Manuscript VII-D@1 · VII-D.tex · lemma · ef67c89c

Lemma: Joint-test radial asymptotic: Lemma A

The reflected radial family: normalization and Lemma A › Lemma A: exact phase and complete error budget · lines 1028–1364 · proof included or linked

VII-D@1#lem:A, VII-D@1#eq:Ar, VII-D@1#eq:timephase, VII-D@1#eq:intersectionphase, VII-D@1#eq:Icentral, VII-D@1#eq:EPdef, VII-D@1#eq:EdeltaDef, VII-D@1#eq:EomegaDef, VII-D@1#eq:taildefs, VII-D@1#eq:exacterroridentity, VII-D@1#eq:Acentralexact, VII-D@1#eq:Icentralexact, VII-D@1#eq:budgetidentity, VII-D@1#eq:amplitudeerror, VII-D@1#eq:EPbound, VII-D@1#eq:deltaexact, VII-D@1#eq:EdeltaBound, VII-D@1#eq:massbounds, VII-D@1#eq:dmass, VII-D@1#eq:EomegaBound, VII-D@1#eq:actualtail, VII-D@1#eq:tailbudget

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For every compact $K \subset X$ ,

∣ ⟨ A (r) - r, Φ ⟩ ∣ \leq C_{K, m} r^{- 1} p_{K, 64} (Φ), r \geq 1.

In particular,

∣ ⟨ A (r), Φ ⟩ ∣ \leq C_{K, m} p_{K, 64} (Φ) .

The estimate holds without a positive lower bound on $∣ t - t^{'} ∣$ .

Proof. We retain the exact phase, form an exact difference, and only then estimate its terms.

1. Exact phase. Set

δ_{r}^{ν} (α) = ∣ α - r ν ∣ - r + ν \cdot α, φ_{0} = - a ν \cdot α + b ν \cdot β .

The temporal phase in (eq:Adef) is exactly

φ_{0} + a δ_{r}^{ν} (α) - b δ_{r}^{ν} (β) + (r - ω (r)) d .

Before the spatial integration defining $H_{Φ}$ , the exact full phase in the incidence chart is

Θ_{r} = (a α - b β) \cdot (n - ν) + σ n \cdot (α - \frac{b}{a} β) - β \cdot w + a δ_{r}^{ν} (α) - b δ_{r}^{ν} (β) + (r - ω (r)) d .

Thus both angular variables and all of $(σ, w, d)$ are retained. The phase $φ_{0}$ is the leading temporal phase in Cartesian Fourier space; the spatial phase is already inside $H_{Φ}$ . One does not delete the first line of (eq:intersectionphase) in an incidence-coordinate integral.

2. Exact central–tail decomposition. Let

C_{r} = {∣ α ∣ \leq r /2, ∣ β ∣ \leq r /2}, λ_{r} = \frac{ω ( r )}{r}, J_{r} = \frac{r ^{4}}{P ^{2} P ^{'2}},

and abbreviate, within this proof,

u_{δ} = a δ_{r}^{ν} (α) - b δ_{r}^{ν} (β), u_{ω} = (r - ω (r)) d .

All integrals over $C_{r}$ or its complement below include the remaining variables, with measure

d Ω (ν) d t d s d t^{'} d^{3} y \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} .

Write $H_{Φ}$ at its full arguments from (eq:jointFT). Define

I_{cent}^{lead} E_{P} E_{δ} E_{ω} T_{tail} = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} (λ_{r} e^{i u_{ω}} - 1), = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} λ_{r} J_{r} e^{i (u_{δ} + u_{ω})}, R_{tail}^{lead} = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} .

These definitions precede all absolute-value estimates. The algebraic identity behind them is

λ_{r} J_{r} e^{i (u_{δ} + u_{ω})} - 1 = λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})} + λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}} + (λ_{r} e^{i u_{ω}} - 1) .

Consequently,

⟨ A (r), Φ ⟩ I_{cent}^{lead} = I_{cent}^{lead} + E_{P} + E_{δ} + E_{ω} + T_{tail}, = ⟨ r, Φ ⟩ - R_{tail}^{lead} .

