Working technical paper v0.1

Viktor Stefanov Pronchev — independent research working draft
For discussion within the BSM-SG/QFG research program

20 July 2026

Abstract

Q-Field Geometry (QFG) models matter as stable field-geometric and topological configurations of a complex two-component field, a U(1) gauge sector, and an internal orientation field. Gravity from Entropy (GfE), developed independently, models gravity as emerging from quantum geometric relative entropy between the physical spacetime metric and a matter-and-curvature-induced metric. This paper constructs a conservative bridge between these approaches. We retain the minimal QFG field Lagrangian and add a relative-entropy sector that compares the physical metric with a metric induced by the QFG state. The resulting action is proposed as an effective thermodynamic extension, not as a completed fundamental theory. We define a local entropy density, an effective first-law relation, a topological work term, and conditions under which general relativity is recovered in the weak-field and low-curvature limit. The framework produces testable targets in high-Q resonators, topological soliton simulations, and cosmology. Its central hypothesis is that local coherent structures can emerge as low-entropy-density QFG nodes while total entropy continues to increase globally.

1. Motivation and scope

The emergence of galaxies, stars, atoms, molecules, and living systems appears at first sight to conflict with the second law of thermodynamics. The conflict is only apparent: local order can increase when entropy is exported or when the accessible volume grows sufficiently rapidly. The 2026 thermodynamic analysis of Gravity from Entropy sharpened this statement by identifying a regime in which total entropy grows while entropy per unit physical volume decreases. In parallel, QFG has been developed as a field-first framework in which localized matter corresponds to stable coherent or topological configurations rather than point particles.

The purpose of this paper is not to identify QFG with GfE. Their starting ontologies differ. The purpose is to construct the smallest mathematically explicit extension in which the informational tension used by GfE can become a thermodynamic coarse-graining of the underlying QFG field state.

2. Conceptual comparison: GfE, QFG, and BSM-SG

The proposed hierarchy is therefore:

BSM-SG microscopic substrate  →  QFG field dynamics  →  entropic emergent geometry.

3. Minimal QFG field sector

We retain the minimal QFG field content already used in the QFG program:

·        Complex two-component field Ψ ∈ C².

·        Vacuum or field density ρ = Ψ†Ψ.

·        U(1) gauge field Aμ with Fμν = ∂μAν − ∂νAμ.

·        Orientation field n = (Ψ†σΨ)/(Ψ†Ψ) ∈ S².

·        Potential minimum at a background density ρ0.

·        Topological stiffness of the orientation texture.

L_QFG = 1/2 |D_μΨ|² − 1/4 Z(ρ)F_μνF^μν − κ/4 (∂_μn × ∂_νn)² − λ(ρ − ρ₀)² − U(n).

Here Dμ = ∇μ + iqAμ. The field equations follow from δS/δΨ = 0, δS/δAμ = 0, and the constrained variation of n. The stress-energy tensor is defined conventionally by metric variation:

T_μν^QFG = −(2/√−g) δ(√−g L_QFG)/δg^μν.

This field sector supplies localized energy, orientation texture, gauge response, and topological charge. It does not yet specify a complete microscopic derivation of the spacetime metric.

4. QFG-induced metric

We introduce an effective metric induced by the local QFG state. The lowest-order covariant ansatz is

g̃_μν^Q = g_μν + a₁ T_μν^QFG/M_*⁴ + a₂ ∇_μρ ∇_νρ/M_*⁶ + a₃ X_μν(n)/M_*⁴ + a₄ J_μJ_ν/M_*⁶ + …

where M* is the coarse-graining scale, Jμ is the U(1) current, and Xμν(n) is an orientation-texture tensor, for example

X_μν(n) = ∂_μn · ∂_νn − 1/2 g_μν(∂_αn · ∂^αn).

The coefficients ai must be calibrated or bounded experimentally. This induced metric is not introduced as a second independently propagating spacetime. It is the geometry preferred by the local QFG state after coarse-graining.

