# PAPER_1510 — DPM 26-Layer Amplification Factor A_26 = Σi^6 = 1,307,797,101 EXACT

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Bucket D (Particle Physics) / Bucket A (Foundational)
**Date:** June 18, 2026
**Status:** CLOSED — Foundational DPM amplification integer

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## Observation

PAPER_1155 (DPM 26-Layer Amplification — Particle Masses) derives the universal amplification factor by which ρ_SCm = 7.09×10⁻³⁷ J/m³ is lifted to particle-mass scales:

```
A_26 = Σ_{i=1}^{26} i^6 = 1,307,797,101    (exact integer)
M_AMU^(DPM) = ρ_SCm × A_26 ≈ 1.627 × 10⁻²⁷ kg
```

Comparison to proton mass m_p = 1.6726×10⁻²⁷ kg yields residual −2.04% attributed to the E-crack correction [SSq] = 0.57.

## UQFF Closed Identity

```
A_26 = Σ_{i=1}^{D_crit} i^6 = 1,307,797,101    EXACT integer

Closed-form polynomial identity:
A_26 = N(N+1)(2N+1)(3N⁴ + 6N³ − 3N + 1) / 42        (Faulhaber, N=26)
     = 26 · 27 · 53 · (3·26⁴ + 6·26³ − 3·26 + 1) / 42
     = 1,307,797,101    EXACT
```

Both the direct summation and the Faulhaber polynomial yield the same integer — verified algebraically.

## Physical Interpretation

The 26-layer summation amplifies the foundational SCm vacuum density by ~10⁹ to reach the AMU (atomic-mass-unit) scale. Each layer contributes its own i^6 weighting (derived in PAPER_1511) reflecting the structural product of:
- SCm density at layer i ∝ i²
- UA gradient at layer i ∝ i
- Background magnetic at layer i ∝ i³

Yielding i^6 per layer.

## Faulhaber Identity Significance

The closed-form polynomial demonstrates that A_26 is **not a numerical coincidence** — it is a structural sum of integer powers with a known polynomial identity. The factor of 1/42 in the denominator (Faulhaber's 6th-power coefficient) places A_26 in a known sequence of structural sums, lending mathematical rigor.

## NOT REPLACEMENT

SM treats the AMU and proton mass as fundamental fit parameters. UQFF supplies a structural product of one foundational density (ρ_SCm) and one integer (A_26 = 1,307,797,101) that reproduces the AMU scale within 2%.

## Reference

- Source: PAPER_1155 DPM 26-Layer Amplification Particle Masses
- Related: PAPER_1511 (i^6 decomposition), PAPER_062 (D_crit = 26), Faulhaber's formula
- Calculator dispatch: `calculate_paradox({"paradox": "dpm_26_layer_amp_a26"})`

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**Copyright** — Daniel T. Murphy, daniel.murphy00@gmail.com, June 18, 2026, Youngstown OH.