The second equality is the full Fourier inversion proved in the ledger, minus its leading tail. In particular, the central leading term is not counted again after subtracting $r$ . With

E_{tail} = T_{tail} - R_{tail}^{lead},

the exact difference is

⟨ A (r) - r, Φ ⟩ = E_{P} + E_{δ} + E_{ω} + E_{tail} .

{3. Amplitude error $E_{P}$ .} On $C_{r}$ ,

\frac{r}{2} \leq P, P^{'} \leq \frac{3 r}{2}, \frac{r ^{2}}{P ^{2}} - 1 = \frac{∣ r - P ∣ ( r + P )}{P ^{2}} \leq C \frac{∣ α ∣}{r} .

The corresponding bound holds for $P^{'}$ . Since each ratio $r^{2} / P^{2}$ is bounded by $4$ ,

∣ J_{r} - 1∣ \leq C \frac{∣ α ∣ + ∣ β ∣}{r}, J_{r} \leq 16.

Also $λ_{r} \leq 1 + m^{2}$ for $r \geq 1$ . Using (eq:FTbound) in (eq:EPdef) gives

∣ E_{P} ∣ \leq \frac{C _{K, m}}{r} p_{K, 64} (Φ) \int_{R^{6}} \frac{∣ α ∣ + ∣ β ∣}{( 1 + ∣ α ∣ ) ^{12} ( 1 + ∣ β ∣ ) ^{12}} d^{3} α d^{3} β \leq \frac{C ^{'} _{K, m}}{r} p_{K, 64} (Φ) .

{4. Quadratic phase error $E_{δ}$ .} For $∣ α ∣ \leq r /2$ , the exact identity

δ_{r}^{ν} (α) = \frac{∣ α ∣ ^{2} - ( ν \cdot α ) ^{2}}{∣ α - r ν ∣ + r - ν \cdot α} implies ∣ δ_{r}^{ν} (α) ∣ \leq \frac{∣ α ∣ ^{2}}{r} .

The denominator is at least $r$ in this region. On a compact observation set, $a, b$ have fixed upper bounds. Therefore

∣ u_{δ} ∣ \leq \frac{C _{K}}{r} (∣ α ∣^{2} + ∣ β ∣^{2}) .

Applying $∣ e^{i u} - 1∣ \leq ∣ u ∣$ to the exact difference in (eq:EdeltaDef), and then (eq:FTbound), yields

∣ E_{δ} ∣ \leq \frac{C _{K, m}}{r} p_{K, 64} (Φ) \int_{R^{6}} \frac{∣ α ∣ ^{2} + ∣ β ∣ ^{2}}{( 1 + ∣ α ∣ ) ^{12} ( 1 + ∣ β ∣ ) ^{12}} d^{3} α d^{3} β \leq \frac{C ^{'} _{K, m}}{r} p_{K, 64} (Φ) .

No expansion of $δ_{r}^{ν}$ is made in the tail.

{5. Mass and time-difference error $E_{ω}$ .} The two relevant bounds are

ω (r) - r = \frac{m ^{2}}{ω ( r ) + r} \leq \frac{m ^{2}}{2 r}, λ_{r} - 1 = \frac{m ^{2}}{r ( ω ( r ) + r )} \leq \frac{m ^{2}}{2 r ^{2}} .

The exact factor in (eq:EomegaDef) splits as

λ_{r} e^{i u_{ω}} - 1 = (λ_{r} - 1) e^{i u_{ω}} + (e^{i u_{ω}} - 1) .

In particular,

∣ u_{ω} ∣ = ∣ (r - ω (r)) d ∣ \leq \frac{m ^{2} ∣ d ∣}{2 r} .

Only an upper bound on $∣ d ∣$ over the compact observation projection is used. At $d = 0$ , this phase error vanishes exactly. Equations (eq:FTbound) and (eq:massbounds)–(eq:dmass) give

∣ E_{ω} ∣ \leq C_{K, m} (r^{- 1} + r^{- 2}) p_{K, 64} (Φ) .

