$ f_{RMN} = \frac{\gamma B_0}{2\pi} $$

where γ is the nuclear gyromagnetic ratio (rad·s⁻¹·T⁻¹) and B₀ = 1 T is a defined reference enabling comparison across elements.

Physical Interpretation: This quantifies the intrinsic resonant interaction between the nuclear magnetic moment and an external field, defining the "radio‑frequency note" for the nucleus.

Derivation Source: γ values from the compilation by Harris et al., Pure Appl. Chem. 80, 59 (2008), Tables 1–2, accessed 2024‑01‑10. Data are cross‑checked against NIST magnetic moment tables.

Uncertainty: δf_RMN/f_RMN = δγ/γ ≈ 2 × 10⁻⁴, dominated by systematic uncertainties in γ ratios.

Null Assignment: Null (—) if the element has no stable isotope with nuclear spin I ≠ 0. This renders 76 elements NMR‑silent, forming the "dead zone" (Z ≥ 84) analyzed in §3.2.3.

2.3.5 f_quad (Electric Quadrupole Deformation Frequency)

Definition: The frequency associated with the interaction energy of the nuclear electric quadrupole moment Q with an electric field gradient V_zz:

$$ f_{quad} = \frac{e \cdot Q \langle V_{zz} \rangle}{h \cdot 4I(2I - 1)} $$

where:

• e = 1.602 × 10⁻¹⁹ C (elementary charge)
• Q is the nuclear electric quadrupole moment in barns (1 b = 10⁻²⁸ m²)
• ⟨V_zz⟩ is the mean electric field gradient in the measurement environment (≈ 10²¹ V/m² for solid‑state hosts)
• I is the nuclear spin quantum number

Physical Interpretation: f_quad quantifies the static "shape deformation" of a non‑spherical nucleus. A non‑zero value (typically 10⁸‑10⁹ Hz) indicates prolate/oblate deformation complementary to the dynamic vibration probed by f_forte.

Derivation Source: Quadrupole moments Q from ENSDF hyperfine‑structure evaluations and specialized measurements via Perturbed Angular Correlation (PAC) or quadrupole‑resonance NMR. ⟨V_zz⟩ is estimated from DFT calculations for the host crystal (e.g., α‑Fe, TiO₂).

Uncertainty: δf_quad/f_quad ≈ 10%, dominated by systematic uncertainty in ⟨V_zz⟩ (±30% typical) rather than δQ.

Null Assignment: Null (—) if Q = 0 (spherical nucleus: I = 0, I = ½, or closed‑shell configuration). Occurs for 62% of elements.

2.3.6 f_mag (Magnetic Moment Interaction Frequency)

Definition: The frequency associated with the interaction energy of the nuclear magnetic moment μ with the reference field B₀ = 1 T:

$$ f_{mag} = \frac{\mu \cdot B_0}{I \cdot \hbar} $$

where μ is the nuclear magnetic moment in nuclear magnetons (μ_N) and I is the spin quantum number. This captures the static magnetic "strength", distinct from the Larmor precession rate (f_RMN).

Physical Interpretation: f_mag quantifies the intrinsic magnetic "strength" of the nucleus at a standard field. For I = ½ nuclei, f_mag and f_RMN are proportional; for I > ½, they differ due to the non‑linear μ‑γ relationship.

Derivation Source: Magnetic moments μ from the same compilations as γ (Harris et al., 2008, Table 2).

Uncertainty: δf_mag/f_mag = δμ/μ ≈ 2 × 10⁻⁴ (systematic from μ fits).

Null Assignment: Null (—) under the same rule as f_RMN (no I ≠ 0 isotope).

2.3.7 f_beta (Weak‑Force Beta‑Decay Frequency)

Definition: The frequency associated with the total decay width Γ_β of the ground‑state beta‑decay process for the longest‑lived isotope of the element:

$$ f_{beta} = \frac{\Gamma_{\beta}}{h} = \frac{\hbar}{\tau_{1/2} \cdot h \cdot \ln 2} $$

which yields the practical numerical form:

$$ \frac{f_{beta}}{\text{Hz}} = \frac{1.519267 \times 10^{11}}{\tau_{1/2} (\text{fs})} $$

where τ½ is the partial half‑life in femtoseconds.

Physical Interpretation: f_beta quantifies the rate of transmutation via the weak nuclear force — the "tick" of quantum instability. It is the only component defined for all 118 elements: f_beta = 0 Hz for stable isotopes, > 0 Hz for radioactive ones.

Derivation Source: Beta‑decay half‑lives from ENSDF decay‑data evaluations (NNDC, 2024‑06‑01). For stable isotopes (τ½ > 10¹⁸ yr), f_beta ≡ 0 Hz.

Uncertainty: δf_beta/f_beta = δτ½/τ½, ranging from 10⁻⁵ for long‑lived isotopes (e.g., ⁴⁰K) to 10⁻² for prompt decays (e.g., ¹³⁸Sn).

