An auditable path through the 2025 variational GME, Weyl semimetry and Bianchi identity to a formalized—but still open—geometry-to-biology bridge.
g_mu_nu = eta_mu_nu + kappa A_mu A_nu
eta = diag(-1,+1,+1,+1)
This is the published starting point of Lindgren, Kovacs and Liukkonen (2025)i, not a result derived by BERM.
A^2 = eta^(mu nu) A_mu A_nu
g^(mu nu) = eta^(mu nu) - kappa A^mu A^nu / (1 + kappa A^2)
det(g) = -(1 + kappa A^2)
sqrt(-det g) = sqrt(1 + kappa A^2)
The inverse requires 1 + kappa A² ≠ 0. A real Lorentzian volume element requires the stronger condition 1 + kappa A² > 0. A² is not a Euclidean magnitude, so timelike and spacelike potentials cannot be merged into one |A| formula without an additional reduction.
delta_g = kappa(A_bio⊗a + a⊗A_bio + a⊗a)
h = kappa(A_bio⊗a + a⊗A_bio) [first order]
D_u sqrt(-det g) = kappa (A·u) / sqrt(1 + kappa A^2)
L2 spatial/scalar choice: q = |A_bar| := sqrt(kappa)s
L1 algebraic formula: chi_geo(q) = q / sqrt(1 + q^2); evaluate chi_geo(|A_bar|) only after that L2 choice
The exact tensor expansion and the directional volume response from geodesic deviation follow from the ansatz as L1 results. In Lorentzian geometry the numerator remains the signed contraction kappa A·u; it is not replaced by an absolute value before an explicit reduction.
Selecting the positive coordinate q=|A_bar| requires a dimensionless, collinear Lorentz-to-Euclidean spatial projection and is an L2 step. The directional derivative then evaluates the always-L1 formula as chi_geo(|A_bar|). Normalizing a membrane or ambient quantity to this coordinate is another open L0→L2 bridge.
S_EH[A] = integral sqrt(-det g(A)) R_LC[g(A)] d^4x
R_EH^mu := delta S_EH / delta A_mu
R_EH^mu = -2 kappa sqrt(-det g) G_LC^(mu nu) A_nu = 0
S_GME[g,nabla_hat] = integral <nabla_hat g, nabla_hat g>_g sqrt(|det g|) d^4x
Box_div T := nabla_hat_sigma(g^(sigma rho) nabla_hat_rho T)
content-bound full Euler-Lagrange attestation for the declared action scope = required
R_harm_mu_nu := Box_div g_mu_nu
R_eta_mu_nu := Box_div eta_mu_nu
R_outer_mu_nu := Box_div(kappa A_mu A_nu) = kappa R_GME_mu_nu
R_GME_mu_nu = A_nu Box_div A_mu + A_mu Box_div A_nu
+ g^(sigma rho)[(nabla_sigma A_nu)(nabla_rho A_mu)
+ (nabla_sigma A_mu)(nabla_rho A_nu)]]
R_decomp := R_harm - R_eta - R_outer = 0
R_harm = R_outer only under the separately checked condition R_eta=0
variational_check := bound full_EL attestation AND R_EH=0 AND R_harm=0
AND R_outer=0 AND R_eta=0 AND R_decomp=0
The Einstein–Hilbert action and the source's metric-gradient/GME action are parallel L0 premises, not stages derived from one another. The EH residual is computed from the supplied A and supplied symmetric Einstein tensor; this numerical API does not reconstruct the Einstein tensor from second metric derivatives. The GME branch uses the locked Weyl divergence operator. Full-variation attestations are bound to the SHA-256 digest of the complete numerical input bundle, but remain caller evidence rather than an independent symbolic derivation. Harmonic, background, outer-product and decomposition residuals remain separate.
Here Box_div denotes the locked operator nabla_hat_sigma(g^(sigma rho)nabla_hat_rho); it must not be confused with the metric perturbation delta g_mu_nu used above.
Q_sigma_mu_nu := tilde_nabla_sigma g_mu_nu - 2 phi_sigma g_mu_nu
Weyl-semimetry contract: Q_sigma_mu_nu = 0
T^lambda_mu_nu := tilde_Gamma^lambda_mu_nu - tilde_Gamma^lambda_nu_mu = 0
tilde_Gamma^lambda_mu_nu = {lambda_mu_nu}_LC
- delta^lambda_mu phi_nu - delta^lambda_nu phi_mu
+ g_mu_nu phi^lambda
R_Weyl := supplied tilde_Gamma - reconstructed tilde_Gamma(g,phi) = 0
Weyl semimetry is not a consequence of Bianchi but a separate geometric structure in the 2025 formulation. In addition to Q=0, the gate checks zero torsion and compares the supplied connection directly with the Weyl connection reconstructed from g, its Levi–Civita connection and phi. A wrong nonsymmetric connection therefore cannot pass through Q=0 alone.