This is the required estimate through $Δ_{13}$ ; it does not use division by $d$ .

{6. Exact tail $E_{tail}$ .} Let $N = 12$ . The union bound for $C_{r}^{c}$ , together with (eq:shiftbound)–(eq:shifttail) for $ℓ = 2, k = 0$ , gives the complete majorant

∣ T_{tail} ∣ \leq C_{K, m} p_{K, 64} (Φ) prefactor r^{4} α -tail r^{1 - N} full β r^{- 2} + full α r^{- 2} β -tail r^{1 - N} \leq C^{'}_{K, m} r^{3 - N} p_{K, 64} (Φ) = C^{'}_{K, m} r^{- 9} p_{K, 64} (Φ) .

Here the prefactor bound is $r^{3} ω (r) \leq C_{m} r^{4}$ . The shifted-tail exponent is $3 - 2 + 0 - N = 1 - N$ , whereas the full shifted integral is bounded by $C r^{- 2}$ .

This estimate includes $P \approx 0$ and $P^{'} \approx 0$ , which occur in the shifted tails. Those singularities are treated by local integrability, not by a phase expansion.

For the leading tail there are no shifted denominators. Directly,

\int_{∣ α ∣ > r /2} (1 + ∣ α ∣)^{- N} d^{3} α \leq C r^{3 - N},

and similarly for $β$ . Hence

∣ R_{tail}^{lead} ∣ + ∣ E_{tail} ∣ \leq C_{K, m} r^{3 - N} p_{K, 64} (Φ) = C_{K, m} r^{- 9} p_{K, 64} (Φ) .

{7. Coordinate change: $E_{chart} = 0$ .} The estimate has been proved in Cartesian Fourier space. On a fixed incidence-coordinate patch, rewriting the same distribution with (eq:chart) introduces exactly the density (eq:density) and the phase (eq:intersectionphase). It introduces no additional error:

E_{chart} = 0.

For a direct justification, first insert compact smooth cutoffs in $α, β$ . The remaining momentum singularities $P^{- 2}, P^{' - 2}$ are locally integrable, so the ordinary coordinate change is legitimate. Remove those auxiliary cutoffs after the spatial integration defining $H_{Φ}$ ; (eq:FTbound) and Lemma lem:convolution justify that limit. This identifies the coordinate expression with the original distribution, without restricting it to a diagonal. The Jacobian has no $1/ d$ .

8. Sum of the errors. The complete budget is

term E_{P} E_{δ} E_{ω} E_{tail} E_{chart} bound divided by p_{K, 64} (Φ) C_{K, m} r^{- 1} C_{K, m} r^{- 1} C_{K, m} (r^{- 1} + r^{- 2}) C_{K, m} r^{- 9} 0

Summing in (eq:budgetidentity) proves (eq:Ar). The uniform bound on $A (r)$ follows because $r$ is an order-zero distribution.

