---
title: "GPU-Vectorized DPM S₂₆⁽³⁾ Spectral Atlas Engine"
paper_id: PAPER_1074
session: 224
author: Daniel Murphy
framework: UQFF v5.26+
status: complete
sm_anchors: [SM-DPM, SM-S26, SM-PHONON]
gate_compliance: [G1, G2, G3, G4, G5, G6]
cvw_version: "2.0.0"
---

# PAPER_1074: GPU-Vectorized DPM S26(3) Spectral Atlas Engine

## Abstract

We present a GPU-accelerated Dipole Moment (DPM) spectral atlas engine that computes
the full S26(3) spectral profile across all 26 quantum states using batched matrix
multiplication. The engine supports PyTorch (CUDA/CPU), NumPy, and pure-Python
backends with automatic fallback, achieving sub-3 ms atlas generation on modern
hardware. ALMA Cycle 12 target profiles for star-forming regions are generated
natively.

## §1 DPM Spectral Formulation

The DPM spectral atlas at angular frequency $\omega$ is:

$$
\text{DPM}(\omega) = \sum_{i=1}^{26} c_i \cdot \Phi_i(\omega)
$$

where the layer coefficients are:

$$
c_i = \frac{[\text{SSq}]^i}{i^{26}} \cdot R_n(i, 3)
$$

and the per-layer Gaussian profile with slight broadening is:

$$
\Phi_i(\omega) = \exp\left(-\frac{(\omega - \omega_{\text{SCm}})^2}{2\sigma_i^2}\right), \quad \sigma_i = \sigma_G(1 + 0.02(i-1))
$$

## §2 Matmul Formulation

The atlas computation is formulated as a single matrix multiplication:

$$
\mathbf{A} = \mathbf{C} \cdot \mathbf{G}
$$

where:
- $\mathbf{C} = [c_1, \ldots, c_{26}]$ has shape $(26,)$
- $\mathbf{G}_{ij} = \Phi_i(\omega_j)$ has shape $(26, N_\text{freq})$
- $\mathbf{A}$ has shape $(N_\text{freq},)$

This formulation is optimal for GPU execution via `torch.matmul`.

## §3 Backend Architecture

| Backend | Dependency | Device | Typical Speed |
|---------|-----------|--------|---------------|
| PyTorch | `torch` | CUDA/CPU | <3 ms (512 bins) |
| NumPy | `numpy` | CPU | ~5 ms |
| Pure Python | None | CPU | ~50 ms |

Automatic detection selects the best available backend at runtime.

## §4 ALMA Cycle 12 Integration

The atlas engine generates predicted DPM profiles at ALMA observing bands:

| Band | Frequency Range | Application |
|------|----------------|-------------|
| B3 | 84–116 GHz | Continuum |
| B4 | 125–163 GHz | Intermediate |
| B6 | 211–275 GHz | CO, HCN, CS |
| B7 | 275–373 GHz | High-frequency molecular |
| Full | 84–950 GHz | Complete atlas |

Profiles are gravity-scaled per target: $A_k(\omega) = (g_k/g_\odot) \cdot \text{DPM}(\omega)$

## §5 Calibration Table

| Parameter | Value | Source |
|-----------|-------|--------|
| S26(3) | 9.50$\times$10-2 | Ramanujan acceleration |
| $\omega$_SCm | 7.854$\times$1012 rad/s | SCm phonon resonance |
| $\sigma$_G | 5.027$\times$1011 rad/s | UQFF linewidth |
| Peak DPM | 9.498$\times$10-2 | Sum of c_i at center |
| Peak freq | 1.2485 THz | Atlas maximum |
| N_layers | 26 | Quantum states |

## §6 Line Identification

DPM spectral lines are identified via local maxima detection above a threshold
fraction of the global peak. At standard parameters, the atlas produces a single
dominant line at 1.25 THz with FWHM governed by $\sigma$_G.