R_B^nu := nabla_mu^LC G_LC^(mu nu) = 0 [contracted Bianchi]
F_mu_nu := partial_mu A_nu - partial_nu A_mu (F = dA)
R_F_mu_nu := F_mu_nu - (partial_mu A_nu - partial_nu A_mu) = 0
R_dF_lambda_mu_nu := partial_lambda F_mu_nu
+ partial_mu F_nu_lambda + partial_nu F_lambda_mu
homogeneous contract: R_dF = dF = d(dA) = 0
for a torsion-free connection: equivalently nabla_[lambda F_mu_nu] = 0
contracted_bianchi_residual checks the supplied LC-divergence vector for zero; the numerical API does not construct it from second metric derivatives. The F-definition residual binds F to the same supplied partial A, and the cyclic exterior derivative checks dF=0 for the supplied partial F. The LC and partial-F attestations are bound to the exact same input-bundle digest. This is an auditable conditional input contract, not an independent differential-geometric proof. None of the conditions alone contains the source current J.
∇^LC G_LC=0 and F=dA ⇒ dF=0 are distinct residual contracts with named connection provenance. dF=0 closes only the homogeneous Maxwell equations; neither one alone produces electric current or charge density.
The 2025 source's Maxwell limit additionally requires the variational principle, the R_GME=0 harmonicity condition, the Q=0 Weyl structure, and identifications relating the potential, Weyl one-form and source. It does not follow from Bianchi alone.
Bianchi branch: R_B=nabla^LC G_LC=0 AND R_F=F-dA=0 AND R_dF=0
required gate: variational_check AND weyl_check AND bianchi_check
sourced limit: delta S/delta A=0 + Q=0 + Bianchi + source identifications
invalid shortcut: Bianchi alone =/=> sourced Maxwell equation
Residuals and attestations grouped into the three AND branches make the source's formal chain testable in code. Every residual has its own scale, atol/rtol threshold and provenance. Passing does not establish that a geometric quantity couples to tissue, an ion channel or another biological observable: that remains the open L2 bridge.
L1 formula: chi_geo(q) = q / sqrt(1 + q^2)
L2 spatial/scalar coordinate choice: q = |A_bar|
L0→L2 biological candidate: delta_V_VGCC = C_bridge chi_geo(q) Delta_V_mem(f)
observable: O_r = integral K_r^(mu nu) delta_g_mu_nu dV
The algebraic chi_geo formula retains its L1 status here: empirical use does not reclassify the derived result. The spatial magnitude projection q=|A_bar| is an L2 choice, and biological identification is a separate open L0→L2 bridge. C_bridge/K_r, units, gauge choice, tissue transfer, channel structure and activation threshold follow neither from the metric determinant nor from the three residual contracts above.
candidate mixed-provenance product (NOT L1 as a whole): w_L(f) = SAR_norm(f) × VGCC_coupling(f) × MOD_envelope(f)
L1 components: SAR form/normalization; Schwan DeltaV=1.5 r E g(f); algebraic product/sum identities
OPEN L2: field/geometry -> tissue and channel observable identification
L3 inputs: numerical tissue dielectric/material parameters for SAR; numerical tau_m for Schwan; VGCC_coupling(f), MOD_envelope(f), fitted endpoint weights
supplied E, r and evaluation f remain variables of the L1 forms; the formulas do not relabel them as L3
L3 DKC: BL(t) = alpha(k_B*FS)(t) + (1-alpha)(k_R*FS)(t)
L3 pathways: VGCC / tissue-response mechanisms
L3 Hill endpoint: Delta Health = -gamma BL^n / (x_half^n + BL^n)
The w_L product is not L1-derived as a whole. The SAR form and normalization are L1; only numerical tissue dielectric/material parameters are L3, and supplied E is not relabelled. Schwan ΔV=1.5rE g(f) is L1; only the numerical value of τ_m is L3, and the presence of r, E or f does not change the formula's level. Joining either form to a tissue or channel observable is open L2; VGCC coupling, the modulation envelope, fitted weights and DKC biology are L3.
chi(0)=0; delta F_ion=C_bridge chi(0) DeltaV_mem=0
x=sqrt(kappa)s; chi(x)=x/sqrt(1+x^2)=kappa^(-1/2)dV/ds approaches 1
D_u sqrt(-det g)=kappa(A·u)/sqrt(1+kappa A^2); on the L2 spatial slice delta F_ion is proportional to cos(theta)
SAR=sigma|E_internal|^2/rho; f_res=c/(4L)=44.1 MHz for L=1.7 m, acceptance band +/-20%
Delta V=1.5 r E/sqrt(1+(2 pi f tau_m)^2)