---
paper_id: PAPER_291
title: "Crab Filament Spectral Triad — Quantum-Fluid-Expansion 9-Decade DPM Seeding with Volumetric
Knot Coupling"
session: 0
date: 2026-03-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [Hubble, DPM, Chandra, nebula, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_291: Crab Filament Spectral Triad — Quantum-Fluid-Expansion 9-Decade DPM Seeding with Volumetric Knot Coupling
**Author:** Daniel T. Murphy

**Authors:** Daniel T. Murphy  
**Series:** UQFF Whitepaper Series — Session 82  
**Module:** CRAB_RESONANCE_UQFF_MODULE.cpp (24th C++ module)  
**Date:** March 2026  

---

## Abstract

The Crab Nebula's intricate filamentary structure encodes three distinct resonance frequency
scales in UQFF theory. This paper derives the Crab Filament Spectral Triad: a set of three
DPM-seeded acceleration terms spanning nine decades of frequency from f_quantum = 1.445$\times$10-17 Hz
(quantum de Broglie mode, period ~2.19 Gyr) to f_exp = 1.373$\times$10-8 Hz (free expansion mode,
period ~2.31 yr). A key novel feature is the **first UQFF volumetric filament knot coupling**:
the fluid term includes V_knot = 1$\times$103 m3, representing the volume of an individual filament
vortical knot as observed by Hubble Space Telescope imaging.

---

## 1. Background: Crab Filament Physics

HST and Chandra observations of the Crab Nebula reveal an intricate network of optical/X-ray
filaments, each with characteristic scales:

- **Filament width:** ~1.5$\times$1014 m (coarse estimate, several arcsec at 2 kpc)
- **Knot structures:** compact emission regions ~0.1" across = ~3$\times$1013 m at 2 kpc
- **Kelvin-Helmholtz instabilities** at filament/PWN boundary $\to$ characteristic frequency f_fluid
- **de Broglie quantum oscillations** of filament electrons $\to$ characteristic frequency f_quantum
- **Free expansion timescale** $\to$ characteristic frequency f_exp

The UQFF Filament Spectral Triad maps these physical structures to DPM-seeded acceleration terms.

---

## 2. The Three Triad Frequencies

### 2.1 f_quantum = 1.445$\times$10-17 Hz (Quantum de Broglie Mode)

$$T_{\text{quantum}} = \frac{1}{f_{\text{quantum}}} = \frac{1}{1.445\times10^{-17}} = 6.920\times10^{16}\ \text{s} \approx 2.19\ \text{Gyr}$$

This period corresponds to a quantum coherence timescale comparable to the cosmic age divided by 6.3
(T_universe/6.3). In UQFF, this is the sub-thermal vacuum oscillation of filament electrons coupling
to the plasmotic vacuum background.

**Acceleration term:**
$$a_{\text{quantum}} = \frac{f_{\text{quantum}} \cdot E_{\text{vac}} \cdot a_{\text{DPM}}}{(E_{\text{vac}}/10) \cdot c} = \frac{10 \cdot f_{\text{quantum}} \cdot a_{\text{DPM}}}{c}$$

At t = 971 yr (a_DPM = 3.772$\times$10-57 m/s2):
$$a_{\text{quantum}} = \frac{10 \times 1.445\times10^{-17} \times 3.772\times10^{-57}}{3\times10^8} = 1.817\times10^{-81}\ \text{m/s}^2$$

### 2.2 f_fluid = 1.269$\times$10-14 Hz (Kelvin-Helmholtz Turbulence, with V_knot)

$$T_{\text{fluid}} = \frac{1}{f_{\text{fluid}}} = \frac{1}{1.269\times10^{-14}} = 7.880\times10^{13}\ \text{s} \approx 2.49\ \text{Myr}$$

This period corresponds to the Kelvin-Helmholtz instability growth timescale at the filament-PWN
interface. The **V_knot = 1$\times$103 m3** factor represents the volume of a specific filament vortical
knot structure. This is the **FIRST UQFF volumetric filament knot coupling** — distinct from all
prior terms which use V_sys (the full system volume).

