Finite Gluing of the Three-Dimensional Ising Free Energy

We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention Φ=+log Z/V, the final object is Φ₃(K,H,L)=Φ₀(K)+ΔΦH(K,H)+ΦFSS(K,H,L), from which M=∂HΦ₃, χ=∂H²Φ₃, and UK=∂KΦ₃ are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite H and finite L. The materials and operations are separated explicitly. A diagnostic dimensional stage uses $d=1,2,4$ anchors in μc=-log Kc to transport critical coordinates and positive film amplitudes toward $d=3$. The practical 3D generator then protects established 3D values of Kc, critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with C² matching. The inner magnetic increment joins an O(K²) high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in K. A recovered z⁴ strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an H-independent residual stabilizer that preserves the $H=0$ axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives Kc(3)=0.22301549 and sub-percent estimates of β, δ, and ν. After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window |K-Kc|≤0.01, H≤0.1, the magnetization has RMS relative error $1.21%$ and maximum error $4.29%$. The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.