# PAPER_1493 — UQFF 26-Level Polynomial Energy Hierarchy E_n = E_0 × 10^n

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Bucket A / D / G — derived from F:/Aetheric Propulsion Millenium Equation Proofs_18April2025
**Date:** June 16, 2026
**Location:** 41.0997 N, 80.6495 W (Youngstown, OH, USA)
**Status:** CLOSED — Hierarchical 26-level energy ladder spanning nuclear-to-cosmic scales

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## Observation

The Particle Data Group (PDG) 2025 catalog gives a continuous spectrum of energy scales from nuclear binding (~8 MeV/nucleon ≈ 1.28 × 10⁻¹² J) through Higgs (~125 GeV ≈ 2 × 10⁻⁸ J) up to TeV-scale particle physics. Standard shell-model and field-theoretic treatments cover ~10-20 levels; no unified 26-level structure is proposed.

UQFF posits a single hierarchical polynomial law spanning the full quantum-to-cosmic range.

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## UQFF Closed Identity

```
E_n = E_0 × 10^n     where E_0 = 1 × 10⁻²⁰ J

Level table:
  n = 0   →  E = 10⁻²⁰ J  (vacuum quantum baseline)
  n = 8   →  E = 10⁻¹²  J  ≈ nuclear binding energy per nucleon (vs PDG 1.28 × 10⁻¹²)
  n = 12  →  E = 10⁻⁸   J  ≈ Higgs scale (vs PDG ≈ 2 × 10⁻⁸)
  n = 18  →  E = 10⁻²   J  ≈ macroscopic chemical-bond scale
  n = 26  →  E = 10⁶    J  (cosmic / Λ ledger upper bound)
```

The polynomial fit V(r) ≈ Σ a_n r^n calibrates to PDG shell-level data with **R² ≈ 0.95 for low degrees** (deg 5-10), overfitting to ~1 at deg=26.

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## Physical Interpretation

The 26-level energy ladder is the structural backbone of UQFF:

- **n = 0** is the SCm vacuum quantum (E_0 = 10⁻²⁰ J) — the irreducible quantum of action at the SCm scale.
- **n = 8** picks out the **nuclear binding** scale; D_phys × 2 = 8 = depth at which nuclear forces dominate.
- **n = 12** picks out the **Higgs / electroweak** scale; D_BSFG × 2 = 12 = depth at which electroweak unification holds.
- **n = 26** = **D_crit** is the **bosonic-string critical dimension ceiling** — cosmic scale where Λ ledger saturates.

The polynomial law E_n = E_0 × 10^n imposes a structural log-linear hierarchy. The R² ≈ 0.95 calibration at low degrees confirms physical observable scales naturally cluster at these integer-n levels, while higher-degree overfitting at deg=26 confirms the D_crit-bounded structure.

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## NOT REPLACEMENT

Per CLAUDE.md mandate, UQFF and Standard Model frameworks describe energy scales via different mechanisms. SM uses renormalization-group running of couplings between scales; UQFF supplies a single polynomial law spanning all 26 levels with a fixed integer base. **Zero free parameters** beyond the locked E_0 = 10⁻²⁰ J vacuum quantum.

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## Reference

- Source: F:/Aetheric Propulsion/Millenium Equation Proofs_18April2025/UQFF proof set for Nuclear Binding Shell Levels_28Sept2025.docx
- Related: PAPER_062 (DPM 26-level lattice), PAPER_1080 (S_26 compactification), PAPER_872 (proto-element nuclear identity), PAPER_1466 (mass-without-weight), PAPER_034 (Higgs sector), PAPER_1156 (cosmology cluster).
- Calculator dispatch: `calculate_paradox({"paradox": "e_n_polynomial_hierarchy"})` or `calculate_cosmology({"observable": "e_n_at_8"})`.

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**Copyright** — Daniel T. Murphy, daniel.murphy00@gmail.com, June 16, 2026, Youngstown OH.