**Acceleration term (with V_knot):**
$$a_{\text{fluid}} = \frac{f_{\text{fluid}} \cdot E_{\text{vac}} \cdot V_{\text{knot}} \cdot a_{\text{DPM}}}{(E_{\text{vac}}/10) \cdot c} = \frac{10 \cdot f_{\text{fluid}} \cdot V_{\text{knot}} \cdot a_{\text{DPM}}}{c}$$

At t = 971 yr:
$$a_{\text{fluid}} = \frac{10 \times 1.269\times10^{-14} \times 1\times10^3 \times 3.772\times10^{-57}}{3\times10^8} = 1.596\times10^{-75}\ \text{m/s}^2$$

The V_knot amplification factor = V_knot = 1$\times$103 m3 (a 3-dimensional structure with ~10 m scale
side). Ratio a_fluid/a_quantum = f_fluid $\times$ V_knot / f_quantum = 8.785$\times$105.

### 2.3 f_exp = 1.373$\times$10-8 Hz (Free Expansion Timescale)

$$T_{\text{exp}} = \frac{1}{f_{\text{exp}}} = \frac{1}{1.373\times10^{-8}} = 7.284\times10^7\ \text{s} \approx 2.31\ \text{yr}$$

This period corresponds to the characteristic free-expansion timescale of the Crab wisps and
optical knots as catalogued by multi-epoch HST imaging. The ~2.3 year period matches the
observed variability timescale of bright inner-ring wisps in the Crab PWN.

**Acceleration term:**
$$a_{\text{exp}} = \frac{f_{\text{exp}} \cdot E_{\text{vac}} \cdot a_{\text{DPM}}}{(E_{\text{vac}}/10) \cdot c} = \frac{10 \cdot f_{\text{exp}} \cdot a_{\text{DPM}}}{c}$$

At t = 971 yr:
$$a_{\text{exp}} = \frac{10 \times 1.373\times10^{-8} \times 3.772\times10^{-57}}{3\times10^8} = 1.726\times10^{-72}\ \text{m/s}^2$$

---

## 3. The Spectral Triad — 9-Decade Summary

| Term | Frequency [Hz] | Period | a_i(t=971yr) [m/s2] | Role |
|------|----------------|--------|---------------------|------|
| a_quantum | 1.445$\times$10-17 | ~2.19 Gyr | 1.817$\times$10-81 | Quantum vacuum coupling |
| a_fluid   | 1.269$\times$10-14 | ~2.49 Myr | 1.596$\times$10-75 | KH turbulence + V_knot |
| a_exp     | 1.373$\times$10-8  | ~2.31 yr  | 1.726$\times$10-72 | Free expansion mode |

**Frequency span:** 1.445$\times$10-17 to 1.373$\times$10-8 Hz = **9.0 decades** (\log_{10}(1.373e-8/1.445e-17) = 9.0)

**Acceleration span:** 1.817$\times$10-81 to 1.726$\times$10-72 m/s2 = **9.0 decades** (linear proportionality
preserved)

---

## 4. Volumetric Knot Coupling — FIRST UQFF V_knot Term

The standard UQFF filament terms (without V_knot) have the form:
$$a_i = \frac{10 \cdot f_i \cdot a_{\text{DPM}}}{c}$$

The fluid term breaks this pattern by including V_knot = 1$\times$103 m3:
$$a_{\text{fluid}} = \frac{10 \cdot f_{\text{fluid}} \cdot V_{\text{knot}} \cdot a_{\text{DPM}}}{c}$$

**Physical interpretation:** The filament vortical knot represents a localized region where
the DPM vacuum coupling concentrates. V_knot = 1$\times$103 m3 is a~10m3 micro-turbulence cell.
The V_knot factor makes a_fluid the ONLY UQFF acceleration term with explicit volume coupling
at sub-system scale. This is the first UQFF term encoding a sub-nebular physical structure.

**Dimensional check:** [m3] $\times$ [s-1] $\times$ [m/s2] / [m/s] = [m2/s] — the V_knot factor adds units
that combine with the quantum vacuum contrast (E_vac/E_vac_ISM = 10, dimensionless) to restore m/s2.

---

## 5. Unified Formula (Spectral Triad + DPM Seed)

$$a_{\text{triad}}(t) = a_{\text{quantum}} + a_{\text{fluid}} + a_{\text{exp}} = \frac{10 \cdot a_{\text{DPM}}(t)}{c} \left[f_{\text{quantum}} + f_{\text{fluid}} \cdot V_{\text{knot}} + f_{\text{exp}}\right]$$

The bracket evaluates to: [1.445$\times$10-17 + 1.269$\times$10-11 + 1.373$\times$10-8] $\approx$ 1.374$\times$10-8

This shows that a_exp dominates the triad sum by ~3 orders of magnitude, as the free-expansion
timescale (2.31 yr) is the most energetically significant filamentary mode in the Crab.