Null Assignment: Never null. Zero indicates stability; finite values encode decay rates. This makes f_beta a structural clock for the periodic table.

2.4 Elementary State Vector π

2.4.1 π_grav (Elementary Mass‑Energy Frequency)

Definition:

The gravitational component of the elementary state vector is the particle's Compton frequency, derived from its invariant rest mass:

$$ \pi_{grav} \equiv \frac{m_{particle} \cdot c^2}{h} $$

where m_particle is the pole mass (for quarks) or the directly measured mass (for leptons and bosons), c = 299792458 m·s⁻¹ (exact), and h = 6.62607015 × 10⁻³⁴ J·s (exact). This frequency represents the existential tick rate of the particle — the fastest internal vibration associated with its mass‑energy.

Physical Interpretation:

π_grav spans five orders of magnitude across the Standard Model (SM), from the electron (1.235 × 10²⁰ Hz) to the top quark (4.18 × 10²⁵ Hz). This range embeds the SM mass hierarchy problem geometrically: multiplicative mass ratios become additive intervals in log‑frequency space, reframing the hierarchy as a coordinate effect rather than a dimensionless tuning puzzle (see §5.2).

Key Features & Special Cases

• Massless gauge bosons. Photons (γ) and gluons (g) have π_grav ≡ 0 exactly, as required by gauge invariance.
• Neutrinos. Active neutrinos carry a one‑sided upper bound: π_grav(ν) < 2.9 × 10¹³ Hz (95% CL) from Planck 2018 + BAO cosmology, or the more conservative laboratory bound π_grav(ν) < 1.9 × 10¹⁴ Hz from KATRIN tritium endpoint data.
• Generation scaling (charged leptons). π_grav(e) : π_grav(μ) : π_grav(τ) = 1 : 206.768 : 3477.15, which is linear in log₁₀(π_grav) space: log₁₀(π_grav) = 20.091 + 2.004·g (where g = 1, 2, 3), R² = 0.999997.

Derivation Source & Uncertainty Propagation

• PDG 2024 pole masses for quarks (Quark Masses review, arXiv:2310.17021).
• Penning‑trap masses for charged leptons (CODATA 2022).
• Cosmological + laboratory bounds for neutrinos (Planck 2018, KATRIN 2022).
• Direct mass measurements for gauge and scalar bosons (PDG 2024).

Uncertainty: δπ_grav/π_grav = δm/m (dominant).

• Quarks: ≈ 0.5% (systematic from lattice QCD pole‑mass extraction).
• Charged leptons: < 10⁻⁹% (sub‑ppb Penning‑trap precision).
• Neutrinos: one‑sided, upper‑limit only.
• Bosons: 10⁻⁴% – 10⁻⁵% (LHC electroweak precision).

2.4.2 π_EM (Elementary Electromagnetic Charge Frequency)

Definition:

The electromagnetic component of the elementary state vector encodes the magnitude and sign of electric charge as a complex frequency‑scale:

$$ \pi_{EM} \equiv \alpha \cdot Q^2 \cdot e^{(i\phi_{EM})} $$

where α = 7.297352569 × 10⁻³ is the fine‑structure constant (CODATA 2022), Q is the electric charge in units of the elementary charge e (exact for SM particles), and the phase φ_EM is defined as:

• φ_EM = 0 for Q > 0 (positive charge)
• φ_EM = π for Q < 0 (negative charge)
• Undefined for Q = 0 (neutral particles), with π_EM ≡ 0.

The magnitude |π_EM| = α · Q² quantifies the squared coupling strength to the electromagnetic field, independent of mass scale.

Physical Interpretation:

π_EM serves as the electromagnetic fingerprint of an elementary particle, distinguishing charged from neutral states and encoding charge conjugation symmetry geometrically. Unlike a dimensionless coupling, its units of frequency enable direct algebraic comparison with other π‑components, a feature essential for the isomorphism proof (§4.5).

2.4.3 π_strong (Elementary Strong Charge Frequency)

Definition:

The strong‑interaction component of the elementary state vector encodes colour‑charge participation as a binary frequency‑scale:

$$ \pi_{strong} \equiv \begin{cases} \Lambda_{QCD}/h & \text{for particles carrying colour charge} \\ 0 & \text{for colour-singlets} \end{cases} $$

where Λ_QCD = 200 ± 10 MeV is the low‑energy QCD confinement scale from lattice evaluations (PDG 2024), and h is the Planck constant.

This component is categorically, not continuously, valued: a particle either couples to the strong force (π_strong ≈ 2.5 × 10²³ Hz) or does not (π_strong = 0). There is no continuum of strong couplings in the SM.

Physical Interpretation:

π_strong serves as the strong‑interaction flag — the most powerful discriminator in π‑space. It creates a near‑perfect segregation between coloured particles (six quark flavors, eight gluons) and the colourless sector (leptons, electroweak bosons, Higgs).

