Purpose: Official website release; systematic account of the model’s first-principles predictions for the dark energy equation-of-state parameter w, including the baseline w≈−1 and possible phantom (w<−1) origin, physical meaning, and relation to observational tests.

Abstract

In the Quantum Narrative Matrix (QNM) framework, cosmological-scale dynamics are driven by unitary evolution of finite-dimensional Hermitian matrices and derived observables (entanglement entropy, complexity). The dark energy equation-of-state parameter w (p = wρ) is not a fitted quantity but is derived from matrix eigenvalue structure and unitarity deviation under first principles. This document gives: (1) the origin of the theoretical baseline w = −1; (2) the theoretical mechanism for phantom (w < −1) in QNM—the correction from the fraction of eigenvalues with negative real part and unitarity deviation; (3) the adoption of the dimensionless coefficient 1/(8π) consistent with geometric/CFT corrections; (4) the testing relation between predictions and current observations (Planck, DESI, Pantheon+): comparison is for falsifiability, not for matching ΛCDM. The model formally adopts the zero-order native prediction for w(z): w≈−1 with no strong evolution in redshift, and discloses that if the theory outputs w<−1, that is a testable prediction to be confronted with observations.

I. Introduction

In Friedmann cosmology, the dark energy equation of state is pDE = w ρDE c², with w dimensionless. A cosmological constant Λ corresponds to w = −1. Current precision cosmology (Planck CMB, DESI BAO, Pantheon+ supernovae, etc.) constrains w very close to −1 and has not yet provided conclusive evidence for significant w(z) evolution with redshift. Phantom dark energy (w < −1) can imply instabilities in classical field theory but is widely discussed in effective field theory and quantum corrections; whether it is realized in nature is a testable physical question.

Stance of this framework: This model derives H(z), w(z) theoretically from first-principles / formal structure (matrix evolution, entanglement entropy S(t), complexity C(t)). We do not fit for the sake of fitting: comparison with observations is for testing and falsifiability, not for tuning to match standard cosmology. If the theory self-consistently gives w < −1, that result is a model prediction, disclosed as such and left to observational test; we do not treat phantom as a defect to be removed.

II. Theoretical Origin of w in the QNM Framework

2.1 Dynamical observables from matrix evolution

In the QNM implementation, the cosmic quantum state is described by an N×N Hermitian matrix evolving with cosmic time t via unitary evolution U(t)=e−iHt/ℏ. Observables are derived from matrix eigenvalues and state-dependent quantities (e.g. reduced entanglement entropy S(t), Fubini–Study complexity C(t)). The Hubble parameter H(z) is obtained from Heff = (1/3)Ċ/C via t↔z mapping and scale constant κ on the adopted complexity-driven path. The equation of state w is, in the default implementation, computed directly from matrix eigenvalue structure and unitarity deviation using first-principles formulae (§2.2–2.3). Thus w is a theoretical output, not a free fitting parameter.

2.2 Baseline w = −1 and unitary evolution

Under pure unitary evolution, Hamiltonian eigenvalues are conserved; the natural theoretical baseline for dark energy as vacuum energy is wbase = −1. This matches the cosmological constant Λ relation p = −ρ. The framework formally adopts this baseline as the public stance (the QNM constant-baseline model): predict w≈−1, consistent with current observations; we do not introduce phenomenological weff(z) to meet metrics.

2.3 First-principles mechanism for phantom (w < −1)

In the core evolution algorithm of QNM, w(t) is given by the following first-principles formulae:

(1) Unitarity deviation — Δuni = |tr(ρρ†) − (tr ρ)²|, where ρ is the density matrix at the current step. For pure-state unitary evolution this is zero; numerics or effective dissipation can make it non-zero. Physical picture: In a standard closed quantum system, evolution is strictly unitary. On cosmological scales, if the effective horizon (e.g. the Hubble horizon) exchanges information or entanglement with the “outside”, the system behaves as an open quantum system, giving rise to unitarity deviation (Δuni > 0). This inflow/outflow of information is physically equivalent to anomalous growth of dark energy density—the microscopic quantum origin of phantom energy in classical cosmology (w < −1, i.e. energy density increasing with expansion).