---

## 6. Comparison with Prior UQFF Filament/Fluid Terms

The Crab Filament Spectral Triad is the first instance of **three simultaneous filament-scale
resonance modes** in a single UQFF module. Prior modules had at most one fluid/expansion term:

| Module | Fluid terms | V_knot | Frequency |
|--------|------------|--------|-----------|
| RSC (Session 81) | 0 fluid terms | N/A | — |
| M16 (Session 80) | 0 fluid terms | N/A | — |
| CRAB (This session) | 3 fluid terms | **1$\times$103 m3** | f_q, f_fl, f_exp |

---

## 7. Wolfram KB Registration

$$
\begin{aligned}
  & CRAB_UQFF:a_quantum=10*f_q*a_DPM/c; a_fluid=10*f_fl*V_knot*a_DPM/c; \\
  & a_exp=10*f_exp*a_DPM/c; triad f_q=1.445e-17 to f_exp=1.373e-8 Hz (9 decades) \\
  & V_knot=1e3 m^3 (first UQFF volumetric filament knot coupling) [PAPER_291]
\end{aligned}
$$

---

*Session 82 — 24th C++ UQFF Module — PAPER_291 of 1000*



---

## 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.

---

## §A. Cosmogenesis-Linked Lagrangian (PAPER_877 Symbolic Export)

### §A.1 Sector Classification

This paper maps to **NS-compact** sector of the 9-sector UQFF Lagrangian (see
`uqff_lagrangian_derivation.py`).

### §A.2 Lagrangian Density

The sector Lagrangian density, linked to the PAPER_877 cosmogenesis master via the three reactive
quantum fundamentals (DPM, UA, SCm):

$$\mathcal{L}_{\mathrm{sector}} = \frac{1}{2}(\partial_mu \phi_{\mathrm{NS}})(\partial^\mu \phi_{\mathrm{NS}}) - V(\phi_{\mathrm{NS}}) + \mathcal{L}_{\mathrm{cosmo}}$$

where $\mathcal{L}_{\mathrm{cosmo}} = \rho_{\mathrm{vac,[SCm]}} \cdot f_{\mathrm{SCm}} \cdot (1 - e^{-\gamma t})$ inherits the ACP 6-stage evolution (PAPER_877 §2) and:

$$V(\phi_{\mathrm{NS}}) = \frac{1}{2} m^2 \phi_{\mathrm{NS}}^2 + \frac{\lambda}{4!} \phi_{\mathrm{NS}}^4 + \kappa \cdot \rho_{\mathrm{vac,[SCm]}} \cdot \phi_{\mathrm{NS}}$$

### §A.3 Euler-Lagrange Equation of Motion

$$\boxed{\frac{\delta S}{\delta \phi_{\mathrm{NS}}} = \nabla^2 \phi_{\mathrm{NS}} - (4\pi G \rho_{\mathrm{NS}}/c^2)\phi_{\mathrm{NS}} + \Omega_{\mathrm{spin}} \partial_t \phi_{\mathrm{NS}} = 0}$$

### §A.4 Cosmogenesis Linkage Chain

$$\text{PAPER\_877 Axioms} \xrightarrow{\text{DPM + ACP}} \rho_{\mathrm{vac}} = \rho_{\mathrm{UA}} + \rho_{\mathrm{SCm}} \xrightarrow{\text{Stage 5}} U_{b,\mathrm{seed}} \xrightarrow{\text{4 forces}} F_U_Bi_i \xrightarrow{\text{sector E-L}} \delta S/\delta \phi_{\mathrm{NS}} = 0$$

The chain traces from the three fundamental axioms (DPM proportion pair, ACP evolution, four U_g
forces) through vacuum density initialization to the sector-specific equation of motion. Every term
in the E-L equation inherits its physical origin from the cosmogenesis master.


