The common source contract for spin two

GRAV.SPIN2.EMERGENCE.01

Original ledger classification: Open (research contract; quadratic witnesses [E] + quadratic Hamiltonian half executed; content pinned on the untruncated form factor; Einstein eq stays a downstream/conditional readout) | Round-3 v1024 supplies finite gap/TT locality no-gos [E]; spin-2 emergence remains Open [O].

Original claim text

THE SPIN-2 EMERGENCE CONTRACT [O] (registered 2026-08-28, review wave 3). DEMAND: from the SAME spectral determinant that carries the gauge fixpoints, a massless transversal spin-2 pole with two helicities, universal T_munu coupling, diffeomorphism Ward identities, and positive H / RP transfer. RELATION: the entanglement-equilibrium Einstein equation (v358/v359) stays a DOWNSTREAM / CONDITIONAL readout -- it presupposes a local QFT and can never be cited as producing one; GRAV.NONCIRCULAR.01 is BINDING (BW/CHM uses remain assumption-declared). KILL: a massive or longitudinal extra mode; non-universal coupling; failed diffeomorphism Ward; negative Hessian / RP violation on the spin-2 sector. EXECUTED UPDATE (2026-08-28, completeness wave, v997): exact Barnes-Rivers quadratic decomposition -- R+R^2 clean (one massless TT pole, two helicities, scalaron at M_scal exact); the LOCAL a4 Weyl^2 truncation necessarily carries an opposite-residue spin-2 ghost -- typed. The contract's content is pinned on the untruncated form factor a(Box)=e^{-Box/M^2}. Einstein eq stays downstream/conditional. Display stays [O]. Python-only / Wolfram mirror deferred. EXECUTED UPDATE (2026-08-29, review wave 4, v1000): quadratic Hamiltonian half executed -- positivity exact, 2 TT helicities at all 5852 nonzero momenta (N=6..12), omega = k - k^3/24 exact (lattice q_i = 2 sin(k_i/2)). Spectral-action Hessian identification, nonlinear Ward and universal coupling stay [O]. Display stays [O]. WAVE-6 (2026-08-30): GRAV.WEINBERG.WITTEN.01 registered as the BINDING firewall BEFORE any spin-2 marker move (microscopic Hamiltonian family must NOT possess a fundamental Lorentz-covariant local conserved T_munu; Lorentz, diffeomorphism Ward identities and the spin-2 mode must emerge in the SAME IR limit). Display stays [O]. EVENING (2026-08-30, v1012): interacting TT half-step executed -- conserved interaction-complete lattice stress ~1e-15, TT spectral positivity exact, sum rules ~1e-15, dominant positive-Z pole persists under phi^4 (gapped, as expected for massive generic matter); Z trend k1 suppressed 0.958 / k2 enhanced 1.033. Missing: TFPT content, k->0 volume scaling, Ward identities. Display stays [O]. EXECUTED UPDATE (2026-09-05, v1024): GRAV.PARENT.GAP.NO_GO.01 proves that a uniform positive full-parent gap excludes a massless spin-2 pole and that the exact TT projector has a direction-dependent k->0 limit, hence cannot be a strictly finite-range direct microscopic projector. A critical physical sequence with local tensor gauge constraints remains the viable open route. No TFPT stress operator, massless pole, Ward identity or universal coupling is constructed. Parent status stays Open [O]. ROUND-4 CURRENT: v1029 gives a local quadratic first-class constraint target with two z=1 nonzero-momentum modes. The full periodic target is unbounded below at k=0; an additional global zero-mode removal is indispensable. No common-parent embedding/nonlinear universal coupling; [O] unchanged.

Original classification detail

Open [O] + quadratic witnesses [E] (v997): R+R^2 clean (massless TT pole, two helicities, scalaron at M_scal exact); local a4 Weyl^2 truncation necessarily carries an opposite-residue spin-2 ghost -- typed. Content pinned on the untruncated form factor. Einstein equation stays downstream/conditional; GRAV.NONCIRCULAR.01 binding. | 2026-08-29 v1000: quadratic Hamiltonian half executed (positivity, 2 helicities at 5852 momenta, omega=k-k^3/24). Display stays [O]. | WAVE-6 (2026-08-30): GRAV.WEINBERG.WITTEN.01 is the binding firewall before any spin-2 marker move; display stays [O] | EVENING (2026-08-30 v1012): interacting TT half-step: conservation/positivity/sum-rules/Z-persistence executed; missing TFPT content, k->0 volume scaling, Ward identities. Display stays [O]. | ROUND3 2026-09-05: uniform full gap and direct finite-range TT are excluded mechanisms; critical gauge-constrained escape remains open. No graviton claim. | ROUND-4 CURRENT: v1029 gives a local quadratic first-class constraint target with two z=1 nonzero-momentum modes. The full periodic target is unbounded below at k=0; an additional global zero-mode removal is indispensable. No common-parent embedding/nonlinear universal coupling; [O] unchanged.

Dependencies as recorded

GRAV.NONCIRCULAR.01; GAUGE.DETLINE.FIXPOINT.01; QFT4D.OS.RECON.01; QG.G2.01