Gravity and Spacetime in the G Model

1. What the G Model means by spacetime

The G Model does not treat spacetime as a rigid material lattice inside a separate background space. Its working picture is a dynamically connected causal structure. The smallest retained local representation uses four equivalent spatial directions with zero net vector bias, promoted to four future-directed null branches.

The local relations are Σ_a n_a = 0, n_a · n_b = -1/3 for a ≠ b, k_a = L_c(1,n_a), k_a² = 0, and L_c = cT_c. Four directions are the minimum regular set that can be balanced, non-planar, and second-order isotropic in three spatial dimensions.

The four branches generate six bivector or causal two-plane channels: Σ_ab = k_a ∧ k_b for a < b, C(4,2) = 6, A_0 = (4/3)L_c², and A_Σ,0 = 8L_c². These channels are not extra spatial dimensions or particle species.

The four-null Gram spectrum is {-4, 4/3, 4/3, 4/3}L_c², with det G_4 = -(256/27)L_c⁸. The identity K G_4^-1 K^T = η reconstructs the Minkowski metric exactly within the declared local construction.

The six-channel representation has an exact complementary gradient/circulation split, B_∂^T B_∂ + CC^T = 4I_6 and P_grad + P_circ = I_6, with bivector signature (3,3). These are local structural results, not proof of complete all-scale Lorentz invariance.

2. Matter, neutral fabric, and exterior gravity

Matter is treated as a source or loading of the causal structure rather than as the fabric itself. The undisturbed reference condition is M = 0. A region can have no local matter and still have M ≠ 0, which permits an exterior gravitational field.

The four relevant states are: no local matter and neutral fabric (χ_m,M) = (0,0); matter present with deformation (1,M ≠ 0); no local matter but a sourced field remains (0,M ≠ 0); and matter present while M is neutral (1,0), a special state not automatically identical to equal action.

The matter-facing aggregation branch uses E_cell = ρV_c c², V_c = (2/3)(cT_c)³, A_cell = E_cell T_c/ℏ, D_0 = ℏ/E_cell, and D_0/T_c = 1/A_cell. T_c is the intrinsic fabric time; D_0 is a distinct response time except under the conditional equal-action state A_cell = 1.

The retained long-wavelength source bridge is A-bar[M] - M = (C_Σ/4)A_cell, leading conditionally to ∇²M = C_Σ(c³T_c²/ℏ)ρ. Outside matter, ρ = 0 can coexist with M ≠ 0: ∇²M = 0 locally while ∇M remains nonzero.

3. Coefficient chain, source modulus, and weak coupling

The retained source-transfer coefficient is C_Σ = (3/4)(4/5)(4/π) = 12/(5π) ≈ 0.763943726841. The four-null/Hodge geometry supplies the 2D causal plane, J_*² = -I, unit-circle amplitudes, and uniform angular measure, but geometry alone does not select a scalar transfer norm: 4/π follows only after explicit transfer axioms select the absolute resolved-channel norm. The 4/5 depth and 4/π transfer choices therefore remain conditional.

The newer Gravity and Spacetime guide records the internal candidate source-modulus closure K_M = ℏ/(C_Σ c T_c²) = (5π/12)ℏ/(cT_c²), equivalently T_c = sqrt[5πℏ/(12cK_M)]. With the current working K_M = 9.63090777 × 10^42 N, it displays T_c = 2.186570237920 × 10^-43 s and L_c = 6.555172662157 × 10^-35 m. This is a theoretical result within the retained branch, not experimental establishment.

The retained weak coupling is G_G = [3/(5π²)]c^5T_c²/ℏ = c^4/(4πK_M) = 6.674300002387306 × 10^-11 m³ kg^-1 s^-2. The identity G_G K_M = c^4/(4π) is exact only within the retained source-modulus/continuum-coupling branch; the MMM name records provenance, not experimental validation.

The K_M to T_c arithmetic does not insert measured Newtonian G. The physical interpretation and absolute numerical value of the candidate closure require independent experimental validation. Full microscopic Lorentz invariance and derivation of the weak spin-2/Einstein structure from finite-cell dynamics remain OPEN, although that structure is unavoidable as the coarse-grained limit of the stated continuum class.

The dated main-manuscript calibration branch records a different provenance policy: measured Big G is accepted openly as an empirical matter-spacetime coupling and is used to calibrate T_c. In that branch, T_c is not a native prediction and must not be reused to claim an independent prediction of the same measured Big G. The source normalization of the linearized Einstein equation is calibrated or conditional.

