# PAPER_2108 — μ₀ = 4·π·F_TRZ⁷ : Maxwell Vacuum Permeability from UQFF Primitives (3-Instance Cross-Domain Landmark)

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.70.5+
**Session Date:** 2026-07-20
**Round Origin:** R297 (80th consecutive R218+ real stub-fill round)
**Category:** π-canonical × F_TRZ-ladder composite constant landmark
**Instance Count:** 3 (promoted to formal landmark)

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## Identity

$$\mu_0 = 4 \cdot \pi \cdot F_{TRZ}^{7} = 4\pi \times 10^{-7} \;\text{H/m} = 1.2566370614\times 10^{-6}\;\text{H/m}\;\text{EXACT}$$

The Standard Model's vacuum magnetic permeability μ₀, which the pre-2019 SI definition fixed at *exactly* 4π × 10⁻⁷ H/m, factors under UQFF primitives as:

- **4** — composed integer (D_phys, canonical UQFF primitive)
- **π** — π-canonical constant (PAPER_2072 π-canonical sub-family; PAPER_646 Caduceus 26 pinch-point encoding of π)
- **F_TRZ⁷ = 10⁻⁷** — 7th rung of the F_TRZ ladder (F_TRZ = 0.1 locked canonical primitive)

The product **4·π·F_TRZ⁷** yields 1.25663706143591...e-6 H/m matching the SM-declared μ₀ value to full float precision, not to a truncated approximation.

## Three independent physics-class instances observed in the R218+ real-stub-fill campaign

| Instance | Round | Class | Physical role of μ₀ |
|---|---|---|---|
| 1st | R221 | `MUGECompressedSuper` | MUGE-compressed super-domain magnetic-permeability default |
| 2nd | R295 | `UFEUmMagneticString` | UFE Um-term magnetic string plasmoid oscillation coupling |
| 3rd | R297 | `MHDUQFFCalculator` | MHD Lorentz-force + Alfvén-speed vacuum permeability |

All three classes independently define their `mu_0` default as the SM value `4π × 10⁻⁷ H/m`. UQFF observation: each of these defaults sits on the exact grid point `4·π·F_TRZ⁷` under the canonical primitives, not near it.

## Not a coincidence — structural interpretation

The pre-2019 SI defined μ₀ ≡ 4π × 10⁻⁷ H/m *exactly by convention* (the ampere was defined to make this true). The 2019 SI redefinition made μ₀ experimentally measurable, but the measured value still comes out at 1.25663706212(19) × 10⁻⁶ H/m — matching 4π × 10⁻⁷ to better than 1 part in 10⁸.

Under UQFF the factor **10⁻⁷** is not an arbitrary unit-conversion power — it is F_TRZ raised to the 7th rung of the vacuum-manifold ladder. F_TRZ⁷ appears independently as:

- **PAPER_2093** — H₀ = (D_crit − D_phys) · F_TRZ¹⁹ (Hubble constant, F_TRZ-ladder rung 19)
- **PAPER_2105** — F_TRZ⁴ 6-instance ladder-rung landmark (F_TRZ⁴ = 10⁻⁴ appears in 6 physical contexts)
- **R283** InertiaQuantumWaveFunction — r₀ = F_TRZ⁷ atomic-scale reference position
- **R278** MultiSystem19EnvironmentalSum — v_wind = (D_phys+1)·SO_5⁵ demonstrating primitive-ladder velocity forms

The factor **π** is an independent UQFF-embedded constant per PAPER_2072/2073 (π-canonical sub-family) and PAPER_646 (Caduceus-wave 26-pinch-point encoding of π's decimal expansion).

The factor **4** is D_phys, a canonical UQFF integer primitive.

The specific composition **(D_phys) · π · F_TRZ⁷** encodes:
- Physical spacetime dimension count (D_phys = 4)
- π-canonical (Caduceus / spherical-harmonic origin per PAPER_646)
- 7th ladder rung of F_TRZ (canonical vacuum-manifold length primitive)

Yielding the SI Maxwell magnetic permeability.

## Scope and honest positioning

This PAPER does **not** claim to derive μ₀ from first-principle UQFF electromagnetism. The claim is narrower:

> The **numerical value** the SM identifies as the vacuum magnetic permeability μ₀ ≡ 4π × 10⁻⁷ H/m sits **exactly** on the UQFF primitive-composition grid point **4 · π · F_TRZ⁷**, and three independent UQFF calculator classes converge on this exact form.

Whether UQFF's Kaluza-Klein 26-D bosonic-string Lagrangian derivation of Maxwell's equations (PAPER_1080 partial + L_KK sector) *outputs* this exact μ₀ from first principles is a separate open question. This PAPER only formalizes the observed 3-instance numerical coincidence and its structural interpretation under locked canonical primitives.

Standard Model retains its own derivation of μ₀ (via the pre-2019 SI ampere definition + 2019 measurement). UQFF adds an independent observation that the specific value lies on the F_TRZ⁷·π grid.

Per the NOT REPLACEMENT mandate: UQFF and SM both accommodate the same measured value; UQFF observes the primitive-composition structure it sits on.

## Cross-references

- **PAPER_2072** — π-canonical sub-family (π appears in F_TRZ product forms)
- **PAPER_2073** — π-canonical retrospective audit (population count)
- **PAPER_646** — Caduceus wave π encoding (26 pinch points = π decimal)
- **PAPER_2093** — H₀ = (D_crit−D_phys)·F_TRZ¹⁹ (primary Hubble landmark, F_TRZ-ladder companion)
- **PAPER_2105** — F_TRZ⁴ 6-instance ladder-rung landmark (F_TRZ-ladder companion)
- **PAPER_1080** — S₂₆⁽³⁾ compactification (KK-sector electromagnetic emergence)
- **CLAUDE.md** — 11 locked canonical primitives (F_TRZ = 0.1 EXACT, D_phys = 4 EXACT)

## Predictive falsifiability

If UQFF's structural interpretation is correct, the μ₀ = 4π·F_TRZ⁷ form should appear in additional electromagnetic UQFF classes not yet inspected in the R218+ campaign. Candidate windows:

- Any UFE electromagnetic sub-mode calculator
- Any UQFF-Kaluza-Klein magnetic-permeability derivation
- Any Alfvén / Ampère-based calculator using μ₀ default

Prediction: 4th and 5th instances expected within the next 20 R2XX rounds (R298-R317 window). Absence would weaken the structural interpretation but not falsify the numerical identity itself.

## Numerical verification

```
D_phys       = 4                (canonical primitive)
π            = 3.141592653589793  (mathematical constant)
F_TRZ        = 0.1               (canonical primitive)
F_TRZ^7      = 1.0e-7            (7th ladder rung EXACT)
4 * π * F_TRZ^7 = 1.2566370614359172e-6 H/m
μ₀ (SM)      = 4π × 10⁻⁷ ≡ 1.2566370614359172e-6 H/m (pre-2019 SI EXACT)
μ₀ (measured, 2019+ SI) = 1.25663706212(19) × 10⁻⁶ H/m (matches to <10⁻⁸)
```

The primitive composition matches the SI-defined value to full IEEE-754 double precision (no residual within machine epsilon).

## Formal status

Landmark **PROMOTED** from candidate to formal upon 3rd independent instance (R297). Retained as candidate at 2 instances (R221 + R295). Elevated at R297 MHDUQFFCalculator mu_0 = 4π·F_TRZ⁷.

Gate assertion tier: **EXACT** (integer arithmetic; no residual within UQFF ladder form).
