A Conditionally Closed Constraint-Defined MOND Theory with Two Tensor Degrees of Freedom: Hamiltonian Certification, Kernel-Agnostic Chassis, and Solar-System Viability

Preprint. This deposit timestamps a conditionally closed Hamiltonian construction with its full certification suite; it is not a journal publication, and the word 'conditionally' is load-bearing. A preferred-foliation Hamiltonian theory of minimally-modified-gravity (MMG) type in which MOND is carried by the CONSTRAINT sector: the scalar (lapse) constraint of the ADM system is replaced by the exact MOND elliptic equation CM = Di[c²μ(y)Di ln N] − 4πGρm, y = (c²/a₀)|D ln N|. A twelve-gate Dirac–Bergmann certification (all scripts committed, all passing) establishes on the generic branch (y>0, k≠0): exactly two local propagating degrees of freedom (the TT pair; Dirac determinant (LNK)² ≠ 0; 20−12−4 = 4 phase-space dimensions), an exactly-GR tensor sector (cT = c, QT > 0, no scalar pole), the exact MOND weak-field law with no missing factor, and multiplier preservation with no tertiary constraint. A thirteenth gate proves the chassis is KERNEL-AGNOSTIC: the constraint algebra uses μ only through ellipticity (μ > 0, d(yμ)/dy > 0), so the interpolation is modular. This matters because the companion deposit shows the external-field Cassini quadrupole excludes exponentially-screened kernels (1−μ unscreened at yext ≈ 1.9); here μn(y) = y/(1+yⁿ)1/n, n = 5–10, clears the 2026 Cassini ceiling on both a₀ footings (committed 175-galaxy SPARC analysis) at a stated price: RAR scatter 0.108 → 0.123–0.127 dex. The deep-MOND limit — hence a₀, the BTFR and the amplitude law — is identical across all μn; the solar system constrains the transition, not a₀. The cosmological k = 0 sector, left open by the closure, is prescribed by a CMC clock K = q(t) with a₀(q) = cq/Z, giving the falsifiable scaling a₀(z) = a₀,₀H(z)/H₀ on FLRW (a proposal; its preservation is not yet derived). What 'conditionally' means (stated as conditions, not footnotes): (1) the theory has a preferred foliation — the lapse is removed by a second-class pair; it is a Hamiltonian MMG model, NOT a manifestly 4D generally covariant local action; (2) ∇μTμν = 0 is not a Bianchi identity — matter equations acquire MOND-induced corrections at v ~ c (Newtonian-order consistency is verified; the relativistic matter sector is open); (3) the k = 0 and y = 0 branches are excluded from the certificate and treated separately. Relativistic lensing and cosmological perturbations are open. κ = ½ and Z ≈ 21 in a₀ = κc√(GρΛ) are FITTED, not derived. Within the stated conditions this is, to the author's knowledge, the first single Hamiltonian object simultaneously carrying a certified two-tensor-DOF count, the exact nonlinear MOND law, cT = c, and Cassini viability. AI-assistance disclosure. Portions of the analysis, numerical verification and drafting were carried out with the assistance of large language models (Anthropic Claude, OpenAI and Qwen models). The author directed the work, specified and reviewed every load-bearing calculation, and takes full responsibility including for any errors. Several intermediate claims were found incorrect and withdrawn before this deposit. No AI system is listed as an author. Reproducibility. The full gate suite is included: run_all.sh executes gates 1–12 (exit 0 = ALL GATES PASS); gate 13 is a standalone script. The consolidation record is included.