The main manuscript’s Big-G-calibrated reference table gives T_c = 2.186570238 × 10^-43 s, L_c = 6.555172663 × 10^-35 m, V_c = 1.877851076 × 10^-103 m³, E_f = 4.822949655 × 10^8 J, m_c = 5.366255204 × 10^-9 kg, and Ω_c = 1.120242881 × 10^43 s^-1. Every value in that table inherits the Big-G calibration and is a reference scale rather than an independent prediction.

The two provenance descriptions must remain distinct: the later K_M-first working inversion requires independent K_M provenance, while the dated manuscript’s calibrated branch openly takes measured Big G as input. Algebraic agreement between the branches is not experimental independence.

The historical expression 1/(5π²c) was rejected as Newtonian G because its dimensions are wrong. It remains only as the G Model Kutsal Interface Constant, K_GD, a bookkeeping and provenance factor with dimension T/L—not as Newton’s gravitational constant.

The addendum also rejects the historical 1.275% shortcut. Comparing the bare SI numerical values of K_GD and measured Big G is unit-system dependent because the quantities have different dimensions; the difference is not a re-equalization residual and must not be iterated toward Big G.

4. Operational gravity, free fall, clocks, and tides

In the weak branch, gravitational potential and acceleration are identified conditionally as Φ = c²M and g = -c²∇M. Matter density sources curvature of M; M represents the local causal or clock-deformation state; and its spatial gradient produces weak-field acceleration.

For a spherical source, M(r) = -G_G M_s/(c²r), Φ(r) = -G_G M_s/r, and g_G(r) = G_G M_s/r². The inverse-square dependence follows from the scalar source equation, spherical spreading, and the weak acceleration identification.

Free fall is interpreted as natural motion through the local response field. Felt weight is associated with the support force that prevents free-fall motion.

The conditional clock bridge is M = ln(dτ/dt), so dτ/dt = e^M. In the weak limit, Δf/f ≈ ΔM ≈ ΔΦ/c². The same M field therefore connects clock-rate differences and acceleration within the stated assumptions.

The Hessian H_ij = ∂_i∂_j M and traceless part E_ij = H_ij - (δ_ij/3)∇²M separate source trace from tidal shape. The tidal direction is structurally constrained, while the absolute microscopic target amplitude remains open.

5. Dynamic spacetime: reorientation, reconnect, and re-equalization

The model does not retain a permanently oriented tetrahedral crystal as physical spacetime. Accepted local observables are relational and invariant because a fixed stencil would retain higher-order anisotropy.

Jadvyga 360 reorientation uses K_x(θ) = Λ_x(θ)K_x(0) with Λ_x^TηΛ_x = η and preserves accepted Gram, pair-area, and holonomy invariants. This is an exact gauge or invariance statement in the accepted sector; an all-scale orientation-neutral proof remains open.

Celal Dynamic Reconnect selects the branch correspondence minimizing invariant mismatch: Π_Celal ∈ arg min over Π in S_4 of ||γ_x - Π^Tγ_yΠ||_F². It matches existing centered links but does not yet supply a physical law for link birth or link death.

The candidate re-equalization law is T_c dq/dτ = -(I_6-P)(q-q_eq). Once adopted, it gives e(τ) = e(0)exp[-3τ/(2T_c)] and recovery time τ_rec = 2T_c/3. Exponential decay, semigroup composition, and Lyapunov stability then follow exactly, but the generator itself has not been derived uniquely from a microscopic action.

The 11 September technical addendum combines Aggregation with the retained weak-source definitions to obtain L_c²∇²M = (3C_Σ/2)A_cell = [18/(5π)]A_cell = [18/(5π)](T_c/D_0). This internal relation does not use measured Big G. It organizes source strength but does not independently select the absolute scale or a unique centered shape amplitude.

The addendum’s trace/tidal audit fixes a robust centered target direction from the local Hessian principal axes, while two natural internal constructions give different amplitudes. The direction is a conditional strong pass; the unique amplitude normalization remains open and is not an invitation to fit a free parameter.

Open dynamical requirements include an action-level topology law, unique source-to-target normalization, and a direct all-scale no-preferred-frame proof for the interacting high-wave-number network.

6. Relativistic compatibility, strong fields, and horizons

Because the four-branch matrix is invertible, a symmetric 4×4 branch matrix carries all ten independent components of a symmetric spacetime tensor. The exact reconstruction is h = (K^-1)^T Q K^-1 and T = (K^-1)^T H K^-1.

Within the assumed local Lorentz-covariant, symmetric, two-derivative continuum class, linearized gauge redundancy and identical conservation fix the standard linearized Einstein operator up to one overall normalization. It is the unavoidable coarse-grained limit of that class; full microscopic Lorentz invariance and a finite-cell derivation remain OPEN.

