Manuscript X-C@3 · X-C-v3.tex · section · 2260f5f8

Frozen sheet and C1: the copied second vertex

Frozen sheet and C1: the copied second vertex · lines 570–669

X-C@3#eq:density, X-C@3#eq:vertices, X-C@3#eq:Euler-coefficients, X-C@3#eq:Hessians

Write $H = tr_{η} h$ . The copied density coefficients are

b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} H, = \frac{1}{2} H η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} H^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{α} h^{α ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .

Set

A_{2} [h, k] = a_{1} [k] + a_{2} [h], B_{2} [h, k] = b_{1} [k] + b_{2} [h] .

These are the $κ^{2}$ coefficients of $- q q^{- 1}$ and $- q$ , respectively.

The frozen switched density is $(1 - g) L_{η} + g L_{q}$ . Its two interaction coefficients are

V_{1, g} [h, u] V_{2, g} [h, k, u] = \frac{1}{2} \int g (a_{1}^{μν} [h] \partial_{μ} u \partial_{ν} u - m^{2} b_{1} [h] u^{2}) d^{4} x, = \frac{1}{2} \int g (A_{2}^{μν} [h, k] \partial_{μ} u \partial_{ν} u - m^{2} B_{2} [h, k] u^{2}) d^{4} x .

The second vertex is therefore fixed by the supplied density before either new map is defined.

The divergence-form coefficients, including switch derivatives, are

L_{1, g} [h] u = L_{2, g} [h, k] u = g [\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} u) + m^{2} b_{1} [h] u] + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u, g [\partial_{μ} (A_{2}^{μν} [h, k] \partial_{ν} u) + m^{2} B_{2} [h, k] u] + (\partial_{μ} g) A_{2}^{μν} [h, k] \partial_{ν} u .

They are formally self-adjoint, with smooth compact coefficients. Neither is obtained by multiplying an unswitched Euler equation after variation by $g$ .

Compact integration by parts gives the sign ledger

S_{0}^{''} = - P_{m}, V_{1, g}^{'} [u] = - L_{1, g} u, V_{1, g}^{''} = - L_{1, g}, V_{2, g}^{'} [u] = - L_{2, g} u, V_{2, g}^{''} = - L_{2, g} .

The positive Euler coefficients are not the variational Hessians. This resolves the inconsistent Hessian wording in X-A's handoff in accordance with its explicit Euler-sign audit.

{}1.12

Item	Declaration
Regulator	Same smooth ramp, same fixed $ε > 0$
Field content	Scalar quantum; $h, k$ classical and off shell
Second vertex	$V_{2, g}$ in (eq:vertices), switched before variation
New maps	$R_{1} (V_{2, g}; F_{c})$ and $\frac{1}{2} R_{2} (V_{1, g}, V_{1, g}; F_{c})$ , on linear representatives
Normalization	Their sum equals the copied $\int c ρ_{2, g}$
Ward order	Completed observable through $κ^{2}$ on $g_{U}$
Topology	Smooth backgrounds and compact-test topology
Not opened	Nonlinear-insertion $R_{2}$ , $T_{2}$ , $T_{3}$ , quantum metric, sharp limit, two-point responses

Displayed formulas in this passage

X-C@3#eq:density · line 575
$b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} H, = \frac{1}{2} H η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} H^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{α} h^{α ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .$
X-C@3#eq:vertices · line 598
$V_{1, g} [h, u] V_{2, g} [h, k, u] = \frac{1}{2} \int g (a_{1}^{μν} [h] \partial_{μ} u \partial_{ν} u - m^{2} b_{1} [h] u^{2}) d^{4} x, = \frac{1}{2} \int g (A_{2}^{μν} [h, k] \partial_{μ} u \partial_{ν} u - m^{2} B_{2} [h, k] u^{2}) d^{4} x .$
X-C@3#eq:Euler-coefficients · line 615
$L_{1, g} [h] u = L_{2, g} [h, k] u = g [\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} u) + m^{2} b_{1} [h] u] + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u, g [\partial_{μ} (A_{2}^{μν} [h, k] \partial_{ν} u) + m^{2} B_{2} [h, k] u] + (\partial_{μ} g) A_{2}^{μν} [h, k] \partial_{ν} u .$
X-C@3#eq:Hessians · line 634
$S_{0}^{''} = - P_{m}, V_{1, g}^{'} [u] = - L_{1, g} u, V_{1, g}^{''} = - L_{1, g}, V_{2, g}^{'} [u] = - L_{2, g} u, V_{2, g}^{''} = - L_{2, g} .$
unlabeled · line 588
$A_{2} [h, k] = a_{1} [k] + a_{2} [h], B_{2} [h, k] = b_{1} [k] + b_{2} [h] .$