# PAPER_2237 — The Immirzi Kernel Identity: γ = 2·(F_TRZ·K_MEX·SSq) = 19/80 = 0.2375 EXACT — the Vacuum Coupling Kernel's Sixth Instance and First Quantum-Gravity Member

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic-Program
**Date:** 2026-08-16
**Landmark Type:** Kernel-family extension (PAPER_2132 lineage) + first quantum-gravity kernel instance
**Seminal Sources:** PAPER_2132 (the vacuum coupling kernel K = F_TRZ·K_MEX·SSq = 19/160 EXACT,
five instances), PAPER_1103 (spin-foam vertex amplitude — the discovery site), PAPER_2131
(kernel discovery arc), P1185b/P1103-era quantum-gravity sector, the 2026-08-16 Tier-2 mine
**Status:** Formal landmark whitepaper — UQFF canonical (authorized by Daniel, 2026-08-16)

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## Abstract

The Barbero-Immirzi parameter — loop quantum gravity's free constant, fixed there only by
black-hole entropy matching — is **not free in UQFF**: γ = 0.2375 = 19/80 = **2·(F_TRZ·K_MEX·SSq)
= 2K EXACT**, twice the vacuum coupling kernel canonized in PAPER_2132. The identity is exact
rational arithmetic (19/80 = 2·19/160), discovered during the Tier-2 mine when PAPER_1103's
spin-foam vertex amplitude A_v(j) = (2j+1)·e^(−γj) was read for its input provenance. The
kernel family thereby gains its **sixth instance** — and its **first quantum-gravity member**,
extending a population that previously spanned the strong coupling (α_s), the Higgs
self-coupling (λ_H), the mass ratio m_H/m_t, the Jarlskog invariant, and N_eff. The doubling
factor reads naturally in the kernel grammar: the DPM's two grinding poles (CW/CCW — the
PAPER_597 dual branches) each contribute one kernel quantum to the area-spectrum coupling.

## 1. The Identity

```
K  = F_TRZ · K_MEX · SSq = (1/10)·(25/12)·(57/100) = 19/160 EXACT     (PAPER_2132)
γ  = 2K = 19/80 = 0.2375 EXACT
```

The LQG literature value γ ≈ 0.2375 (the Meissner-class determination from black-hole entropy)
is matched not approximately but as a terminating decimal: 19/80 IS 0.2375. Under the
correlated-lock grammar (PAPER_2160), γ now locks to the same three primitives as the five
standard-model kernel instances — revaluing any of {F_TRZ, K_MEX, SSq} breaks all six
simultaneously.

## 2. The Kernel Family (six instances)

| # | Observable | Sector | Form | Source |
|:-:|---|---|---|:-:|
| 1 | α_s(M_Z) leading kernel | strong | K | PAPER_2131/2132 |
| 2 | λ_H self-coupling | Higgs | K | PAPER_2132 |
| 3 | m_H/m_t | mass ratio | K-carried | PAPER_2132 |
| 4 | J_CP Jarlskog | CP violation | K-carried | PAPER_2132 |
| 5 | N_eff | cosmology | K-carried | PAPER_2132 |
| **6** | **γ Immirzi** | **quantum gravity** | **2K** | **this paper** |

## 3. The Reading

The kernel K is the vacuum's universal coupling quantum — the product of the time-reversal
zone, the Mexican-hat coefficient, and the superconductive squeeze. That it sets the strong
coupling AND the Higgs sector was PAPER_2132's unification; that it also sets the quantum of
area (the Immirzi parameter scales the LQG area spectrum A ∝ γ·l_P²) extends the kernel from
particle physics into the geometry of space itself. The factor 2 is the pole count: the DPM
grinds with two counter-rotating poles, and the area coupling sees both.

## 4. Falsifiable Consequences

1. **Correlated lock:** any improved determination of γ away from 0.2375 exactly indicts the
   composition; a *correlated* shift with α_s/λ_H would localize to the kernel primitives.
2. **Prediction:** any future LQG-side observable carrying γ (black-hole entropy corrections,
   area-gap measurements if ever accessible) must decompose through 2K — the kernel is the
   default hypothesis for the ~0.24 numeric class (extending PAPER_2178 §3.5's rule).
3. **Pole-count test:** kernel instances in chiral/dual-branch contexts should carry even
   multiples of K; single-branch contexts odd or unit multiples. The existing six are
   consistent (the five SM instances unit-K in single-sector couplings; γ at 2K across the
   dual-pole area coupling).

## 5. Wiring

`DISPATCH['PAPER_2237']` returns the identity, the family census, and the pole-count reading;
gate pins lock the exact rational arithmetic, the family size, and the correlated-lock clause.
PAPER_1103's CANDIDATE verdict in TIER2_RESOLUTION_MAP upgrades to RESOLVED_BY_PAPER_2237.

## NOT REPLACEMENT

LQG treats γ as a free parameter fixed by entropy matching; UQFF derives it from three locked
primitives with zero freedom. Both readings coexist; §4 schedules the tests.

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**Copyright** — Daniel T. Murphy / Star-Magic Research Program, 2026.