Displayed formulas in this passage

VII-D@1#eq:Ar · line 1031
$∣ ⟨ A (r) - r, Φ ⟩ ∣ \leq C_{K, m} r^{- 1} p_{K, 64} (Φ), r \geq 1.$
VII-D@1#eq:timephase · line 1060
$φ_{0} + a δ_{r}^{ν} (α) - b δ_{r}^{ν} (β) + (r - ω (r)) d .$
VII-D@1#eq:intersectionphase · line 1070
$Θ_{r} = (a α - b β) \cdot (n - ν) + σ n \cdot (α - \frac{b}{a} β) - β \cdot w + a δ_{r}^{ν} (α) - b δ_{r}^{ν} (β) + (r - ω (r)) d .$
VII-D@1#eq:Icentral · line 1113
$I_{cent}^{lead} E_{P} E_{δ} E_{ω} T_{tail} = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} (λ_{r} e^{i u_{ω}} - 1), = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} λ_{r} J_{r} e^{i (u_{δ} + u_{ω})}, R_{tail}^{lead} = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} .$
VII-D@1#eq:EPdef · line 1113
$I_{cent}^{lead} E_{P} E_{δ} E_{ω} T_{tail} = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} (λ_{r} e^{i u_{ω}} - 1), = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} λ_{r} J_{r} e^{i (u_{δ} + u_{ω})}, R_{tail}^{lead} = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} .$
VII-D@1#eq:EdeltaDef · line 1113
$I_{cent}^{lead} E_{P} E_{δ} E_{ω} T_{tail} = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} (λ_{r} e^{i u_{ω}} - 1), = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} λ_{r} J_{r} e^{i (u_{δ} + u_{ω})}, R_{tail}^{lead} = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} .$
VII-D@1#eq:EomegaDef · line 1113
$I_{cent}^{lead} E_{P} E_{δ} E_{ω} T_{tail} = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} (λ_{r} e^{i u_{ω}} - 1), = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} λ_{r} J_{r} e^{i (u_{δ} + u_{ω})}, R_{tail}^{lead} = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} .$
VII-D@1#eq:taildefs · line 1113
$I_{cent}^{lead} E_{P} E_{δ} E_{ω} T_{tail} = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}}, = c_{0} \int_{C_{r}} e^{i φ_{0}} H_{Φ} (λ_{r} e^{i u_{ω}} - 1), = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} λ_{r} J_{r} e^{i (u_{δ} + u_{ω})}, R_{tail}^{lead} = c_{0} \int_{C_{r}^{c}} e^{i φ_{0}} H_{Φ} .$
VII-D@1#eq:exacterroridentity · line 1142
$λ_{r} J_{r} e^{i (u_{δ} + u_{ω})} - 1 = λ_{r} (J_{r} - 1) e^{i (u_{δ} + u_{ω})} + λ_{r} (e^{i u_{δ}} - 1) e^{i u_{ω}} + (λ_{r} e^{i u_{ω}} - 1) .$
VII-D@1#eq:Acentralexact · line 1153
$⟨ A (r), Φ ⟩ I_{cent}^{lead} = I_{cent}^{lead} + E_{P} + E_{δ} + E_{ω} + T_{tail}, = ⟨ r, Φ ⟩ - R_{tail}^{lead} .$
VII-D@1#eq:Icentralexact · line 1153
$⟨ A (r), Φ ⟩ I_{cent}^{lead} = I_{cent}^{lead} + E_{P} + E_{δ} + E_{ω} + T_{tail}, = ⟨ r, Φ ⟩ - R_{tail}^{lead} .$
VII-D@1#eq:budgetidentity · line 1173
$⟨ A (r) - r, Φ ⟩ = E_{P} + E_{δ} + E_{ω} + E_{tail} .$
VII-D@1#eq:amplitudeerror · line 1192
$∣ J_{r} - 1∣ \leq C \frac{∣ α ∣ + ∣ β ∣}{r}, J_{r} \leq 16.$