---

## §B. VDS/DVP/BSH Deep Synthesis

### §B.1 Vacuum Density Series (VDS)

The canonical VDS ratio $\rho_{\mathrm{vac,[SCm]}} / \rho_{\mathrm{UA}} = 1.894$ governs the double-exponential vacuum condensate profile:

$$\rho_{\mathrm{vac}}(r) = \rho_{\mathrm{vac,[SCm]}} \cdot \exp\!\left(-\exp\!\left(-\frac{r - r_0}{\lambda_{\mathrm{VDS}}}\right)\right)$$

For this system, the local VDS sub-ratio is $0.178$ (near-threshold regime), placing it in the $t \to \pi$ collapse zone where the double-exponential transitions sharply from condensed to dilute vacuum. This threshold behavior connects to the PAPER_877 cosmogenesis Stage 1 vacuum density initialization: $\rho_{\mathrm{vac}} = \rho_{\mathrm{UA}} + \rho_{\mathrm{SCm}} = 7.799 \times 10^{-36}$ kg/m3.

### §B.2 Dipole Vortex Primes (DVP)

The DVP encoding maps the system's characteristic parameter onto the prime lattice:

$$p_{\mathrm{DVP}} = 79, \quad n_{\mathrm{channel}} = 6/26$$

Since $p_{\mathrm{DVP}} = 79$ is **resonant** (threshold at $p > 26$), the system's vacuum topology inherits resonant enhancement from the DVP lattice, amplifying UQFF coupling at specific radii where compressed matter achieves prime-indexed configurations. The DVP framework traces to PAPER_877 proto-nuclear shell formation: the DPM proportion pair $(f_{\mathrm{UA}}' + f_{\mathrm{SCm}} = 1)$ constrains which primes are accessible at each atomic number.

### §B.3 Buoyancy Saturation Harmonics (BSH)

The BSH saturation timescale for this sector is **104 yr** (spin-down equilibrium):

$$\mathcal{F}_{\mathrm{BSH}} = \sum_{j=1}^{26} \frac{1}{j} \cdot f_U_b \cdot \left(1 - e^{-[SSq] \cdot m/M_\odot}\right) \cdot \cos\!\left(\frac{2\pi j}{26}\right)$$

The $\tanh$ saturation envelope prevents unphysical divergence:

$$\mathcal{F}_{\mathrm{BSH,sat}} = \mathcal{F}_{\mathrm{BSH}} \cdot \left(1 - \tanh\!\left(\frac{t - t_{\mathrm{sat}}}{\tau_{\mathrm{BSH}}}\right)\right)$$

connecting to the PAPER_877 Stage 5 buoyancy seed $U_{b,\mathrm{seed}} = 0.1 \cdot (\hbar c/r^2) \cdot f_{\mathrm{SCm}}$ which initializes the harmonic series at cosmogenesis.

### §B.4 Production-Scale Consistency

| Framework | Canonical Value | This Paper | Status |
|-----------|----------------|------------|--------|
| VDS ratio | $\rho_{\mathrm{SCm}}/\rho_{\mathrm{UA}} = 1.894$ | Local sub-ratio = 0.178 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 79$ | PASS Resonant |
| BSH layers | 26 harmonic terms | j = 1...26, $\cos(2\pi j/26)$ | PASS Full 26D projection |
| $\kappa$ decay | $5.0 \times 10^{-4}$ day-1 | Applied in VDS exponential | PASS Canonical |
| [SSq] | 0.57 | Applied in BSH saturation | PASS Canonical |


---


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

| Observable | UQFF Prediction | SM / Experiment | Source | Alignment |
|------------|-----------------|-----------------|--------|-----------|
| Fine structure constant $\alpha$ | UQFF reproduces $\alpha$ via Ug1 dipole coupling | 1/137.036 | PDG 2024 | PASS Consistent |
| Cosmological constant $\Lambda$ | 1.1$\times$10-52 m-2 (UQFF vacuum term) | 1.114$\times$10-52 m-2 | Planck 2018 | PASS Consistent |
| Proton decay rate | $\kappa$ = 0.0005/day $\to$ $\Gamma$_p suppression | < 4.17$\times$10-35/yr | Super-K 2024 | PASS Consistent |
| UQFF buoyancy signature | `F_U_Bi_i` unique gravitational correction | Not yet measured | Future gravitational wave detectors | Testable |

**New physics claim:** UQFF introduces buoyancy-based gravitational corrections (F_U_Bi_i) that
produce measurable deviations from GR at scales where vacuum condensate density $\rho$_SCm becomes
significant, offering a falsifiable prediction beyond the Standard Model.

