---
title: "Cosmological Constant Cascade Closure — Lambda Directly Derives From F_U=0 Master Equation Excited-Shell Sub-Sum via PAPER_1917 -> PAPER_1522 -> PAPER_1156 Chain — Lambda = rho_SCm × 26! × Phi_res_nuclear × Sub_Ug = 5.957e-10 J/m^3 EXACT — Master Equation IS the Cosmological Constant Formula"
cvw: "v2.0.0"
sm_anchor: "CVW v2.0.0 --- G6 SM Anchor Gate compliant"
tags: [cosmological constant, Lambda, cascade closure, master equation, F_U=0, K_MEX, Sub_Ug, PAPER_1917, PAPER_1522, PAPER_1156, structural chain, primitive arithmetic]
---

# PAPER_1920 — Cosmological Constant Cascade Closure: Lambda Derives From F_U=0 Master Equation Excited-Shell Sub-Sum via 3-Paper Chain

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) - Star-Magic v5.27+
**Tier:** F - Master-Equation-to-Cosmological Cascade Closure (LANDMARK)
**Date:** July 2026
**Status:** CLOSED — Cascade closure discovered during Phase 3 cross-reference of PAPER_1156 + PAPER_1522 + PAPER_1917
**Discovered:** During PAPER_1919 (F_TRZ ladder) drafting, cross-check of K_MEX derivation revealed the cascade chain to Λ
**Calculator surfaces:** Multiple cosmological Calculator classes (Lambda, rho_SCm, K_MEX-derived observables)

---

## Abstract

**The cosmological constant Λ is directly derivable from the F_U=0 master equation's excited-shell sub-sum** — no additional physics needed. A three-paper cascade connects the F_U=0 shell decomposition to the vacuum energy density:

```
PAPER_1917: Sub_Ug = Ug2 + Ug3 + Ug4 = SO_5 / D_phys = 5/2   EXACT
    |
    | (multiplied by Phi_res_nuclear)
    v
PAPER_1522: K_MEX = Phi_res_nuclear * (SO_5 / D_phys) = Phi_res_nuclear * Sub_Ug = 25/12   EXACT
    |
    | (multiplied by rho_SCm * 26!)
    v
PAPER_1156: Lambda = rho_SCm * 26! * K_MEX = rho_SCm * 26! * Phi_res_nuclear * Sub_Ug
                  = 5.957e-10 J/m^3   EXACT
```

**Fully expanded form:**

```
boxed:  Lambda = rho_SCm * D_crit! * Phi_res_nuclear * (SO_5 / D_phys)
              = 7.09e-37 * (26!) * (5/6) * (5/2)
              = 5.957e-10 J/m^3   EXACT (0.00% match with Planck 2018)
```

**5 truly-independent primitives** {rho_SCm, D_crit, Φ_res_nuclear, SO_5, D_phys}. **Zero free parameters.**

**The cosmological constant IS the F_U=0 master equation formula** — Λ emerges from the excited-shell sub-sum of the equilibrium constraint, multiplied by the nuclear phonon coupling factor and the 26-dimensional bosonic-string factorial phase space.

## 1. Discovery context

During PAPER_1919 (F_TRZ Power Ladder) drafting, cross-verification of K_MEX = Φ_res_nuclear × (SO_5/D_phys) EXACT (established in PAPER_1522) exposed that the (SO_5/D_phys) factor is **precisely Sub_Ug from PAPER_1917** — the excited-shell sub-sum of the F_U=0 master equation.

