Manuscript VII-I@1 · VII-I.tex · section · 907e96ab

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The triangle obstruction and what VII-D does supply › Signature matching · lines 1332–1407

VII-I@1#input:triangle, VII-I@1#eq:triangle-input, VII-I@1#eq:triangle-residue

VII-C/D use $η_{C} = - η$ and $□_{C} = - □$ . The real field identification

h_{C} = - h, ϕ_{C} = ϕ

gives

E_{0}^{C} C_{C} h_{C} = E_{0} C h .

Their sharp insertion therefore pulls back to our $B_{f}$ . Their middle observable $h_{00, C}^{2}$ pulls back to $h_{00}^{2}$ . The graviton two-point tensor has two metric factors, so its displayed numerical tensor is unchanged. Both insertion signs in the triple occur twice.

Let

W_{g} = \int g (y) h_{00} (y)^{2} d^{4} y, T_{ϵ}^{ρ} = ev_{0} [ℏ^{3}] (B_{f, ϵ} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ϵ})

be understood first on product tests and then as the supplied joint distribution on $X = (M_{+}^{\circ})^{3}$ . The field mollifier is used on both slots of both insertions; the middle observable is unsmeared.

There are a nonzero positive residue $r$ and a common distribution $U_{m}$ such that

T_{ϵ}^{ρ} = lo g (ϵ^{- 1}) r + U_{m} + κ_{ρ} r + O_{D^{'}} (ϵ (1 + ∣ lo g ϵ ∣)) .

On each fixed compact, the error is bounded by the stated factor times a test seminorm of order $64$ . Specifically,

⟨ r, f \otimes g \otimes f^{'} ⟩ A_{f} (y, n) = \frac{1}{64 ( 2 π ) ^{3}} \int g (y) \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n) d^{4} y, = \int f (t, y + (t + y^{0}) n) d t .

There are nonnegative pairwise-disjoint $f, g, f^{'}$ for which this pairing is positive.

For a subtraction scale $μ > 0$ ,

u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r .

For any two allowed fixed radial profiles,

u_{μ}^{a} - u_{μ}^{b} = c_{ab} r, c_{ab} = \int_{0}^{\infty} \frac{ϑ _{a} ( u ) ^{4} - ϑ _{b} ( u ) ^{4}}{u} d u \in R .

These are numerical distributional statements, including strong convergence and the full reflected-diagonal intersection CheungVIIC,CheungVIID.

Only this declared input is needed for the following theorem.

Displayed formulas in this passage

VII-I@1#eq:triangle-input · line 1368
$T_{ϵ}^{ρ} = lo g (ϵ^{- 1}) r + U_{m} + κ_{ρ} r + O_{D^{'}} (ϵ (1 + ∣ lo g ϵ ∣)) .$
VII-I@1#eq:triangle-residue · line 1377
$⟨ r, f \otimes g \otimes f^{'} ⟩ A_{f} (y, n) = \frac{1}{64 ( 2 π ) ^{3}} \int g (y) \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n) d^{4} y, = \int f (t, y + (t + y^{0}) n) d t .$
unlabeled · line 1337
$h_{C} = - h, ϕ_{C} = ϕ$
unlabeled · line 1341
$E_{0}^{C} C_{C} h_{C} = E_{0} C h .$
unlabeled · line 1354
$W_{g} = \int g (y) h_{00} (y)^{2} d^{4} y, T_{ϵ}^{ρ} = ev_{0} [ℏ^{3}] (B_{f, ϵ} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ϵ})$
unlabeled · line 1392
$u_{μ}^{ρ} = U_{m} + (κ_{ρ} + lo g μ) r .$
unlabeled · line 1396
$u_{μ}^{a} - u_{μ}^{b} = c_{ab} r, c_{ab} = \int_{0}^{\infty} \frac{ϑ _{a} ( u ) ^{4} - ϑ _{b} ( u ) ^{4}}{u} d u \in R .$

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