Original ledger status
GRAV.LOOP.FINITE.01
GRAV.LOOP.FINITE.01
The entire-form-factor graviton is UV-finite at the LOOP level by power counting (2026-06-24) -- the honest next step past v386 (which established the tree amplitude), extending v304/v370/v380/v386 from the propagator to the superficial degree of divergence of loop graphs. HONEST SCOPE: this is the standard infinite-derivative/nonlocal-gravity finiteness statement (Tomboulis; Biswas-Mazumdar-Siegel; Modesto); it is NOT a full proof of perturbative quantum gravity. [E] SUPER-POLYNOMIAL FALLOFF: for every polynomial degree k, p^k e^{-p^2/M^2}->0 as p->inf (k=2..20), so the dressed propagator cannot leave a power-counting divergence. [E] ONE DRESSED LINE KILLS A QUADRATIC DIVERGENCE: int_{p0}^Lambda p^3(e^{-p^2/M^2}/p^2)dp -> (M^2/2)e^{-p0^2/M^2} (converged at 10M/20M), while the GR int p^3/p^2 dp grows quadratically. [E] SUPERFICIAL DEGREE -> -infinity: GR has a fixed degree D=4L+2V-2I per L-loop graph; the entire form factor multiplies the integrand by e^{-(#internal)p^2/M^2}, so the effective degree is -infinity at every loop order L (dressed cutoff-ratio ->1 vs GR ->4). [E] the regulator is the SEAM scale M=M_scal=c3^{7/2}Mbar~3.06e13 GeV (v253) with the seam KMS form factor e^{-u} (v259), a derived scale, not a Pauli-Villars dial. [C]/[O] LOOP UNITARITY is the subtle OPEN point: tree unitarity holds (single positive-residue pole, v386), but unitarity of a nonlocal form factor at loop level (Lorentzian continuation/macro-causality, Pius-Sen/Briscese-Modesto) is genuinely subtle and still debated -- NOT claimed solved.
Recorded status: Perturbative graviton loop UV-finiteness [E] (power counting; super-polynomial falloff, degree -> -infinity at every loop order, seam-regulated) + loop unitarity [C]/[O] (the honest open point, not claimed solved)
Dependencies, status detail and source
Canonical status: [E] the Euclidean power-counting UV-finiteness at every loop order (super-polynomial falloff; one dressed line turns a quadratic divergence into a finite integral; the superficial degree -> -infinity), regulated by the derived seam scale M_scal (v253/v259); [C]/[O] full loop UNITARITY of the nonlocal form factor (the Lorentzian continuation / macro-causality, the genuinely open point), NOT claimed solved. Extends v386 from tree to loop FINITENESS only; introduces no new number.
Dependencies: v253; v259; v304; v370; v380; v386
Scripts: v389_grav_loop_finiteness.py
Location: tfpt_4_frontier
Original CSV, line 452