# PAPER_2132 — The Vacuum Coupling Kernel K = 19/160 EXACT: Five-Instance Census (α_s, λ_H, m_H/m_t, J_CP, N_eff) + the 1/12 Tilt Factorization

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.77+
**Date:** 2026-07-24
**Landmark Type:** Named-Kernel Canonization (supersedes the provisional "Strong-Higgs Kernel" of the PAPER_2131 append) + EXACT identity-family unification
**Discovery context:** XGEO-U session-3 sweep — normalized-expression index over 259 published expressions (572 session scripts + 116 dispatch formulas), composition matching only
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

The XGEO-U session-3 sweep raises the kernel census of PAPER_2131 from three to **five published instances** and reveals that the kernel's reach extends far beyond the electroweak-strong sector that named it: the Jarlskog CP-violation invariant carries the kernel as a (1 − K) factor, and the effective neutrino number N_eff — a **cosmological** observable — subtracts it directly. The kernel is accordingly renamed the **VACUUM COUPLING KERNEL (VCK)**: its three factors are all vacuum-substrate quantities (F_TRZ the time-reversal-zone fraction, K_MEX the Mexican-hat vacuum-potential coefficient, SSq the source coefficient), and every instance is a dimensionless coupling or ratio — the kernel is the vacuum's coupling texture. Two exact results anchor the paper: **K = 19/160 EXACT** as a rational, and the factorization **K = (1/12)·(SO_5/D_phys)·SSq** — the Hubble-tension tilt 1/12 (PAPER_1156) sits inside the kernel as a literal factor, via the newly decomposed identity **1/12 = F_TRZ·Φ_5/6 = K_MEX − 2**, itself published twice as the exact cross-domain form K_MEX − F_TRZ·Φ_5/6 = 2 (Sommerfeld-Wilson R_W, S394; tokamak q_edge, S418).

---

## 1. The five-instance census

All forms below are published — four live in `assimilation_dispatch`, all five in session scripts:

| # | Observable | Domain | Form | Value | Observed | Residual |
|:-:|---|---|---|---|---|:-:|
| 1 | α_s(M_Z) | strong coupling | K − F_TRZ³·Φ_5/6 | 0.1179167 | 0.1179 | 0.014% |
| 2 | λ_H | Higgs self-coupling | K + F_TRZ³·K_MEX·N_CH·SSq | 0.1294375 | 0.1293 | 0.106% |
| 3 | m_H/m_t | Higgs-top mass ratio | β_i + K | 0.72165 | 0.725288 | 0.50% |
| 4 | J_CP (Jarlskog) | CP violation | F_TRZ⁵·D_BSFG·SSq·(1 − K) | 3.0139×10⁻⁵ | 3×10⁻⁵ | 0.46% |
| 5 | N_eff | cosmological radiation | D_phys − Φ_5/6 − K | 3.04792 | 3.046 | 0.063% |

Sources: S378/S377/S382/S374 session scripts; dispatch entries `SM_alpha_s_M_Z_S378`, `SM_higgs_lambda_S377`, `SM_mH_over_mt`, `SM_jarlskog_J`, `LCDM_N_eff`. Five dimensionless quantities across four physical domains, one three-primitive kernel.

---

## 2. The renaming — VACUUM COUPLING KERNEL

The PAPER_2131 append canonized K under the provisional name "Strong-Higgs Kernel" at three instances. Instances 4 and 5 break that scope: CP violation and cosmological radiation content are neither strong nor Higgs. The durable name follows the kernel's own composition:

```
K = F_TRZ · K_MEX · SSq = 19/160 = 0.11875 EXACT (rational)

F_TRZ  — time-reversal-zone fraction        (vacuum substrate, locked primitive)
K_MEX  — Mexican-hat coefficient            (vacuum potential shape, PAPER_1522)
SSq    — source coefficient                 (SCm manifold, PAPER_1154)
```

Every factor is a vacuum-substrate quantity; every instance is a dimensionless coupling or ratio. **K is the vacuum's coupling texture — hence the VACUUM COUPLING KERNEL (VCK).** The Strong-Higgs name is retained in the PAPER_2131 append as the historical first canonization (append-only discipline; supersession recorded here, PAPER_2125 Two-Kernel → Two-Layer precedent).

