# UTFANSWF v23 — Pre-PDF Global Cross-Sector Closure

**Date:** 2026-08-20  
**Role:** pre-PDF research hardening / global closure  
**Production status:** protected v23 bundle, `main.tex`, and `main.pdf` unchanged

## Verdict

`PASS_PRE_PDF_GLOBAL_CROSS_SECTOR_CLOSURE`

Fresh canonical regression after the working-copy integration: **18/18 PASS, 0 blocking failures**.

## 1. Axion ↔ GW echo universality repair

The legacy scalar identification

\[
\mathcal R_a=\mathcal R_{\rm echo}=\Xi
\]

is too strong once the canonical TT/GW projection is derived to vanish. v23 therefore advances the universality statement from equality of unrelated scalar channel outputs to a common **response-operator law**.

Let the normalized global HB/NZH response be a passive positive operator \(\mathbb X_\Sigma\) on the direct-sum channel space, with sector projector \(P_c\). Then

\[
\boxed{\Xi_c=
\frac{\langle u_c|P_c\mathbb X_\Sigma P_c|u_c\rangle}
{\langle u_c|u_c\rangle}}
\]

and

\[
\boxed{\dot S_c=\frac{A_{\Sigma,c}\kappa_{{\rm eff},c}}{2\pi}\,\Xi_c.}
\]

The universal object is the response law/operator, not the numerical equality of every projected channel. Equality between two channel scalars requires an independently derived intertwiner/symmetry and mapped source states. None is currently derived between the axion and TT/GW channels.

Consequences:

- canonical stationary Kerr TT projection: \(P_{TT}\mathbb X_\Sigma P_{TT}=0\);
- Row 23 therefore has \(\mathcal R_{\rm NZH}=0\), \(\epsilon_1=0\), \(\Upsilon=0\);
- Row 21 remains an independent axion channel and is **not** forced to zero;
- the old direct axion→echo amplitude back-prediction is retired unless a genuine cross-channel intertwiner is later derived.

## 2. Canonical PPN/GW tensor reduction

For the canonical no-preferred-frame EH+\(R^2\) branch:

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

Thus the old tensor speed/mass scan variables are not independent canonical parameters.

## 3. Nonstationary FLRW GW-friction map

For

\[
f(R)=\frac{M_{\rm Pl}^2}{2}R+\frac{\alpha_R}{12M^2}R^2,
\]

\[
M_*^2=2f_R
=M_{\rm Pl}^2+\frac{\alpha_R R}{3M^2}.
\]

Using the scalaron mass ownership

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

this becomes

\[
\boxed{M_*^2(z)=M_{\rm Pl}^2\left(1+\frac{R(z)}{3m_s^2}\right)}.
\]

Therefore the GW friction index is not an independent parameter:

\[
\boxed{\nu(z)=\frac{d\ln M_*^2}{d\ln a}
=-(1+z)\frac{R'(z)}{3m_s^2+R(z)}}.
\]

The luminosity-distance map follows exactly:

\[
\boxed{
\frac{d_L^{GW}(z)}{d_L^{EM}(z)}
=
\sqrt{\frac{1+R(0)/(3m_s^2)}{1+R(z)/(3m_s^2)}}.}
\]

For the stationary/Ricci-flat black-hole limit, \(\nu=0\) and the ratio is unity.

For flat \(\Lambda\)CDM,

\[
R(z)=3H_0^2\left[\Omega_m(1+z)^3+4\Omega_\Lambda\right],
\qquad
x_R\equiv\frac{H_0^2}{m_s^2}.
\]

The functional propagation map is now closed. The **numerical IR scalaron mass \(m_s\)** remains unfrozen because current v23 materials do not contain a source-complete selector for it.

## 4. Row 25 global stress consequence

The canonical theory no longer requires a detectable echo. Row 25 therefore consumes:

- Row 23 Kerr QNM benchmark: PASS;
- canonical echo amplitude: exactly zero;
- nonzero reflective echo branch: optional/noncanonical extension, not an active canonical kill-shot.

Row 25 remains PASS.

## 5. Parameter reduction

The former canonical GW set

\[
\{\alpha_1,c_T,m_T,M_*^2,\nu,\mathcal R_{\rm echo}\}
\]

reduces to

\[
\boxed{
\alpha_1=0,\quad c_T=1,\quad m_T=0,\quad
M_*^2,\nu\leftarrow m_s,\quad
\mathcal R_{\rm echo}=0.}
\]

Thus only the already-owned scalaron scale remains to set the size of nonstationary amplitude friction.

## 6. Remaining defensible boundaries

These were **not** forced closed:

- numerical IR scalaron mass \(m_s\): no source-complete selector in current v23 materials;
- generalized \(R_{\mu\nu}R^{\mu\nu}\) and nonminimal \(\phi R\) FRG: source-incomplete and not required by the canonical branch;
- nonzero reflective echo response: requires genuinely new derived boundary dynamics;
- historical Row-25 joint CCG and CIG uncertainty-width subtests: missing source-complete estimators/logs;
- Row-21 direct experimental comparison table: covers 10–60 \(\mu\mathrm{eV}\), not the full declared 6–60 \(\mu\mathrm{eV}\) band.

These are explicit future/external boundaries, not unresolved algebra in the canonical gravity/GW chain.

## Artifacts

- `tools/row19/gravity_scalar_gw_block_v23.py`
- `research/pre_pdf_global_closure/global_cross_sector_closure_v1.py`
- `results/pre_pdf_global_cross_sector_closure_v1.json`
- `harness/gate_row23_ringdown.py` (working-copy integration)
- `harness/gate_row25_stress.py` (working-copy integration)
- `full_global_closure_regression_v2.log`