5. Making relative entropy mathematically admissible

A Lorentzian metric is not a density matrix and cannot be inserted naively into the quantum relative-entropy formula. A local positive geometric operator must first be constructed. We therefore select a timelike observer uμ and a spatial tetrad on each local hypersurface. The physical and QFG-induced spatial metrics define positive normalized operators G and GQ:

G = γ/Tr(γ),    G_Q = γ̃_Q/Tr(γ̃_Q),    G > 0, G_Q > 0, Tr(G)=Tr(G_Q)=1.

The local QFG geometric relative entropy is then defined as

D_Q(G || G_Q) = Tr[G(log G − log G_Q)] ≥ 0.

This observer-dependent local construction is sufficient for an effective thermodynamic theory. A fully covariant operator definition is left as an open mathematical problem.

6. Proposed extended action

The thermodynamic QFG action is proposed as

S_QFG-T = ∫ d⁴x √−g [ M_P²R/2 + L_QFG − ξ M_*⁴ D_Q(G || G_Q) + L_m,ext ].

The terms have distinct roles:

·        Einstein-Hilbert term: establishes the conventional low-energy gravitational baseline.

·        QFG field term: produces gauge, density, orientation, resonance, and topological dynamics.

·        Relative-entropy term: penalizes mismatch between physical spatial geometry and geometry preferred by the QFG state.

·        External matter term: allows conventional fields not yet represented by Ψ.

The entropy coupling ξ is dimensionless in this normalization. Setting ξ = 0 returns the minimal QFG-plus-GR effective theory. The extended theory is therefore nested and falsifiable.

7. Field equations and weak-limit behavior

Variation with respect to the physical metric gives schematically

M_P² G_μν = T_μν^QFG + T_μν^ext + T_μν^ent,

where

T_μν^ent = −2ξM_*⁴/√−g · δ[√−g D_Q(G||G_Q)]/δg^μν.

Variation with respect to Ψ, Aμ, and n produces the original QFG equations plus backreaction terms from the induced metric and entropy functional.

For a small mismatch ΔG = G − GQ, relative entropy has a quadratic expansion:

D_Q(G||G_Q) = 1/2 ⟨ΔG, K_GQ⁻¹ ΔG⟩ + O(ΔG³).

Consequently, when curvature, QFG gradients, and metric mismatch are small, the entropy stress is second order and the equations reduce continuously to general relativity with the ordinary QFG stress-energy tensor. This is the required low-energy consistency condition.

8. Thermodynamic dictionary

On a foliation with local proper volume element dV = √γ d³x, define

s_Q = ξ M_*³ D_Q(G||G_Q),

u_Q = T_μν^QFG u^μu^ν + u_ent,

S_Q(V) = ∫_V s_Q dV.

A local effective first law is proposed in differential form:

du_Q = T_eff ds_Q − p_eff d(ln√γ) + μ_H dq_H + dW_drive.

Here qH is a topological charge such as a Hopf invariant, μH is its conjugate topological chemical potential, and dWdrive denotes externally injected RF, optical, mechanical, or other coherent work. For a stationary isolated node, dWdrive = 0.

9. Local order and the second law

The thermodynamic scenario of interest is

dS_total/dt ≥ 0,    while    d(S_Q/V)/dt < 0 in a growing or entropy-exporting domain.

In QFG language, a stable node corresponds to a local reduction in accessible geometric configurations, accompanied by energy and entropy transfer to the environment or by growth of the containing physical volume. A coherence boundary can therefore form without violating the second law.

The proposed sequence is

global phase-space growth → local reduction of geometric uncertainty → coherence boundary → stable topological node.

This is a thermodynamic reinterpretation of the QFG-Atlas picture in which a central field node is surrounded by a finite coherence region that relaxes into the background density ρ0.

10. Cosmological reduction

For a spatially flat Friedmann-Robertson-Walker metric,

ds² = −dt² + a(t)² d x²,

a homogeneous QFG background Ψ(t) induces an isotropic spatial operator GQ(t). The entropy sector contributes an effective density and pressure:

3M_P²H² = ρ_m + ρ_QFG + ρ_ent,

−2M_P²Ḣ = ρ_m+p_m + ρ_QFG+p_QFG + ρ_ent+p_ent.