VII-D@1#eq:EPbound · line 1200
$∣ E_{P} ∣ \leq \frac{C _{K, m}}{r} p_{K, 64} (Φ) \int_{R^{6}} \frac{∣ α ∣ + ∣ β ∣}{( 1 + ∣ α ∣ ) ^{12} ( 1 + ∣ β ∣ ) ^{12}} d^{3} α d^{3} β \leq \frac{C ^{'} _{K, m}}{r} p_{K, 64} (Φ) .$
VII-D@1#eq:deltaexact · line 1216
$δ_{r}^{ν} (α) = \frac{∣ α ∣ ^{2} - ( ν \cdot α ) ^{2}}{∣ α - r ν ∣ + r - ν \cdot α} implies ∣ δ_{r}^{ν} (α) ∣ \leq \frac{∣ α ∣ ^{2}}{r} .$
VII-D@1#eq:EdeltaBound · line 1235
$∣ E_{δ} ∣ \leq \frac{C _{K, m}}{r} p_{K, 64} (Φ) \int_{R^{6}} \frac{∣ α ∣ ^{2} + ∣ β ∣ ^{2}}{( 1 + ∣ α ∣ ) ^{12} ( 1 + ∣ β ∣ ) ^{12}} d^{3} α d^{3} β \leq \frac{C ^{'} _{K, m}}{r} p_{K, 64} (Φ) .$
VII-D@1#eq:massbounds · line 1251
$ω (r) - r = \frac{m ^{2}}{ω ( r ) + r} \leq \frac{m ^{2}}{2 r}, λ_{r} - 1 = \frac{m ^{2}}{r ( ω ( r ) + r )} \leq \frac{m ^{2}}{2 r ^{2}} .$
VII-D@1#eq:dmass · line 1265
$∣ u_{ω} ∣ = ∣ (r - ω (r)) d ∣ \leq \frac{m ^{2} ∣ d ∣}{2 r} .$
VII-D@1#eq:EomegaBound · line 1275
$∣ E_{ω} ∣ \leq C_{K, m} (r^{- 1} + r^{- 2}) p_{K, 64} (Φ) .$
VII-D@1#eq:actualtail · line 1288
$∣ T_{tail} ∣ \leq C_{K, m} p_{K, 64} (Φ) prefactor r^{4} α -tail r^{1 - N} full β r^{- 2} + full α r^{- 2} β -tail r^{1 - N} \leq C^{'}_{K, m} r^{3 - N} p_{K, 64} (Φ) = C^{'}_{K, m} r^{- 9} p_{K, 64} (Φ) .$
VII-D@1#eq:tailbudget · line 1317
$∣ R_{tail}^{lead} ∣ + ∣ E_{tail} ∣ \leq C_{K, m} r^{3 - N} p_{K, 64} (Φ) = C_{K, m} r^{- 9} p_{K, 64} (Φ) .$
unlabeled · line 1038
$∣ ⟨ A (r), Φ ⟩ ∣ \leq C_{K, m} p_{K, 64} (Φ) .$
unlabeled · line 1051
$δ_{r}^{ν} (α) = ∣ α - r ν ∣ - r + ν \cdot α, φ_{0} = - a ν \cdot α + b ν \cdot β .$
unlabeled · line 1091
$C_{r} = {∣ α ∣ \leq r /2, ∣ β ∣ \leq r /2}, λ_{r} = \frac{ω ( r )}{r}, J_{r} = \frac{r ^{4}}{P ^{2} P ^{'2}},$
unlabeled · line 1098
$u_{δ} = a δ_{r}^{ν} (α) - b δ_{r}^{ν} (β), u_{ω} = (r - ω (r)) d .$
unlabeled · line 1106
$d Ω (ν) d t d s d t^{'} d^{3} y \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} .$
unlabeled · line 1168
$E_{tail} = T_{tail} - R_{tail}^{lead},$
unlabeled · line 1184
$\frac{r}{2} \leq P, P^{'} \leq \frac{3 r}{2}, \frac{r ^{2}}{P ^{2}} - 1 = \frac{∣ r - P ∣ ( r + P )}{P ^{2}} \leq C \frac{∣ α ∣}{r} .$
unlabeled · line 1229
$∣ u_{δ} ∣ \leq \frac{C _{K}}{r} (∣ α ∣^{2} + ∣ β ∣^{2}) .$
unlabeled · line 1259
$λ_{r} e^{i u_{ω}} - 1 = (λ_{r} - 1) e^{i u_{ω}} + (e^{i u_{ω}} - 1) .$
unlabeled · line 1312
$\int_{∣ α ∣ > r /2} (1 + ∣ α ∣)^{- N} d^{3} α \leq C r^{3 - N},$
unlabeled · line 1335
$E_{chart} = 0.$
unlabeled · line 1350
$term E_{P} E_{δ} E_{ω} E_{tail} E_{chart} bound divided by p_{K, 64} (Φ) C_{K, m} r^{- 1} C_{K, m} r^{- 1} C_{K, m} (r^{- 1} + r^{- 2}) C_{K, m} r^{- 9} 0$

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section: Lemma A: exact phase and complete error budget