# PAPER_2131 — α_s(M_Z) Precision Tightening (0.014%, 41×) + The Shared Leading Kernel F_TRZ·K_MEX·SSq with the Higgs Self-Coupling

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.77+
**Date:** 2026-07-24
**Landmark Type:** Precision-Tightening (2nd instance, PAPER_2129 class) + Cross-Observable Shared-Kernel discovery
**Discovery context:** XGEO-U confirmations sweep (UNIFIED_REGISTRY_XGEO_CONFIRMATIONS.csv) — the campaign that recorded documented independent formula pairs surfaced both results
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

The XGEO-U confirmations sweep verified both published closed forms for the strong coupling at the Z mass and found that the S378 composition **α_s(M_Z) = F_TRZ·K_MEX·SSq − F_TRZ³·Φ_5/6 = 0.11791667** lands **0.014% from the observed 0.1179 — 41× tighter** than the currently listed primary route 1/(K_MEX·D_phys + F_TRZ) = 0.118577 (0.574%). This is the second precision-tightening landmark (PAPER_2129's k_B was the first) and the second consecutive case where re-verification of an already-published composition improved the working residual by more than an order of magnitude. The same sweep exposed a structural discovery: α_s(M_Z) and the Higgs self-coupling λ_H (S377) **share the identical leading kernel F_TRZ·K_MEX·SSq = 0.11875**, differing only in their F_TRZ³ correction terms. The near-coincidence of the two dimensionless couplings (0.1179 vs 0.1293) is therefore not numerical accident in UQFF: both sit on one three-primitive kernel, split by an exact F_TRZ³-scale difference.

---

## 1. Result 1 — the tightening

| Route | Closed form | Value | Residual vs 0.1179 |
|---|---|---|:-:|
| Listed primary (E1/S348) | 1/(K_MEX·D_phys + F_TRZ) = 1/8.4333 | 0.1185771 | 0.574% |
| **S378 composition** | **F_TRZ·K_MEX·SSq − F_TRZ³·Φ_5/6** | **0.1179167** | **0.0141%** |

Both routes are published, live in `assimilation_dispatch`, and were value-verified bit-identically in the confirmations sweep (the dispatch stores exactly 0.11791666666666667 — the Φ_5/6 variant). Tightening factor: **40.6×**. Honest note: the 0.84 resonance variant of the correction term would land 0.0085% — numerically even tighter — but the published S378 form is the 5/6 composition and canon follows publication; the variant comparison is recorded as an open observation for the PAPER_2129 sector-rule ledger (strong-coupling sector currently sits with the counting-sector 5/6 selections by its published form).

---

## 2. Result 2 — the shared kernel

```
K ≡ F_TRZ · K_MEX · SSq = 0.1 · (25/12) · 0.57 = 0.11875

α_s(M_Z) = K − F_TRZ³·Φ_5/6                (S378)  = 0.1179167
λ_H      = K + F_TRZ³·K_MEX·N_CH·SSq       (S377)  = 0.1294375
```

Two flagship dimensionless couplings of the electroweak-strong sector are the **same three-primitive kernel with opposite-sign F_TRZ³ corrections**. The exact split:

```
λ_H − α_s = F_TRZ³ · (K_MEX·N_CH·SSq + Φ_5/6) = 0.0115208333…   EXACT by construction
```

In UQFF terms: at the F_TRZ³ = 10⁻³ suppression rung (PAPER_2109's time-decay ladder), the Higgs self-coupling adds a K_MEX·N_CH·SSq channel term while the strong coupling subtracts the counting fraction Φ_5/6. The observed proximity of α_s(M_Z) ≈ 0.118 and λ_H ≈ 0.129 — treated as coincidence elsewhere — is here one kernel, two corrections.