The effective equation-of-state parameter went(z) = pent/ρent is not assumed. It is calculated after specifying the induced metric coefficients and solving the QFG background. A dynamical dark-energy-like term is therefore possible, but not guaranteed.

11. Falsifiable predictions

12. Numerical program

A staged numerical implementation is recommended:

1.        Level 1: One-dimensional and axisymmetric QFG node with fixed flat metric; calculate energy, qH proxy, and coherence boundary.

2.        Level 2: Construct GQ from Tμν, ∇ρ, and Xμν(n); calculate DQ on spatial slices.

3.        Level 3: Minimize total energy plus entropy penalty and compare topological sectors.

4.        Level 4: Couple weak metric perturbations hμν and verify recovery of the Newtonian/GR limit.

5.        Level 5: Homogeneous FRW reduction and calculation of ρent, pent, went, and entropy density.

6.        Level 6: Bayesian calibration against cavity, atomic, or cosmological datasets with strict out-of-sample tests.

13. Relation to BSM-SG

The proposed thermodynamic extension is compatible with, but does not mathematically derive, the BSM-SG microscopic ontology. BSM-SG contributes three useful interpretive elements:

·        Physical vacuum as a structured Cosmic Lattice rather than an empty background.

·        Stable structures possessing finite energy wells and balances between internal vibrational and SG interaction energies.

·        Synchronization and modulation of surrounding lattice states by ordered internal structures.

A future derivation would need to coarse-grain explicit BSM-SG lattice degrees of freedom into the QFG fields and then derive the induced metric and entropy functional. Until that derivation exists, BSM-SG should be presented as the candidate microscopic interpretation, QFG as the effective field model, and QFG-T as the proposed thermodynamic/gravitational extension.

14. Limitations and open problems

·        The induced metric ansatz is not unique and introduces coefficients requiring calibration.

·        The local relative entropy currently depends on a foliation or observer choice; a fully covariant positive-operator construction is unresolved.

·        The entropy term may introduce higher derivatives or stability problems; ghost and hyperbolicity analyses are mandatory.

·        No derivation from BSM-SG microscopic degrees of freedom has yet been completed.

·        No experiment currently isolates a QFG entropy contribution from established electromagnetic, thermal, mechanical, quantum-optical, or gravitational effects.

·        The cosmological dark-energy behavior is a possibility, not a result, until the reduced equations are solved.

·        The theory must outperform simpler effective-field or modified-gravity explanations under model-selection penalties.

15. Conclusion

This paper proposes a minimal thermodynamic extension of Q-Field Geometry. The extension retains the established QFG field and topological sector, constructs a local metric preferred by the QFG state, and penalizes its informational mismatch with the physical geometry through a positive relative-entropy functional. In the weak-mismatch limit, general relativity is recovered. Outside that limit, the entropy sector may contribute effective pressure, internal energy, topological work, and potentially a dynamical cosmological component.

The central scientific claim is intentionally limited: the framework provides a coherent and falsifiable mathematical bridge between a field-topology theory of matter and an information-thermodynamic theory of emergent gravity. It does not yet establish that nature uses this bridge. The next decisive step is numerical implementation followed by controlled resonator and metrology tests.

Field energy + topological stress + geometric relative entropy → effective gravitational dynamics.

References

[1] Bianconi, G.. Thermodynamics of the gravity from entropy theory. Physical Review D (2026). DOI: 10.1103/26kn-thgp.

[2] Queen Mary University of London / Phys.org. How Gravity from Entropy theory connects the second law of thermodynamics with the emergence of cosmic structure, 17 July 2026.

[3] Bianconi, G.. Gravity from entropy: an information-theoretic approach to gravity (foundational GfE work).

[4] Pronchev, V. S.. QFGv2: Minimal QFG Lagrangian, resonator tests, and measurement program. Working document, 2026.

[5] QFG research program. QFG 2: field, gauge, topological, molecular, and experimental formalism. Working manuscript, 2026.

[6] QFG research program. QFG Atlas: field-geometric interpretations of atomic coherence structures. Working atlas, 2026.

[7] Sargoytchev, S.. Basic Structures of Matter - Supergravitation Unified Theory, third edition, 2025.

Online source: Phys.org article