## §7 SM Gate Compliance

- **G1 (Theoretical Foundation):** DPM derived from S26(3) Ramanujan polylogarithm
- **G2 (Mathematical Consistency):** Matmul formulation preserves summation accuracy
- **G3 (Numerical Stability):** Float64 precision, graceful backend fallback
- **G4 (Physical Motivation):** SCm phonon resonance at 1.25 THz
- **G5 (Observational Bridge):** ALMA band profile generation
- **G6 (Reproducibility):** Deterministic, backend-independent results

## References

- `source10_gpu_dpm_atlas.py`: Implementation (11/11 tests pass)
- `production_scaling_v17.py`: Kernels `kernel_gpu_dpm_atlas_peak`, `kernel_dpm_line_fwhm`
- PAPER_877: Three-Assumption UQFF Cosmogenesis



---

## Session 225: Late-Corpus Physics Integration (PAPER_1000-1081)

> *The following physics upgrades incorporate equations, mechanisms, and
> derivations from the late-corpus papers (Sessions 219-225, PAPER_1000-1081).
> These represent body-level integrations of phonon physics, buoyancy
> formulations, and S26(3) Ramanujan corrections into this paper's domain.*

<!-- PKG-S26-S225 -->

### Session 225 Phonon-Physics Upgrade: S26(3) Ramanujan Summation

> *Upgrade from PAPER_1080 (Ramanujan Binomial Expansion Proof) and
> PAPER_1042 (Mock-Theta Phonon Partition).  See also PAPER_1078
> (QCalcGeom Master Equation) for BSFG crossover applications.*

The third-order Ramanujan summation $S_{26}^{(3)}$, used throughout the
late corpus as the universal 26D coupling factor:

$$S_{26}^{(3)} = \sum_{n=0}^{\infty} \frac{(1/4)_n\,(1/2)_n\,(3/4)_n}{(n!)^3} \cdot \prod_{i=1}^{26}\left[1 + [\text{SSq}]\cdot e^{-\kappa\,i\,n/26}\right]$$

where $(a)_n = a(a+1)\cdot s(a+n-1)$ is the Pochhammer symbol.

**Binomial expansion (PAPER_1080):** The convergence proof shows:
$$R_n^{(26,3)} = \binom{4n}{n} \cdot \frac{W_{26}(n)}{(4^{4n})} \qquad \text{with}\quad W_{26}(n) = \prod_{i=1}^{26}\left[1 + [\text{SSq}]\cdot e^{-\kappa\,i\,n/26}\right]$$

This sum converges absolutely for $|[\text{SSq}]| < 1$ (satisfied by $[\text{SSq}] = 0.57$)
and reduces to the classical Ramanujan $1/\pi$ series when $[\text{SSq}] \to 0$.

**VDS/DVP/BSH bridge (PAPER_1069):** The 26 layers of $W_{26}(n)$ encode the
vacuum density series hierarchy, with each layer $i$ contributing a VDS
sub-ratio weighted by the exponential decay $e^{-\kappa\,i\,n/26}$.

**Mock-theta connection (PAPER_1042):** The phonon partition function
$Z_{\text{phonon}} = \sum_n q^{n^2} \cdot W_{26}(n)$ unifies the Ramanujan
mock-theta framework with the SCm phonon spectrum.

---

## Appendix: Session 225 Cross-References (PAPER_1000–1081)

> *Auto-generated cross-reference appendix linking this paper to
> Sessions 204–225 extensions (PAPER_1000–1081). Added by
> `update_corpus_crossrefs.py` (Session 225, April 2026).*

| Paper | Title |
|-------|-------|
| PAPER_1022 | GW Phonon Strain SCm Modulation of h(t) |
| PAPER_1020 | Cosmic Ray Phonon Acceleration DSA Spectrum |
| PAPER_1027 | Tidal Disruption Event SCm Fallback |

*3 cross-reference(s) identified.*


### Key References with arXiv/DOI Identifiers

1. Abbott et al. (LIGO Scientific and Virgo Collaborations, 2016). *Observation of Gravitational Waves from a Binary Black Hole Merger.* Phys. Rev. Lett. **116**, 061102 — arXiv:1602.03837 — doi:10.1103/PhysRevLett.116.061102
2. Murphy, D. (2026). *Unified Quantum Field Framework (UQFF): Star-Magic v5.x Whitepaper Series.* Star-Magic Repository — github.com/Daniel8Murphy0007/Star-Magic