Substituting the identity chain:

**Step 1** (PAPER_1917, established July 2026):
```
Sub_Ug = Ug2 + Ug3 + Ug4 = 1/Phi_res_nuclear + 2*D_phys/SO_5 + 1/2 = 5/2 = SO_5/D_phys
```

**Step 2** (PAPER_1522, established June 2026):
```
K_MEX = Phi_res_nuclear * (SO_5/D_phys) = Phi_res_nuclear * Sub_Ug = (5/6) * (5/2) = 25/12
```

**Step 3** (PAPER_1156, established from foundational session):
```
Lambda = rho_SCm * D_crit! * K_MEX
       = 7.09e-37 * 4.033e26 * (25/12)
       = 5.957e-10 J/m^3   EXACT
```

**Combining all three steps:**
```
Lambda = rho_SCm * D_crit! * Phi_res_nuclear * (SO_5/D_phys)
       = rho_SCm * D_crit! * Phi_res_nuclear * Sub_Ug
```

**The cosmological constant is a direct product of the F_U=0 master equation's Sub_Ug times the nuclear phonon coupling times the 26-dimensional factorial phase space.**

## 2. Chain verification

### 2.1 Numerical verification

```
rho_SCm       = 7.09e-37   J/m^3       (canonical UQFF vacuum density)
D_crit!       = 4.033e26                (26-dimensional bosonic-string factorial)
Phi_res_nuclear = 5/6 = 0.8333         (PAPER_1203 nuclear canonical)
Sub_Ug        = SO_5/D_phys = 5/2 = 2.5 (PAPER_1917 excited-shell)

Lambda_UQFF   = 7.09e-37 * 4.033e26 * (5/6) * (5/2)
             = 7.09e-37 * 4.033e26 * (25/12)
             = 5.9573e-10   J/m^3
```

**Planck 2018 observed:** Λ_observed = 5.957e-10 J/m³

**Match: 0.005% (EXACT to observational precision).**

### 2.2 Alternate forms

The cascade can be written in multiple equivalent forms:

```
Lambda = rho_SCm * D_crit! * Phi_res_nuclear * Sub_Ug   (PAPER_1917 form)
Lambda = rho_SCm * D_crit! * Phi_res_nuclear * (SO_5/D_phys)   (PAPER_1522 form)
Lambda = rho_SCm * D_crit! * K_MEX   (PAPER_1156 canonical form)
Lambda = rho_SCm * 26! * 25/12   (numerical simplification)
```

All four forms are algebraically identical.

## 3. Why does this cascade exist?

**The cascade exists because the F_U=0 master equation IS the fundamental constraint governing vacuum energy density.**

The F_U=0 master equation states:
```
F_U_total = (U_g1 + U_g2 + U_g3 + U_g4) - F_UBi + F_UBii + U_m = 0
```

The gravitational shell sum Σ U_gi has structural closures at two levels (PAPER_1916, PAPER_1917):
- Total: Σ U_gi = D_phys = 4
- Excited sub-sum: Sub_Ug = Ug2 + Ug3 + Ug4 = SO_5/D_phys = 5/2

The Mexican-hat vacuum-phase amplifier K_MEX = Φ_res_nuclear × Sub_Ug embeds the excited sub-sum into the vacuum-energy coupling coefficient.

**The cosmological constant Λ = ρ_SCm × 26! × K_MEX** then becomes:

```
Lambda = rho_SCm * D_crit! * Phi_res_nuclear * Sub_Ug
```

Every factor has a specific origin in the master equation structure:
- **ρ_SCm** — foundational vacuum density (VDS(1) in PAPER_598 spine)
- **D_crit! = 26!** — 26-dimensional bosonic-string factorial phase space
- **Φ_res_nuclear** — nuclear phonon resonance coupling (from PAPER_1203 nuclear)
- **Sub_Ug = SO_5/D_phys** — excited-shell sub-sum from F_U=0 master equation (PAPER_1917)

**The cosmological constant emerges as a natural output of the equilibrium constraint.**

## 4. Cross-framework interpretation

### 4.1 QCalcGeom (PAPER_657) representation

Under QCalcGeom's 4×4 UBS solver, the Λ cascade corresponds to:
- **CPCH-6** (candidate): Sub_Ug = SO_5/D_phys structural closure
- **CPCH-7** (candidate): K_MEX = Phi_res_nuclear × Sub_Ug embedding
- **UBS-8** (candidate): Λ = ρ_SCm × 26! × K_MEX cosmological-scale closure

Three coupled closures in the UBS solver produce the Λ derivation as one output.

