EHestablished
Einstein–Hilbert
S_EH = (1/16πG) ∫ R √−g. Standard GR curvature, plus GHY on ∂M.
Illustrative prototype. Explore toy models and save demo settings. Browser results cannot authorize controlled execution or support research claims. View repository gates · Open notebook
Fully variational field system
Diffeomorphism-invariant field theory on (M, g) with scalar φ, thermodynamic map Y^A : M → 𝒯, and two-form B. ω is closed and cohomologically nontrivial — it is not d(dℳ), because d² = 0.
SRTCH = SEH + Sφ + SY + SB + Stop + Sm + SGHY
λ_Q
1.00
topological source
β_Q
0.35
local coupling
α_Q
0.350
β_Q / Λ_Q⁴
⟨A⟩
1.000
matter conformal factor
F_Q on a two-cycle · 𝒥_Q heatmap
Q ≈ 0.00 · ⟨𝒥_Q⟩ 0.000
φestablished
Scalar
S_φ = −∫ [½ (∇φ)² + V(φ)] √−g. Sourced by α_φ T_m whenever A depends on φ.
Yestablished
Thermodynamic sigma model
S_Y = −∫ [½ G_AB ∂Y^A ∂Y^B + U(Y)] √−g. Target Christoffel Γ^A_BC enter the EL equation.
Bestablished
Two-form kinetic
S_B = −½ ∫ H ∧ *H with H = dB. Gauge B → B + dΛ leaves H invariant.
topconjectural
Topological coupling
S_top = λ_Q ∫ B ∧ F_Q. Metric-independent: no direct T_μν, yet it sources B and Y.
mconjectural
Matter
S_m[Ψ_m, Ã = A²(φ, ℳ, 𝒥_Q) g]. Conserved w.r.t. the physical metric, not the Einstein frame.
Parameters
λ_Q1.00
β_Q0.35
Λ_Q1.00
winding n1
winding m1
Topology on
Conformal A
Euler–Lagrange
Einstein
G_μν = 8πG (Tᵐ + Tᵠ + Tʸ + Tᴮ + T^{Q,m})_{μν}
Scalar
□φ − V'(φ) + α_φ T_m = 0
Two-form
d * H + λ_Q F_Q = 0
Bianchi
dH = 0, dF_Q = 0
Exchange
∇_μ T_m^{μν} = T_m ∇^ν ln A