(2) Phantom eigenvalue fraction — fph = (number of eigenvalues with Re(λi) < 0) / N. In the dark-energy epoch, matrix construction introduces a first-principles coefficient 1/(8π) consistent with geometric/CFT corrections, so that some eigenvalues have negative real parts (“phantom energy” structure).

(3) Analytic form of w — Baseline wbase = −1. When fph > 1/(8π), the phantom correction is applied: w = wbase + Δwph, Δwph = −2 fph tanh(Δuni). Since Δwph ≤ 0, we have w ≤ −1; i.e. the theory naturally gives w<−1 (phantom) when the above conditions hold.

(4) Dimensionless threshold 1/(8π) — The algorithm uses αph = 1/(8π) as the threshold for phantom to be “significant”. This coefficient is not a freely tuned parameter: it corresponds deeply to the Einstein–Hilbert action coupling constant in general relativity (κ = 8πG) and to area-law geometric factors in black-hole thermodynamics and holographic entanglement entropy. Using it as the phantom phase-transition threshold marks the intrinsic unity of quantum matrix information dynamics with macroscopic gravitational geometry. The phantom correction is applied only when fph > αph; otherwise w = −1 is kept.

2.4 Consistency with H(z)

In the dark-energy epoch, phantom eigenvalues are also used in extracting the Hubble parameter H: H is multiplied by 1 + αph·(phantom count/N), same origin as the w phantom mechanism, ensuring a consistent dark-energy–expansion relation.

III. Summary of Predictions and Testability

ItemContentBaselinew ≈ −1, consistent with current observations; from vacuum/ground-state correspondence under unitary evolution.Phantom possibilityWhen fph > 1/(8π) and unitarity deviation is present, the theory predicts w < −1.Along trajectoryw(z) comes from per-step eigenvalue derivation; under unitary evolution it is approximately conserved, so w(z) is approximately constant in redshift (zero-order native prediction).CoefficientPhantom threshold and correction use 1/(8π), consistent with geometric/CFT corrections, not data fitting.

Relation to observational tests

We compare theory output w with current observational constraints (e.g. Planck, DESI, Pantheon+ on w) for falsifiability, not to match ΛCDM exactly. If the theory gives w<−1 while current data cluster near w≈−1, that difference is recorded as a theoretical prediction for future, more precise or high-redshift observations (e.g. Euclid). Tuning parameters, changing formulae, or downplaying phantom results to make curves “look like ΛCDM” is prohibited. Decisive future test: If next-generation dark energy surveys (e.g. the Euclid space telescope or higher-precision DESI data) statistically rule out strict w = −1 and find the Universe in the w < −1 phantom regime, that would constitute strong observational evidence for the QNM open-quantum-system / unitarity-deviation mechanism. Conversely, the model’s theoretical predictions remain open to stringent falsification by future data.

IV. Implementation and Disclosure

w(t) is computed at each step of the QNM core evolution algorithm from matrix eigenvalues and unitarity deviation using the formulae in §2.3; output w(z) is obtained by t↔z mapping. All reports state that “w(z) is derived from eigenvalues at each evolution step”. Any exploratory weff(z) is separately labeled as exploratory, not first-principles. We have formally adopted the zero-order native prediction for w(z) (w≈ constant with full disclosure); we do not claim observable w(z) evolution with redshift unless future observations warrant it and we adopt an extended scheme.

V. Conclusion

In the QNM framework, the dark energy equation of state w is a theoretically derived quantity: the baseline w=−1 comes from unitary evolution and vacuum-energy correspondence; phantom (w&lt;−1) arises naturally when the matrix has a sufficient fraction of eigenvalues with negative real part and unitarity deviation is present, with threshold and correction using 1/(8π) consistent with geometry/CFT. The model does not fit; both w≈−1 and possible w<−1 predictions are compared with observations for testing and falsifiability, in line with academic integrity and reproducible research.

References and technical support documentation

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