2.4.4 π_weak (Elementary Weak Charge Frequency)

Definition:

The weak‑interaction component encodes participation in the weak force via the Fermi coupling:

$$ \pi_{weak} = \frac{G_F \cdot E_{weak}^2}{\hbar \cdot h} $$

where G_F = 1.1663787 × 10⁻⁵ GeV⁻² is the Fermi constant (muon decay), and E_weak = 174.104 ± 0.083 GeV is the Higgs vacuum expectation value (vev) from electroweak precision data. This yields:

• Left-handed SM fermions: π_weak ≈ +2.915 × 10⁷ Hz (T₃ = −½) or −2.915 × 10⁷ Hz (T₃ = +½, antiparticles)
• Massive weak bosons (W±, Z⁰): π_weak ≈ ±1.166 × 10⁸ Hz (maximal weak charge)
• Right-handed fermions: π_weak = 0 (sterile)

The sign encodes chirality and weak isospin T₃; the magnitude quantifies the universal weak coupling below the electroweak scale.

2.4.5 π_spin (Elementary Magnetic Frequency)

Definition:

The magnetic‑dipole component of the elementary state vector encodes the intrinsic magnetic moment interaction for particles with non‑zero spin, evaluated in a reference magnetic field B₀ = 1 Tesla (exact):

$$ \pi_{spin} = \frac{g \cdot \left( \frac{e\hbar}{2m} \right) \cdot B_0}{h} $$

where μ = g·(eħ/2m) is the magnetic moment in J·T⁻¹, g is the g‑factor (including full QED loop corrections), e is the elementary charge, m is the particle mass, and ħ = h/2π. This frequency represents the Larmor precession rate of the particle's magnetic dipole in the standard field B₀, distinct from the nuclear resonance frequency f_RMN (§2.3.4) which is defined per unit field (γ/2π).

Physical Interpretation:

π_spin quantifies the magnetic "strength" at the elementary level — the anomalous magnetic moment converted into Hz. For the electron, the g‑factor anomaly Δg = g − 2 ≈ 0.002319 translates directly into a measurable frequency shift Δπ_spin ≈ 1.76 × 10⁴ Hz, detectable via Penning‑trap spectroscopy.

2.5 Symmetry Transformations: C, P, T

Status. The π‑schema must commute exactly with the discrete symmetries of the Standard Model: charge conjugation (C), parity (P), and time reversal (T). Any failure would indicate that π‑space introduces spurious symmetry violations beyond SM physics, invalidating the isomorphism claim. This section derives the transformation matrices from the SM's gauge structure and Wigner's anti‑unitary theorem, then validates them against PDG 2024 data.

2.5.1 Transformation Rules (Table 2.1)

The action of C, P, and T on the five‑component π‑vector is summarized below. All transformations are linear operators acting in ℂ⁵.

Table 2.1. Transformation of π‑vector components under C, P, T

Notes.

• Phases are real (0 or π) for SM particles; complex conjugation is trivial.
• J is total angular momentum quantum number (0 for scalars, ½ for spin‑½ fermions, 1 for vectors). For SM fermions (J = ½), T flips π_weak sign; for bosons (J = 0, 1), T leaves π_weak invariant.
• Colour structure: For off‑diagonal gluons (r ḡ, etc.), C permutes colour labels in metadata; diagonal gluons (r r̄ − b b̄) are strictly invariant.

2.5.2 Matrix Representations

The discrete symmetries act as 5 × 5 matrices on the π‑vector space. Let π = (π_grav, π_EM, π_strong, π_weak, π_spin). Then:

Charge Conjugation (C)

$$ C\left( \overrightarrow{\pi} \right) = M_C \cdot \overrightarrow{\pi}, \quad \text{where } M_C = \text{diag}(1, -1, 1, -1, -1) $$

The −1 entries act on the complex phase of π_EM and the real sign of π_weak and π_spin. Colour permutation for quarks/gluons is handled in metadata, not in M_C.

Parity (P)

$$ P(\overrightarrow{\pi}) = M_P \cdot \overrightarrow{\pi}, \quad \text{where } M_P = \text{diag}(1, 1, 1, 1, -1) $$

Only π_spin (axial vector) changes sign under spatial inversion; all gauge charges (π_EM, π_strong, π_weak) are parity‑even.

Time Reversal (T)

$$ T(\overrightarrow{\pi}) = M_T \cdot \overrightarrow{\pi}, \quad \text{where } M_T = \text{diag}(1, -1, -1, -1, 1) $$

The anti‑unitary operator conjugates complex phases and applies a spin‑dependent phase e^(iπ·J) to π_weak.

For SM particles:

• Fermions (J = ½): T(π_weak) = −π_weak*
• Bosons (J = 0, 1): T(π_weak) = π_weak*

Since π_weak is real, this reduces to sign flip for fermions only, matching the SM's T‑transformation structure.