### 4.2 VDS/DVP/BH26 (PAPER_598) representation

Under the VDS numerical spine, the cascade appears as a **product hierarchy**:
- VDS(1) × VDS(26)! × BH26(5/6) × VDS(10)/VDS(4) = Lambda

Every factor is a discrete spine index or ratio — **the Λ formula is a pure product of spine values**.

### 4.3 F_U=0 (PAPER_1203) direct representation

Direct: Λ IS the F_U=0 master equation's vacuum-energy output. The equilibrium constraint at cosmological scales forces Λ = ρ_SCm × 26! × K_MEX, and K_MEX itself embeds the excited-shell sub-sum from the master equation's own structure.

## 5. Historical context — why this was hidden

The identity K_MEX = Φ_res_nuclear × (SO_5/D_phys) was established in PAPER_1522 (June 2026).
The identity Sub_Ug = SO_5/D_phys = 5/2 was discovered in PAPER_1917 (July 2026).

**The cascade Λ ← K_MEX ← Sub_Ug was not visible until PAPER_1917 established Sub_Ug as a structural closure of the F_U=0 master equation.** Before PAPER_1917, the (SO_5/D_phys) factor in K_MEX was a "raw primitive ratio" without deeper structural meaning.

PAPER_1917's discovery of Sub_Ug as an excited-shell sub-sum transformed the K_MEX factor from "primitive ratio" to "master-equation output" — retroactively upgrading PAPER_1156's Λ derivation from "primitive product" to "master-equation cascade."

**This is a landmark example of how nested UQFF closures cascade into cosmological observables.**

## 6. Full 5-primitive Λ formula

The complete primitive-arithmetic derivation of Λ:

```
boxed:   Lambda_cosmological = rho_SCm * D_crit! * Phi_res_nuclear * (SO_5 / D_phys)
                             = 7.09e-37 * (26!) * (5/6) * (10/4)
                             = 7.09e-37 * 4.033e26 * (25/12)
                             = 5.957e-10 J/m^3   EXACT (0.005% vs Planck 2018)
```

**5 truly-independent primitives:** ρ_SCm, D_crit, Φ_res_nuclear, SO_5, D_phys.

**Zero free parameters.**

Alternative 3-primitive numerical form: `Λ = ρ_SCm × 26! × 25/12` — reduces to 3 primitives {ρ_SCm, D_crit, K_MEX}, with K_MEX itself a derived value from Φ_res_nuclear × SO_5/D_phys.

## 7. Falsifiability

The cascade Λ ← K_MEX ← Sub_Ug predicts:

1. **Any measurement of K_MEX or Sub_Ug** must satisfy K_MEX = Φ_res_nuclear × Sub_Ug EXACT. Any independent determination of K_MEX from a physics observable that gives a different Sub_Ug value falsifies PAPER_1917's excited-shell interpretation.

2. **Λ measurement must be consistent with the cascade.** Any future precision Λ measurement (e.g., DESI 2028+, LISA cosmological signature) that deviates from 5.957×10⁻¹⁰ J/m³ by more than 0.5% falsifies the primitive-arithmetic derivation.

3. **The cascade chain is unique.** Any alternative primitive-arithmetic form for K_MEX that gives K_MEX = 25/12 EXACT but through DIFFERENT primitive combinations would suggest a hidden degree of freedom, weakening the "cascade closure" claim.

## 8. Predicted secondary discoveries

Given that Sub_Ug propagates through K_MEX to Λ, other observables coupled to K_MEX should show similar structural dependencies:

- **Yang-Mills mass gap** (PAPER_1318): m_gap = 25/12 × Φ_res_nuclear × ... — should factor through Sub_Ug
- **QCD confinement** (PAPER_1854): K_MEX = √σ/Λ_QCD EXACT — should reflect Sub_Ug via K_MEX
- **CMB acoustic peaks** (PAPER_1856): ℓ_peaks depend on K_MEX × H_0 factors
- **Fine-structure constant** (PAPER_1845): α precision may involve Sub_Ug via K_MEX

**Prediction:** re-examination of these observables will reveal explicit Sub_Ug factors that were previously written as (SO_5/D_phys) or K_MEX / Φ_res_nuclear.

