"""v304 -- QGAMB.IDG.01: the KMS spectral-action cutoff suggests a NONLOCAL (infinite-derivative) graviton form factor; under entire analyticity the Stelle ghost is a TRUNCATION ARTEFACT. This is a CONDITIONAL narrowing of the red-team rt_F perturbative-gravity attack (the R^2/Weyl^2 Stelle ghost), NOT a closure of QG.AMB.01. It introduces no new parameter: the cutoff is the seam KMS weight already derived in v259 (PS.SPECACT.02). Background. rt_F / v278 correctly flag that the local R + R^2 / Weyl^2 spectral- action gravity sector has a four-derivative propagator 1/(p^2(p^2+M^2)) with a negative-residue massive spin-2 mode -- the Stelle ghost -- so the *perturbative* gravity S-matrix is non-unitary. The honest TFPT position keeps QG.AMB.01 [O]. This script asks ONE sharply-bounded question on the theory's own terms: the R + R^2 sector arises by TRUNCATING the spectral action Tr f(D^2/Lambda^2) at the a_4 Seeley-DeWitt order; what happens if one keeps the seam-fixed exponential cutoff f(u) = e^{-u} (v259) WITHOUT truncating? [E] 1. STELLE GHOST = TRUNCATION ARTEFACT. the local four-derivative propagator 1/(p^2(p^2+M^2)) partial-fractions to (1/M^2)[1/p^2 - 1/(p^2+M^2)]: a HEALTHY massless graviton (residue +1/M^2 at p^2=0) plus a massive spin-2 mode with a NEGATIVE residue -1/M^2 at p^2=-M^2 (the Stelle ghost) -- reproducing rt_F / v278 exactly. [E] 2. ENTIRE FORM FACTOR HAS NO EXTRA POLE. for an infinite-derivative kinetic operator p^2 a(p^2) with a(z) = e^{z/M^2} -- an ENTIRE function, nowhere zero (Weierstrass: exp has no zeros) -- the dressed propagator 1/(p^2 a(p^2)) has its ONLY pole at p^2 = 0 (the massless graviton): no new pole, hence no ghost (Tomboulis 1997; Biswas-Mazumdar-Siegel 2006; Biswas-Gerwick-Koivisto-Mazumdar 2012). [E] 3. NEG CONTROL: A POLYNOMIAL FORM FACTOR RE-INTRODUCES THE GHOST. a finite truncation a(z) = 1 + z/M^2 (the R + R^2 order) is a polynomial with a real zero at z = -M^2 -- exactly the ghost pole. So "entire vs polynomial" is the discriminating property: only the UN-truncated (entire) form factor is ghost-free; every finite-derivative truncation carries a zero = a ghost. [C] 4. THE CUTOFF IS THE SEAM'S OWN WEIGHT, NOT A CHOICE. v259 fixes the spectral-action cutoff to the beta=1 seam KMS weight f(u) = e^{-u} (Tomita-Takesaki + the seam unit 2 pi = 1/(4 c3), v58) -- no new parameter. The UN-truncated trace Tr e^{-D^2/Lambda^2} is the nonlocal object; truncating it at a_4 is what produces the local R + R^2 and hence the Stelle ghost, so reading the spectral action at the seam-fixed cutoff WITHOUT truncating is the natural theory-internal move. [C] 5. CONDITIONAL NARROWING (the honest claim). IF the resummed graviton form factor of the un-truncated KMS spectral action is an ENTIRE function of bounded type (the Biswas-Mazumdar-Siegel class), THEN by items 2-3 the perturbative graviton is ghost-free and the rt_F Stelle-ghost attack is a truncation artefact -- a perturbative NARROWING of the objection. [O] 6. WHAT STAYS OPEN (no overclaim). two gaps remain: (i) the trace regulator f in Tr f(D^2/Lambda^2) is NOT identical to the kinetic form factor a(Box) of the graviton propagator -- that the heat-kernel resummation yields an ENTIRE a(Box) (rather than a generic nonlocal / non-analytic tail) is an UNPROVEN analyticity assumption; (ii) even a ghost-free PERTURBATIVE graviton is not the NONPERTURBATIVE ambient measure that QG.AMB.01 demands. So QGAMB.IDG.01 stays [C]/[O]: a conditional narrowing of rt_F, consistent with the SEAM.EQUIV.01 firewall, NOT a TOE closure. Status: [E] the truncation-artefact algebra (items 1-3, exact sympy); [C] the KMS-cutoff link + the conditional ghost-freedom (items 4-5); [O] the entire- analyticity assumption + the nonperturbative gap (item 6). Does NOT change QG.AMB.01's [O] status. Python-only (sympy; elementary pole algebra, like v259). """ import sympy as sp from tfpt_constants import check, summary, reset p2, z = sp.symbols("p2 z", real=True) M2 = sp.symbols("M2", positive=True) # M^2 > 0 (the ghost mass-squared scale) def run(): reset() print("v304 QGAMB.IDG.01: KMS cutoff -> nonlocal graviton form factor; entire => no Stelle ghost (conditional)") # ---- 1. Stelle ghost = truncation artefact (local R+R^2 four-derivative propagator) ---- local_prop = 1 / (p2 * (p2 + M2)) ap = sp.apart(local_prop, p2) res_massless = sp.limit(p2 * local_prop, p2, 0) # residue at p^2 = 0 res_massive = sp.limit((p2 + M2) * local_prop, p2, -M2) # residue at p^2 = -M^2 check("STELLE GHOST = TRUNCATION ARTEFACT [E]: the local four-derivative " "propagator 1/(p^2(p^2+M^2)) = %s has a HEALTHY massless graviton " "(residue %s at p^2=0) and a massive spin-2 mode with NEGATIVE residue " "%s at p^2=-M^2 (the Stelle ghost) -- exactly rt_F / v278" % (ap, res_massless, res_massive), sp.simplify(res_massless - 1 / M2) == 0 and sp.simplify(res_massive + 1 / M2) == 0 and res_massless > 0 and res_massive < 0, exact=True) # ---- 2. entire form factor a(z)=e^{z/M^2} has no zeros => dressed prop has only the p^2=0 pole ---- a_entire = sp.exp(z / M2) zeros_entire = sp.solve(a_entire, z) # exp never vanishes -> [] denom_entire = p2 * sp.exp(p2 / M2) poles_entire = sp.solve(denom_entire, p2) # only p^2 = 0 check("ENTIRE FORM FACTOR HAS NO EXTRA POLE [E]: a(z)=e^{z/M^2} is entire and " "nowhere zero (solve(e^{z/M^2})=%s), so the dressed IDG propagator " "1/(p^2 e^{p^2/M^2}) has its ONLY pole at p^2=%s -- no new pole, no ghost " "(Tomboulis 1997; Biswas-Mazumdar-Siegel 2006)" % (zeros_entire, poles_entire), zeros_entire == [] and poles_entire == [sp.Integer(0)], exact=True) # ---- 3. neg control: a polynomial (truncated) form factor re-introduces the ghost ---- a_poly = 1 + z / M2 zeros_poly = sp.solve(a_poly, z) # z = -M^2 (the ghost) poles_poly = sp.solve(p2 * (1 + p2 / M2), p2) # {0, -M^2}: graviton + ghost check("NEG CONTROL -- POLYNOMIAL FORM FACTOR RE-INTRODUCES THE GHOST [E]: a " "finite truncation a(z)=1+z/M^2 (the R+R^2 order) is a polynomial with a " "real zero at z=%s, so the propagator denominator p^2(1+p^2/M^2) has poles " "%s (graviton + ghost) -- 'entire vs polynomial' is the discriminating " "property: only the un-truncated (entire) factor is ghost-free" % (zeros_poly, poles_poly), zeros_poly == [-M2] and set(poles_poly) == {sp.Integer(0), -M2}, exact=True) # ---- 4. the cutoff is the seam KMS weight, not a choice (v259, recorded) ---- check("CUTOFF = SEAM KMS WEIGHT, NOT A CHOICE [C]: v259 (PS.SPECACT.02) fixes " "the spectral-action cutoff to the beta=1 seam KMS weight f(u)=e^{-u} " "(Tomita-Takesaki + the seam unit 2pi=1/(4 c3), v58) -- no new parameter; " "the UN-truncated trace Tr e^{-D^2/Lambda^2} is the nonlocal object, and " "truncating it at a_4 is what produces the local R+R^2 and the Stelle " "ghost (rt_F)", True) # ---- 5. conditional narrowing (the honest claim) ---- check("CONDITIONAL NARROWING [C]: IF the resummed graviton form factor of the " "un-truncated KMS spectral action is an ENTIRE function of bounded type " "(Biswas-Mazumdar-Siegel class), THEN by items 2-3 the perturbative " "graviton is ghost-free and the rt_F Stelle-ghost attack is a truncation " "artefact -- a perturbative narrowing of the objection, not a closure", True) # ---- 6. what stays open (no overclaim) ---- check("WHAT STAYS OPEN [O]: (i) the trace regulator f in Tr f(D^2/Lambda^2) is " "NOT identical to the kinetic form factor a(Box) -- that the heat-kernel " "resummation yields an ENTIRE a(Box) is an UNPROVEN analyticity " "assumption; (ii) a ghost-free PERTURBATIVE graviton is not the " "NONPERTURBATIVE ambient measure QG.AMB.01 demands. So QGAMB.IDG.01 stays " "[C]/[O]: a conditional narrowing of rt_F, consistent with the " "SEAM.EQUIV.01 firewall, NOT a TOE closure", True) return summary("v304 QGAMB.IDG.01: KMS cutoff -> nonlocal form factor; entire => no Stelle ghost (conditional narrowing of rt_F, NOT a QG.AMB.01 closure)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)