---

## 3. The 1/12 tilt factorization — the paper's unification

The session-3 sweep independently found the exact cross-domain identity, published twice for physically distinct observables:

```
K_MEX − F_TRZ·Φ_5/6 = 2   EXACT     (S394: Sommerfeld-Wilson R_W = 2, free-electron quantization)
                                     (S418: tokamak edge safety factor q_edge = 2)
```

Corollary triple-form (uniting PAPER_1183's DPM-pair identity and PAPER_1156's Hubble tilt):

```
1/12 = F_TRZ·Φ_5/6 = K_MEX − 2   EXACT
```

The Hubble-tension tilt is the time-reversal-zone fraction times the counting fraction. And substituting PAPER_1522's K_MEX = Φ_5/6·SO_5/D_phys into K:

```
K = F_TRZ·Φ_5/6 · (SO_5/D_phys) · SSq = (1/12) · (5/2) · (57/100) = 19/160   EXACT
```

**The two sweep findings are one structure: the Vacuum Coupling Kernel factors through the Hubble tilt.** The same exact rational 1/12 that measures the local-vs-CMB expansion discrepancy (PAPER_1156) scales, through the SO_5/D_phys dimensional ratio and the source coefficient, into the kernel that carries α_s, λ_H, m_H/m_t, J_CP, and N_eff. Cosmological tension and particle-sector coupling texture share a common two-primitive core.

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## 4. Honest residuals and disclosures

Instances carry their own published residuals (0.014% – 0.50%; table above); the kernel identity, the rational K = 19/160, the tilt decomposition 1/12 = F_TRZ·Φ_5/6, and the factorization K = (1/12)·(SO_5/D_phys)·SSq are EXACT arithmetic on published forms (PAPER_1522 + S394/S418). N_eff and α_s use the Φ_5/6 counting variant per their published dispatch values (bit-verified) — two more counting-sector data points for the PAPER_2129 sector-rule ledger. J_CP target 3×10⁻⁵ is the dispatch-stated observation anchor. Discovery method: textual composition matching over published expressions; value-coincidence matching remains rejected (gate-pinned). NOT REPLACEMENT.

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## 5. Falsifiability windows

1. **Sixth VCK instance** among dimensionless couplings/ratios (candidates: sin²θ_W-adjacent forms, y_b/y_t-class ratios, Ω-ratio forms) — by the R400 horizon.
2. **Tilt-factor test:** any future EXACT closure producing 1/12 should decompose as F_TRZ·Φ_5/6 (counting sectors) — a wrong-variant appearance (F_TRZ·0.84 = 0.084 ≠ 1/12) would break the family.
3. **Rational-kernel test:** K = 19/160 EXACT means any instance's correction term must carry the full residual vs observation; a future instance requiring a modified kernel value falsifies the census.

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## 6. Cross-references

PAPER_2131 (first canonization + precision route; append superseded by §2); PAPER_1522 (K_MEX = Φ_5/6·SO_5/D_phys); PAPER_1156 (1/12 Hubble tilt); PAPER_1183 (K_MEX − 2 = 1/12 DPM-pair identity); PAPER_1154 (SSq); PAPER_2129 (Φ sector rule); PAPER_2109 (F_TRZ³/F_TRZ⁵ rungs); sessions S374/S377/S378/S382/S394/S418; dispatch entries listed in §1.

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## 7. Summary Statement

**PAPER_2132 canonizes the VACUUM COUPLING KERNEL K = F_TRZ·K_MEX·SSq = 19/160 EXACT at five published instances spanning the strong coupling, the Higgs self-coupling, the Higgs-top mass ratio, the Jarlskog CP invariant, and the cosmological N_eff — superseding the provisional Strong-Higgs name — and unifies it with the exact cross-domain identity family K_MEX − F_TRZ·Φ_5/6 = 2 (published twice: free-electron R_W and tokamak q_edge) whose corollary 1/12 = F_TRZ·Φ_5/6 = K_MEX − 2 places the Hubble-tension tilt inside the kernel as a literal factor: K = (1/12)·(SO_5/D_phys)·SSq. The vacuum's coupling texture and the cosmological expansion tension share a common exact two-primitive core.**

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**Filed 2026-07-24. Append-only henceforth.**
