A registered result in its source context
Original ledger classification: 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)
Recorded statement: 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…
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Paper context: Frontier and open physical interfaces
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GRAV.LOOP.FINITE.01
Original ledger classification: 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)
Original claim text
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.
Original classification detail
[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 as recorded
v253; v259; v304; v370; v380; v386
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- Frontier and open physical interfaces
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- 2026-06-24 (Graviton loop finiteness: v389 the entire form factor is UV-finite at the loop level by power counting)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- 2026-06-24 (Website: an animated ``Universal Gap Lab'' visualising v383– v391)Research changelog; TFPT — Introduction and reading guide
- SummaryFrontier and open physical interfaces
Registered verification modules
- v389_grav_loop_finiteness.pyRegistered verification module · frontier
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GRAV.LOOP.FINITE.01: the entire-form-factor graviton is UV-finite at the LOOP level by power counting -- the honest next step past v386 (tree). Extends v304/v370/v380/v386 from the propagator to the superficial degree of divergence of loop graphs. HONEST SCOPE: the standard infinite-derivative/nonlocal-gravity finiteness statement (Tomboulis; Biswas-Mazumdar-Siegel; Modesto), NOT a full proof of perturbative QG. [E] super-polynomial falloff p^k e^{-p^2/M^2}->0 for every k. [E] one dressed line turns a quadratic divergence into the finite (M^2/2)e^{-p0^2/M^2}; GR int p^3/p^2 dp diverges quadratically. [E] superficial degree -> -infinity at every loop order (dressed cutoff-ratio ->1 vs GR ->4). [E] the regulator is the SEAM scale M=M_scal=c3^{7/2}Mbar (v253/v259), not a dial. [C]/[O] loop UNITARITY of the nonlocal form factor (Lorentzian continuation/macro-causality) stays the honest open point -- tree unitarity is v386; NOT claimed solved. No new number.
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