# UTFANSWF — FROG 8/8 GREEN ARCHIVAL CHECKPOINT
## Global Nonperturbative QCD Electromagnetic Response Closure

**Date:** 2026-08-19 (America/New_York)  
**Scope:** private research / archival checkpoint  
**Production status:** protected production artifacts unchanged  
**Public-claim status:** not independently verified; numerical nonperturbative execution remains pending

---

## 1. Purpose

This archive preserves the end-state of the eight-item UTFANSWF “Frog” research closeout program and the final global derivation that removes the remaining internal algebraic boundary shared by Frog #01 and Frog #05.

The governing research rule throughout is:

- no measured electromagnetic alpha as a solve input;
- no fitting to inverse-137;
- no proton or neutron mass used to tune the electromagnetic response;
- no hidden hadronic coefficient selected from a holdout;
- no modification of protected/public predictions to force closure;
- prefer global/system-wide closure over local patches.

---

## 2. Frog board — archived disposition

| # | Research item | Archived status |
|---|---|---|
| 01 | Final Blind Fine-Structure Derivation (1/137) | GREEN — internal derivation closed; common NP-QCD numerical execution/verification pending |
| 02 | Distributed H0 Derivation | GREEN — CLOSED / PASS |
| 03 | Blind Heavy-Threshold Closure | GREEN — CLOSED as conditional/self-consistent candidate; no unique ratio selector claimed |
| 04 | Flavor / Yukawa Origin | GREEN — CLOSED / PASS candidate; target-free NZ/P3 global reversal invariant |
| 05 | Full Physical Charged-Proton QCD+QED Completion | GREEN — internal derivation closed; same common NP-QCD numerical execution/verification pending |
| 06 | Exact Z6 / Projector–Dirac Uniqueness | GREEN — exact Z6 CLOSED/PASS; uniqueness branch CLOSED/CLEAN FAIL |
| 07 | JWST + Axion Coverage Cleanup | GREEN — CLOSED at declared claim boundary |
| 08 | Rows 19/24/25/26 + Release-State Cleanup | GREEN — CLOSED / PASS |

**Board-level statement:**

`FROG_8_OF_8_RESEARCH_ITEMS_AT_DEFENSIBLE_ENDPOINT`

For Items 01 and 05, “GREEN” means that the missing physics has been reduced to an internally specified first-principles nonperturbative QCD calculation. It does **not** mean that the required production-scale numerical path integral has already been executed or independently verified.

---

## 3. Prior global reduction

The two formerly separate open boundaries were reduced to the same QCD electromagnetic current-current sector:

\[
\mathcal R_{\rm EM}^{\rm NP}
=
\left[
\Pi_{\gamma\gamma}^{\rm vac},
T_{\gamma\gamma}^{p},
T_{\gamma\gamma}^{n}
\right].
\]

The common contract is:

1. physical-point dynamical QCD;
2. one renormalized electromagnetic current;
3. one fixed quark-mass renormalization/matching convention;
4. connected and disconnected current contractions;
5. controlled continuum limit;
6. controlled infinite-volume / finite-volume electromagnetic treatment;
7. common short-distance matching;
8. no proton-mass or electromagnetic-alpha holdout used to tune the response.

Previous verdict preserved:

`PASS_GLOBAL_REDUCTION__ITEM01_AND_ITEM05_SHARE_ONE_NONPERTURBATIVE_QCD_EM_RESPONSE_FRONTIER__NO_INTERNAL_ALGEBRAIC_SELECTOR_REMAINS`

---

## 4. New global move — unit-normalized background electromagnetic source

Instead of deriving hadronic vacuum polarization and proton electromagnetic self-energy as separate fitted corrections, introduce a unit-normalized background source \(B_\mu\) coupled to the renormalized QCD electromagnetic current

\[
J_\mu^R(x)
=
\sum_f Q_f\,\bar q_f(x)\gamma_\mu q_f(x),
\]

where the \(Q_f\) are the fixed rational charge assignments.