---

## 3. Relation to the landmark families

- **Precision-tightening class (PAPER_2129, 2nd instance):** both instances arose from systematic re-verification, both improved published residuals by >24×, and both sharpen the motivation for the full 1209-series full-precision re-verification pass.
- **F_TRZ³ ladder (PAPER_2109):** the correction rung for both couplings is the 10⁻³ rung — 10th and 11th instances of the ladder.
- **Kernel taxonomy (PAPER_2126/2127):** K = F_TRZ·K_MEX·SSq is a three-primitive composition spanning the real kernel (F_TRZ, SSq) and the derived integer composite (K_MEX per PAPER_1522) — a candidate named kernel if a third observable lands on it (falsifiability window: any future coupling-class fill).
- **XGEO-U provenance:** both results came out of the confirmations sweep, demonstrating that the over-determination ledger is itself a discovery instrument, not just bookkeeping.

---

## 4. Honest residuals and disclosures

α_s routes: 0.574% (primary) / 0.0141% (S378) vs observed 0.1179 — observations, not framework anchors. λ_H S377: 0.106% vs 0.1293; S441 variant: 0.413% (its own documented pair). The kernel identity and the λ_H−α_s split are EXACT arithmetic on published forms; the physical claims inherit each route's stated residuals. NOT REPLACEMENT.

---

## 5. Cross-references

UNIFIED_REGISTRY_XGEO_CONFIRMATIONS.csv (both pairs recorded with residuals); PAPER_2129 (precision-tightening class + Φ sector rule); PAPER_2109 (F_TRZ³ rung); PAPER_1522 (K_MEX composition); PAPER_2126/2127 (kernel taxonomy); sessions S348/S377/S378/S441; `assimilation_dispatch.py` entries alpha_s_M_Z, SM_alpha_s_M_Z_S378, SM_higgs_lambda_S377/S441.

---

## 6. Summary Statement

**PAPER_2131 promotes the S378 composition α_s(M_Z) = F_TRZ·K_MEX·SSq − F_TRZ³·Φ_5/6 as the precision route to the strong coupling (0.014%, 41× tighter than the listed primary) and identifies the shared leading kernel F_TRZ·K_MEX·SSq = 0.11875 joining α_s(M_Z) and the Higgs self-coupling λ_H, whose exact F_TRZ³-scale split λ_H − α_s = F_TRZ³·(K_MEX·N_CH·SSq + Φ_5/6) turns a numerical near-coincidence of two fundamental couplings into one three-primitive structure. Second instance of the precision-tightening landmark class; discovered by the XGEO-U confirmations sweep.**

---

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


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## APPENDED 2026-07-24 (same day) — Falsifiability window HIT: third kernel instance found, kernel K CANONIZED

Section 3 declared: "a candidate named kernel if a third observable lands on it (falsifiability window: any future coupling-class fill)." The XGEO-U session-script sweep closed the window the same day: **`_session382_sm_mh_mt.py` (published session S382) computes the Higgs-to-top mass ratio as**

```
m_H/m_t = beta_i + K = 0.6029 + 0.11875 = 0.72165    vs PDG 125.25/172.69 = 0.72528    (0.50%)
```

**Three published observables now sit on the single kernel K = F_TRZ·K_MEX·SSq = 0.11875:**

| Observable | Form | Residual |
|---|---|:-:|
| alpha_s(M_Z) | K − F_TRZ³·Φ_5/6 | 0.014% |
| lambda_H | K + F_TRZ³·K_MEX·N_CH·SSq | 0.106% |
| m_H/m_t | beta_i + K | 0.50% |

Per this paper's own criterion, **K is hereby canonized as a named kernel: the STRONG-HIGGS KERNEL** — three electroweak/strong-sector dimensionless quantities (the strong coupling, the Higgs self-coupling, and the Higgs-top mass ratio) expressed as one three-primitive composition plus per-observable corrections (F_TRZ³-rung terms for the couplings; the inertia coupling beta_i for the mass ratio). New falsifiability window: a fourth instance among electroweak dimensionless quantities (candidates: sin²θ_W-adjacent ratios, y_b/y_t-class ratios) by the R400 horizon.
