# UTFANSWF v23 — Row 19 → Row 23 Canonical Gravity/Scalar/GW Closeout

Date: 2026-08-20 (America/New_York)
Role: integrated pre-PDF reproducibility hardening; protected main.tex/main.pdf unchanged.

## Verdict

`PASS_ROW19_TO_ROW23_CANONICAL_GRAVITY_SCALAR_GW_BLOCK`

The canonical black-hole tensor branch is now connected executable end-to-end from the audited Row-19 Einstein-Hilbert FRG fixed point through the scalar-sector stability projection to the Row-23 Kerr-QNM benchmark.

## 1. Audited gravity fixed point

The active v23 Row-19 Einstein-Hilbert/Litim executable gives

\[
g_*=0.875888555049176,
\qquad
\lambda_*=0.156604266763380,
\]

with two UV-relevant directions and zero beta-function residual at numerical precision.

## 2. Scalar portal projection

The explicit v23 scalar beta functions were evaluated on the audited gravity fixed point rather than on the inconsistent later printed gravity subsystem.

Two scalar fixed points are recovered. Exactly one is bounded below in the printed quartic truncation:

\[
\tilde\lambda_{\phi *}=1.78396271477,
\qquad
\tilde m_*^2=-0.179781132338.
\]

It has a stable stationary pair of minima with positive fluctuation mass. These UV fixed-point coordinates are used only to establish an admissible stable scalar sector; they are not identified with numerical astrophysical IR vacuum parameters.

## 3. R^2 metric-variation correction

The canonical action contains

\[
\mathcal L_{R^2}=\frac{\alpha_R}{12M^2}R^2.
\]

For general \(f(R)\),

\[
\delta(\sqrt{-g}f)
=\sqrt{-g}\left[
f'(R)R_{\mu\nu}-\frac12 f(R)g_{\mu\nu}
-\nabla_\mu\nabla_\nu f'(R)+g_{\mu\nu}\Box f'(R)
\right]\delta g^{\mu\nu}.
\]

Substitution therefore gives

\[
\boxed{
\mathcal H^{(R^2)}_{\mu\nu}
=\frac{\alpha_R}{12M^2}
\left[
2RR_{\mu\nu}-\frac12R^2g_{\mu\nu}
-2\nabla_\mu\nabla_\nu R+2g_{\mu\nu}\Box R
\right]
}
\]

or equivalently

\[
\boxed{
\mathcal H^{(R^2)}_{\mu\nu}
=\frac{\alpha_R}{6M^2}
\left[
RR_{\mu\nu}-\frac14R^2g_{\mu\nu}
-\nabla_\mu\nabla_\nu R+g_{\mu\nu}\Box R
\right].
}
\]

The older appendix prints the first bracket with prefactor \(\alpha_R/(6M^2)\), which is a factor of two too large. This is a manuscript correction for v23. It does not alter the TT Kerr conclusion because the entire \(R^2\) contribution vanishes in the Ricci-flat TT projection \(R=\delta R=0\).

The corrected vacuum trace equation carries the scalaron mass normalization

\[
\boxed{m_s^2=\frac{M_{\rm Pl}^2M^2}{\alpha_R}}
\]

under the canonical normalization.

## 4. Canonical tensor map

For a stationary constant-scalar background,

\[
\nabla_\mu\phi_0=0,
\qquad
V'(\phi_0)=0,
\]

with the constant vacuum energy absorbed into the effective cosmological constant. The canonical action is Lorentz covariant and contains \(R^2\), but no \(R_{\mu\nu}R^{\mu\nu}\) term in the equation of record.

For the canonical tensor channel:

\[
\boxed{\alpha_1=0},
\qquad
\boxed{c_T^2=1},
\qquad
\boxed{m_T^2=0}.
\]

For constant background curvature \(R_0\),

\[
\boxed{M_*^2=M_{\rm Pl}^2+\frac{\alpha_RR_0}{3M^2}}.
\]

On the Ricci-flat Kerr benchmark \(R_0=0\),

\[
\boxed{M_*^2=M_{\rm Pl}^2},
\qquad
\boxed{\nu=0}
\]

for the stationary background, and the TT propagation equation reduces to

\[
\boxed{\delta G_{\mu\nu}^{TT}=0}.
\]

The \(R^2\) sector adds a scalaron trace mode, not a new massive spin-2 pole.

Thus the canonical tensor propagation speed is exactly luminal and the leading relative spin-2 propagation delay is zero. The older schematic detector-band \(c_T\) correction should not be presented as the canonical \(R+R^2\) result unless an additional higher-derivative tensor operator is explicitly introduced and derived.

## 5. Row 23 handoff

The regular future-horizon condition remains purely ingoing:

\[
\mathcal D_{\rm HB}=-i\hat\omega.
\]

Therefore

\[
\boxed{\mathcal R_{\rm NZH}=0},
\qquad
\boxed{\epsilon_1=0},
\qquad
\boxed{\Upsilon=0}
\]

on the canonical branch.

The Row-23 Kerr QNM benchmark remains PASS.

A nonzero reflective NZH response would require additional independently derived dynamics and is not part of this canonical branch.

## 6. Global regression

The working-copy integration changed no thresholds and no protected production artifact.

Fresh command:

`python -m harness.run_all_gates`

Result:

- canonical gates: 18
- PASS: 18
- non-PASS: 0
- overall blocking verdict: PASS

Key statuses:

- Row 19: `PASS_ROW19_EXECUTABLE_CORE__CANONICAL_GRAVITY_SCALAR_GW_MAP_CLOSED__FULL_CURVATURE_JACOBIAN_AND_GENERALIZED_GW_EXTENSION_OPEN`
- Row 22: `PASS_NZ_SLACKS_AND_SPECTRAL_GUARD`
- Row 23: `PASS_KERR_QNM_BENCHMARK__CANONICAL_ZERO_ECHO_AMPLITUDE_CLOSED`

## 7. Remaining boundary

This closeout is deliberately scoped. It does not claim a general arbitrary-background FRG solution containing \(R_{\mu\nu}R^{\mu\nu}\) and a source-complete nonminimal \(\phi R\) coupling. Those generalized extensions are not required for the canonical Row-23 black-hole tensor branch and may remain future research.

## Final disposition

`PASS_CANONICAL_ROW19_TO_ROW23_GRAVITY_SCALAR_GW_CLOSURE__GENERALIZED_CURVATURE_NONMINIMAL_EXTENSION_NOT_REQUIRED_FOR_V23_BLACK_HOLE_BRANCH`