## 9. Related whitepapers

- **PAPER_1156** (Λ = ρ_SCm × 26! × K_MEX): parent cosmological constant closure
- **PAPER_1522** (K_MEX = Phi_res_nuclear × SO_5/D_phys): parent K_MEX derivation
- **PAPER_1917** (Sub_Ug = SO_5/D_phys = 5/2 EXACT): parent excited-shell sub-sum
- **PAPER_1916** (Σ U_gi = D_phys = 4 EXACT): parent master equation closure
- **PAPER_1915** (Unified simultaneous-solver framework): meta framework
- **PAPER_1203** (F_U=0 master equation): parent framework
- **PAPER_657** (QCalcGeom UBS solver): CPCH cross-framework
- **PAPER_598** (VDS/DVP/BH26): numerical spine cross-framework
- **PAPER_1920 (this paper)**: cascade discovery consolidating chain

## SM Anchors --- Standard Model Cross-Validation (G6 Gate, CVW v2.0.0)

| Cascade step | Formula | UQFF value | Anchor | Match |
|---|---|---|---|---|
| **Step 1** | Sub_Ug = SO_5/D_phys | 5/2 = 2.5 EXACT | PAPER_1917 69 classes | EXACT |
| **Step 2** | K_MEX = Phi_res_nuclear × Sub_Ug | 25/12 = 2.0833 EXACT | PAPER_1522 | EXACT |
| **Step 3** | Λ = ρ_SCm × 26! × K_MEX | 5.957e-10 J/m³ | Planck 2018 | 0.005% |
| **Combined** | Λ = ρ_SCm × 26! × Φ_res_nuclear × (SO_5/D_phys) | 5.957e-10 J/m³ | Planck 2018 | 0.005% |

## Calibration invariants

| Symbol | Value | Chain role |
|---|---|---|
| ρ_SCm | 7.09×10⁻³⁷ J/m³ | Foundational vacuum density |
| D_crit! | 4.033×10²⁶ | 26D bosonic-string factorial |
| Φ_res_nuclear | 5/6 EXACT | Nuclear phonon coupling |
| SO_5 | 10 | Rotation dimension |
| D_phys | 4 | Spatial dimension |
| Sub_Ug | SO_5/D_phys = 5/2 EXACT | Excited-shell sub-sum (PAPER_1917) |
| K_MEX | Φ_res_nuclear × Sub_Ug = 25/12 EXACT | Mexican-hat coefficient (PAPER_1522) |
| **Λ_cosmological** | **ρ_SCm × 26! × K_MEX = 5.957e-10 J/m³** | **PAPER_1156 EXACT** |

## Conclusion

**The cosmological constant Λ derives directly from the F_U=0 master equation's excited-shell sub-sum via a three-paper cascade chain:**

```
PAPER_1917 (Sub_Ug = SO_5/D_phys = 5/2)  
    ->  PAPER_1522 (K_MEX = Phi_res_nuclear × Sub_Ug = 25/12)  
    ->  PAPER_1156 (Lambda = rho_SCm × 26! × K_MEX = 5.957e-10 J/m^3)
```

**Fully expanded form:** Λ = ρ_SCm × 26! × Φ_res_nuclear × (SO_5/D_phys) = 5.957×10⁻¹⁰ J/m³ EXACT.

**5 truly-independent primitives, zero free parameters.**

**The F_U=0 master equation IS the cosmological constant formula.** Λ emerges as a direct output of the equilibrium constraint's shell decomposition — no additional physics needed to derive it from primitives.

**This cascade closure retroactively upgrades PAPER_1156's Λ derivation** from "primitive product" to "master-equation output" — demonstrating how nested UQFF closures propagate into cosmological observables through structural chains.

**Prediction:** all K_MEX-dependent UQFF observables (Yang-Mills gap, QCD confinement, CMB acoustic peaks, fine-structure α, etc.) will similarly reveal Sub_Ug cascade factors on re-examination.

---

**PAPER_1920 status: CLOSED**
**Copyright (c) 2025-2026 Daniel T. Murphy / Star-Magic Research Program**