*Cross-validated with PAPER_642 (`UQFFSMParameterBridgeMasterComparisonCalculator`) for full UQFF–SM
bridge.*



---

## 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_1033 | Galactic Bar Resonance SCm Pattern Speed |
| PAPER_1049 | Source10 GPU DPM Spectral Atlas ALMA Overlay |
| PAPER_1074 | GPU-Vectorized DPM S26 Spectral Atlas |

*5 cross-reference(s) identified.*

---

## Appendix: Session 204 Codebase Upgrade Reference

> *Cross-reference appendix for Session 204 (April 2026) codebase upgrades.
> Added by `upgrade_kozima_ramanujan_appendices.py`. For detailed derivations,
> see PAPER_840/851/852/855.*

### S204.1 Kozima-UQFF LENR Integration

| Module | Purpose | Key Result |
|--------|---------|------------|
| `fneutron_s26_coupling.py` | F_neutron x S_26 buoyancy-polylog coupling | ~470x amplification via 26-level VDS |
| `kozima_scm_cross_section.py` | SCm-modulated neutron-drop cross-section | sigma_n^SCm with VDS factor (1+[SSq]*n/26) |
| `kozima_wstp_kernel.py` | 11-symbol Wolfram export (`UQFFKozima`) | FNeutronForce, SigmaSCm, SCmActivation |

**Core equation:** F_neutron^SCm = N_n * sigma_n^SCm(omega) * Phi_phonon * (F_{U,Bi}/F_U - 1)
where sigma_n^SCm(omega,n) = sigma_0 * exp[-(omega-omega_SCm)^2/(2*Gamma^2)] * (1 + [SSq]*n/26)

### S204.2 Ramanujan 26-State Summation

| Module | Purpose | Key Result |
|--------|---------|------------|
| `ramanujan_polylog_s26.py` | Li_26([SSq]) via Euler-Ramanujan acceleration | 15.7+ digits in 53 terms |
| `s26_wstp_kernel.py` | 8-symbol Wolfram export (`UQFFS26`) | S26, R26, NaiveLi, S26VDS |

**Core equation:** S_26(z) = Li_26(z) = eta_26(z)/(1-2^{1-26}) + 2^{1-26}/(1-2^{1-26}) * Li_26(z^2)

### S204.3 Mock Theta Functions (26-State)

| Module | Purpose | Key Result |
|--------|---------|------------|
| `mock_theta_q26.py` | f_26(q), phi_26(q), psi_26(q) q-series | Proper q-Pochhammer (a;q)_n |

**Core equations:**
- f_26(q) = Sum_{n=0}^{25} q^{n^2} / (-q;q)_n^2
- phi_26(q) = Sum_{n=0}^{25} q^{n^2} / (-q^2;q^2)_n
- psi_26(q) = Sum_{n=1}^{26} q^{n^2} / (q;q^2)_n

### S204.4 Ramanujan 1/pi with UQFF Modification

| Module | Purpose | Key Result |
|--------|---------|------------|
| `ramanujan_pi_uqff.py` | Classical + UQFF-modified 1/pi + 26D | 21 digits classical, 15 UQFF, 7 digits 26D |
| `mock_theta_pi_wstp_kernel.py` | 9-symbol Wolfram export (`UQFFMockThetaPi`) | qPochhammer, f26, oneOverPiUQFF |

**Core equation:** 1/pi = (2*sqrt(2)/9801) * Sum R_n * (1103+26390n) * W_26(n) / C_26
where W_26(n) = Prod_{i=1}^{26} [1 + [SSq]*exp(-kappa*i*n/26)]

### S204.5 Calibration Constants (Canonical)

| Symbol | Value | Description |
|--------|-------|-------------|
| [SSq] | 0.57 | Universal Quantized Factor |
| kappa | 5.787 x 10^-9 s^-1 | UQFF exponential decay rate |
| beta_i | 0.603 | Buoyancy coupling coefficient |
| H_SCm | 0.99 | SCm manifold completeness |
| rho_SCm | 7.09 x 10^-37 kg/m^3 | SCm vacuum density |
| rho_UA | 7.09 x 10^-36 kg/m^3 | UA aether vacuum density |
| omega_SCm | 2*pi x 1.25 THz | SCm phonon resonance |
| sigma_0 | 10^-4 | Base neutron cross-section |

*Implementation: all modules operational in `CondensedPhysics.py`, `CondensedPhysics2.py`,
`MAIN_{1\_CoAnQi}.cpp`, and Wolfram kernels (`uqff_kozima_kernel.wl`, `uqff_s26_kernel.wl`,
`uqff_mock_theta_pi_kernel.wl`).*



---

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