The Mekšriūnas Geometric Cancer Model: A Formal Mathematical Framework

This deposit presents the complete formal mathematical framework for the geometric cancer theory originally proposed by Mekšriūnas. The model describes tumour–immune dynamics through a minimal dynamical core — one spatially distributed PDE governing tumour activity and one global ODE governing immune activation — modulated by four mechanism layers: hysteretic encoding, antigenic holonomy, immune criticality, and least-action invasion. The framework classifies behaviour into healthy, dormant, and malignant regimes via invariant sets and attractor structure, and formalises the malignant phase transition as a fold bifurcation with basin switching and irreversibility. Four coherence constraints — monotonicity, boundedness, minimality, and structural invariance — are stated as provable mathematical conditions. The mathematical formalisation presented here is the work of Mekšriūnas. This deposit serves to give that work the formal platform it deserves, rendering the original conceptual theory into rigorous notation with full existence constraints, well-posedness results, and cross-referenced internal structure. Keywords: geometric cancer model, tumour–immune dynamics, fold bifurcation, immune criticality, hysteretic encoding, antigenic holonomy, least-action invasion, malignant phase transition ============================================================FULL MEKŠRIŪNAS SYSTEM — FORMAL EQUATION ============================================================ ------------------------------------------------------------TUMOUR ACTIVITY PDE------------------------------------------------------------ \[∂ T/∂ t= ∇ · ( Deff(G,H)\, ∇ T )+ ∇ · ( T\, M⁻¹(x,G)\, ∇ P(x,G) )+ reff(x,H)\, T ( 1 - {T}{Keff(G,H)} )- μ_0\, I(t)\, R[γ](t)\, T\] ------------------------------------------------------------IMMUNE ACTIVATION ODE------------------------------------------------------------ \[dI/dt= α_I I(1 - {I}{Iₘₐₓ})+ Φ(B(t))- Ψ(I, B(t))\] \[B(t) = ∫Ω T(t,x)\, dx\] \[Φ(B) = φ_0 {B^{nHill}}{B^{nHill} + B₅₀^{nHill}}\] \[Ψ(I,B) = ψ_0\, I\, Θ_s\!(I - I_c(B))\] ------------------------------------------------------------ANTIGENIC DRIFT ODE------------------------------------------------------------ \[dγ/dt= A\, γ + b\, I(t)\, B(t)\] ------------------------------------------------------------MECHANICAL HISTORY (CONVOLUTION)------------------------------------------------------------ \[H(t,x)= ∫_0^t K(t-s)\, Σ(s,x)\, ds\] \[K(τ)= K∞ + (K0,mem - K∞)\, e-τ/θ_H\] ------------------------------------------------------------MECHANICAL STRESS SOURCE------------------------------------------------------------ \[Σ(t,x)= σ_0\, |∇ T(t,x)|^2+ σ_1\, |κ(x)|\, T(t,x)\] ------------------------------------------------------------RECOGNITION EFFICIENCY (HOLONOMY)------------------------------------------------------------ \[Mₘᵢₛ(t)= ∫_0^t (Aform)_i(γ(s))\, dγ_i/ds\, ds\] \[R[γ](t)= exp\!( -\| Mₘᵢₛ(t) \|^2W )\] ------------------------------------------------------------LEAST-ACTION INVASION — GEODESIC EQUATION------------------------------------------------------------ Action functional: \[S[γᵢₙᵥ]= ∫_0^1 L\!(γᵢₙᵥ(τ),{dγᵢₙᵥ}{dτ}, G) dτ\] Lagrangian: \[L(x,v,G)= 1/2 v^T M(x,G) v - P(x,G)\] Metric tensor: \[M(x,G)= m_0\, eα_m σ(x)(I_n + β_m\, Gconn(x))\] Propagation potential: \[P(x,G)= p_0\, eβ_p λ(x)\] Geodesic equation: \[M(x)\, d^2 x/dτ^2+ Γ_M(x)\!(dx/dτ, dx/dτ)+ ∇_x P(x)= 0\] ------------------------------------------------------------INVASION FLUX (ENTERING THE PDE)------------------------------------------------------------ \[Jᵢₙᵥ(x,t)= -T(t,x)\, M⁻¹(x,G)\, ∇ P(x,G)\] \[∇ · Jᵢₙᵥ= -∇ · ( T\, M⁻¹∇ P )\] ------------------------------------------------------------EFFECTIVE COEFFICIENTS------------------------------------------------------------ Diffusion: \[Deff(G,H)(x)= D_0\, e-α_d σ(x) + β_d λ(x)- ν_0\, H(t,x)\] Proliferation: \[reff(x,H)= r_0\, e-α_r |κ(x)|+ σ_V\, H(t,x)\] Carrying capacity: \[Keff(G,H)(x)= K_0\, eβ_k λ(x)+ KV0\, H(t,x)\] ------------------------------------------------------------SIGMOID & CRITICALITY THRESHOLD------------------------------------------------------------ \[Θ_s(z)= 1/2(1 + \!(z/ε_s))\] \[I_c(B) defined implicitly by J_I(I_c,B) = 0\] ============================================================ "This formalisation is a direct mathematical rendering of the theory presented in: Mekšriūnas, G. (2026). The Topology of Malignancy: Metastasis as Holonomy, Dormancy as Spectral Trapping, and the Geometric Shape of Why Cancer Resists Cure. Omuo Systems, MB." Contact: For enquiries or research questions related to this work, email matthewcarlo.research@gmail.com