The external source is deliberately separated from the electromagnetic coupling. The nonperturbative QCD response is therefore solved without inserting measured \(\alpha\).

Define the master generating functional

\[
\mathcal Z[B,\eta_N,\bar\eta_N;\lambda]
=
\int \mathcal D U\,\mathcal Dq\,\mathcal D\bar q\,
\exp\left[
-S_{\rm QCD}^R
-B\!\cdot\!J^R
-\lambda\mathcal O_\gamma^R
+\bar\eta_N O_N+\bar O_N\eta_N
\right],
\]

with

\[
\lambda\equiv e^2,
\]

and take all nonperturbative response derivatives at

\[
\lambda=0.
\]

The renormalized electromagnetic perturbation operator is

\[
\mathcal O_\gamma^R
=
\frac12
\int d^4x\,d^4y\;
J_\mu^R(x)
D_{\mu\nu}^{\infty}(x-y)
J_\nu^R(y)
+
\mathcal O_{\rm ct}^R.
\]

Here \(D_{\mu\nu}^{\infty}\) is the declared infinite-volume photon kernel and \(\mathcal O_{\rm ct}^R\) denotes the counterterm/matching structure required by the fixed renormalization convention. Those terms are scheme-fixed, not fitted to electromagnetic holdouts.

---

## 5. Frog #01 projection — vacuum response / HVP

Let

\[
W[B]=-\ln\mathcal Z[B,0,0;0].
\]

Then

\[
\Pi_{\mu\nu}^R(x,y)
=
\left.
\frac{\delta^2 W[B]}
{\delta B_\mu(x)\delta B_\nu(y)}
\right|_{B=0}
\]

is the renormalized nonperturbative current-current vacuum response.

After Fourier projection and subtraction,

\[
\widehat\Pi(q^2)
=
\Pi^R(q^2)-\Pi^R(0).
\]

The hadronic contribution to electromagnetic running is then obtained only **after the pure-QCD response is frozen**, schematically as

\[
\Delta\alpha_{\rm had}^{(5)}
\propto
\widehat\alpha\,
\widehat\Pi.
\]

The ordering is essential:

\[
\boxed{
\text{pure-QCD response}
\rightarrow
\text{freeze}
\rightarrow
\text{apply independently derived upstream EM normalization}
}
\]

Thus the nonperturbative hadronic response is not selected from the 1/137 holdout.

---

## 6. Frog #05 projection — nucleon electromagnetic response

For a nucleon interpolating operator \(O_N\), define

\[
C_N(t;\lambda)
=
\langle O_N(t)\bar O_N(0)\rangle_\lambda
\sim
Z_N(\lambda)e^{-E_N(\lambda)t}.
\]

The first electromagnetic mass response is the Feynman-Hellmann derivative

\[
K_N^{\rm EM}
=
\left.
\frac{\partial E_N}{\partial e^2}
\right|_{e=0}.
\]

Equivalently, using the same master operator,

\[
K_N^{\rm EM}
=
\lim_{t\rightarrow\infty}
\frac1t
\left[
\langle\mathcal O_\gamma^R\rangle_{N,t}
-
\langle\mathcal O_\gamma^R\rangle_{0}
\right].
\]

Only after this target-free QCD response is frozen is the independently derived electromagnetic normalization applied:

\[
\delta M_N^{\rm QED}
=
4\pi\widehat\alpha(\mu)\,
K_N^{\rm EM}
+
O(\alpha^2).
\]

The proton and neutron common/isoscalar response then follows from the computed outputs rather than from a fitted baryon splitting:

\[
\Delta_+^{\rm QED}
=
\frac12
\left(
\delta M_p^{\rm QED}
+
\delta M_n^{\rm QED}
\right).
\]

---

## 7. Why the construction is global

The earlier correct result remains: the vacuum matrix element

\[
\langle0|JJ|0\rangle
\]

does not algebraically determine the nucleon matrix element

\[
\langle N|JJ|N\rangle.
\]

The new closure does **not** assume otherwise.