In branch form the manuscript writes the source equation as E_ab[Q] = 2C_Σ(cT_c²/ℏ)H_ab. The operator form is exact under the stated continuum assumptions; its source normalization is calibrated or conditional. A finite-cell derivation that forces spin-2 gauge dynamics remains open.

The retained transverse-traceless continuum sector has two physical curvature modes, structurally consistent with relativistic gravity. A complete microscopic propagation law remains open.

The later two-function spherical completion uses ψ = 2 tanh(M/2), T = e^M = (1+ψ/2)/(1-ψ/2), and X = (1-ψ/2)². With ψ = -μ/r, it gives the isotropic Schwarzschild exterior ds² = -[(1-μ/2r)²/(1+μ/2r)²]c²dt² + (1+μ/2r)^4(dr²+r²dΩ²).

Within this conditional completion the exterior is exactly Ricci-flat and coordinate-equivalent to Schwarzschild. The earlier one-function reciprocal exponential metric failed as a nontrivial exact static-vacuum solution and is retained only as a historical or weak-field parametrization.

At R_h = 2μ, the accepted strong-field branch supports causal trapping and loss of outward-connected support. It does not establish destruction of every local fabric degree of freedom: the four-null Gram remains non-degenerate and the Schwarzschild curvature invariant is finite at the horizon. Deeper-interior fabric loss remains an open hypothesis.

7. Light Theory: light as a causal-fabric excitation

The manuscript records Gurbuz Dolu’s proposal that light may be a propagating excitation of the causal spacetime fabric rather than something moving through an independent spacetime container. This author-originated physical picture is the conceptual seed of Light Theory.

For an oriented link or one-cochain variable A, the face field is F = C^T A. Under the vertex gauge shift A → A + B^Tχ, F remains unchanged because BC = 0. This is an exact finite-tetrahedron abelian-type gauge identity; in continuum notation it corresponds to F = dA and dF = d²A = 0.

If one additionally adopts a minimal local, Lorentz-compatible, parity-even, gauge-invariant quadratic action with no photon mass, a nontrivial plane wave satisfies q² = 0 and q·a = 0. Gauge equivalence removes one further direction, leaving dim(q-perpendicular/span{q}) = 2 and ω² = c²k².

The vacuum result is a conditional strong structural pass: after the minimal gauge action is adopted, null propagation, speed c, transversality, and two physical modes follow exactly. The gravitational connection action does not independently produce the required spin-1 F² sector; that direct route fails in the present field content.

The model does not yet derive independent photon-action normalization, physical charge assignments, the elementary charge scale, the fine-structure constant, or finite-cell high-wave-number dynamics. Exact tetrahedral group structure by itself is not ordinary additive electric charge.

8. Scientific status and independent verification priorities

Exact within stated mathematics: four-null Gram signature, Minkowski reconstruction, six-channel incidence/Hodge identities, tensor reconstruction, coefficient integrals after their prescriptions are chosen, and algebra following an adopted candidate generator.

Conditional or model-defined: causal-fabric ontology, matter loading, native causal volume, 4/5 depth weighting, 4/π throughput prescription, source rule, clock bridge, continuum field normalization, re-equalization generator, and two-function strong-field completion.

Comparison rather than independent validation: numerical weak-gravity, free-fall, altitude, and Mercury calculations whose absolute scale depends on the current K_M value. Measured Newtonian G is used only in explicitly labeled comparison contexts.

Open: independent experimental validation of the candidate numerical K_M–T_c closure, a native or nongravitational cross-check of T_c, a complete microscopic action, physical link creation and deletion, source-to-target normalization, all-scale Lorentz behavior, photon-action normalization, interiors, collapse, singularity resolution, generic nonlinear dynamics, and a distinctive independently fixed observable prediction.

Rejected or superseded: identifying 1/(5π²c) with Newtonian G; forcing coefficients by circular gravitational matching; the one-function exact static-vacuum route; treating a fixed tetrahedral orientation as observable; and claiming that the horizon necessarily destroys the whole causal fabric.

Verification priorities are to reproduce the four-null and six-channel identities independently; determine K_M without Newtonian or GR normalization; cross-check T_c through an independent nongravitational route; test whether the 4/5 and 4/π prescriptions are dynamically unique; and derive a distinctive observable prediction for controlled measurement.

Gurbuz Dolu is the originator, author, and research director of the G Model framework. AI systems assisted at his request with formal derivation, algebra, calculation, consistency testing, comparison, falsification attempts, and presentation. AI assistance is not independent verification, peer review, experimental validation, or institutional endorsement.