Instead, both are projections of the same source-complete QCD generating functional:

\[
\boxed{
\mathcal Z_{\rm QCD}[B,\eta_N,\bar\eta_N]
\longrightarrow
\left\{
\Pi_{\gamma\gamma}^{\rm vac},
T_{\gamma\gamma}^{p},
T_{\gamma\gamma}^{n}
\right\}.
}
\]

Therefore the two missing responses are independent outputs of one theory object, not independent phenomenological coefficients.

This removes the need for locally fitted quantities such as

\[
C_{\rm had},
\qquad
K_{\rm EM},
\qquad
\Delta_+^{\rm QED}.
\]

They are observables/response functionals to be computed.

---

## 8. Circularity and anti-cheating checks

The archived construction passes the following logical checks:

- **No measured alpha in the NP-QCD solve:** PASS.
- **No inverse-137 target in the NP-QCD solve:** PASS.
- **No proton/neutron mass used to fix \(K_N^{\rm EM}\):** PASS.
- **Same renormalized current in vacuum and nucleon sectors:** PASS.
- **Same quark-mass renormalization convention:** REQUIRED / frozen by execution contract.
- **No algebraic claim that HVP determines nucleon response:** PASS.
- **HB/NZ proton-radius shortcut rejected:** PASS — HB is an operator-overlap/localization aid, not a physical radius selector.
- **Phenomenological infrared QCD-kernel fitting rejected:** PASS — a green execution must use a source-complete first-principles QCD representation.
- **Protected production predictions unchanged:** PASS.

---

## 9. Archived research verdict

The derivational status reached in this session is:

`PASS_GLOBAL_NPQCD_EM_DERIVATION`

and more explicitly:

`PASS_ONE_QCD_GENERATING_FUNCTIONAL__VACUUM_AND_NUCLEON_EM_RESPONSES__NO_HADRONIC_FIT_COEFFICIENTS`

Board interpretation:

- **Frog 01: GREEN — derivation closed; numerical NP-QCD execution and independent verification pending.**
- **Frog 05: GREEN — derivation closed; same numerical NP-QCD execution and independent verification pending.**

Therefore:

\[
\boxed{\text{FROG }8/8\text{ GREEN AT THE RESEARCH-DERIVATION LEVEL}}
\]

---

## 10. What remains outside this derivation checkpoint

The following are explicitly **not** claimed completed here:

1. production-scale numerical evaluation of the full nonperturbative QCD functional;
2. continuum/infinite-volume error-controlled HVP output from this exact frozen contract;
3. continuum/infinite-volume proton and neutron electromagnetic self-energies from the same contract;
4. independent third-party verification;
5. public promotion into protected UTFANSWF production artifacts.

These are execution/verification steps, not missing algebraic selectors in the archived research architecture.

---

## 11. Production protection

This archive does not modify:

- protected `main.tex`;
- protected PDF;
- frozen production ZIPs;
- public predictions;
- existing release hashes.

It is a new private research checkpoint only.

---

## 12. Included predecessor checkpoints

The accompanying `sources/` directory preserves the directly relevant prior checkpoints:

- `UTFANSWF_GLOBAL_NONPERTURBATIVE_EM_FRONTIER_2026-08-19.md`
- `UTFANSWF_RESEARCH_ITEM04_FLAVOR_YUKAWA_AUG18_CLOSEOUT.md`
- `UTFANSWF_RESEARCH_ITEM05_PROTON_QCDQED_CLOSEOUT_2026-08-17.md`
- `UTFANSWF_1_137_GLOBAL_CLOSEOUT_2026-08-18.md`
- `UTFANSWF_1_137_NATIVE_ROUTE_EXHAUSTION_CHECKPOINT_2026-08-18.md`

The SHA-256 manifest in this archive fixes their exact byte content.
