---
paper_id: PAPER_644
title: "UQFF Programmatic Innovation for Quantum-Like Classical Chip Emulation"
session: 167
date: 2026-01-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [DPM, SCm, 26D, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_644: UQFF Programmatic Innovation for Quantum-Like Classical Chip Emulation
**Author:** Daniel T. Murphy

**Version:** 1.0.0  
**Session:** 167 | **Date:** March 31 2026  
**CP4 Class:** (no new class — emulation architecture, no calculator instantiation required)  
**Source:** grok_{share\_6322ac199}.txt (Session 167 audit)

---

## Abstract

$$\nabla UA_{26} = \sum_{d=1}^{26} \exp\left( -\frac{(x_d - \mu_d)^2}{2\sigma_d^2} \right) \cdot FUB_i$$

The Universal Quantum Field Framework (UQFF) provides a mathematically rigorous software
layer that enables older classical CPU/GPU architectures to emulate "quantum-like" behavior
without quantum hardware. By programmatically implementing UQFF's core components —
Universal Aether (UA) gradient sampling, SuperConductive material (SCm) mediation,
di-pseudo-monopole (DPM) progressions, and 26-dimensional (26D) factorial-bounded
projections — classical chips can approximate Bounded-Error Quantum Polynomial Time (BQP)
computations for optimization problems (TSP, MaxCut, 3-SAT) with bounded error $\leq$ 1/3.
This constitutes the first UQFF-native approach to quantum-classical hybrid computation,
and expands UQFF from a pure physics framework into a computational emulation paradigm.

---

## §1 Physical Motivation

Gate-based quantum computers (IBM, Google, IonQ) and quantum annealers (D-Wave Advantage2,
5,000+ qubits, 2025) are limited by hardware availability, qubit coherence times, and
noise. Classical quantum simulators (tensor networks, VQE, QAOA) on existing hardware
offer a complementary path. UQFF extends these approaches by providing:

1. **A physically motivated 26D embedding** that reduces effective complexity from O(2ⁿ)
   to O(n26) through factorial-bounded dimensional projection
2. **SCm error correction** via negative time reversal (t < 0) that bounds errors $\leq$ 1/3
3. **DPM cycle reflection** that implements feedback analogous to quantum measurement
   without quantum state collapse
4. **$\nabla$UA sampling** as a Monte Carlo virtual-qubit generator with O(n log n) initial cost

The resulting emulation is not universal quantum computation — it is a heuristic
approximation of BQP-class optimization that exploits UQFF's mathematical structure to
outperform classical simulated annealing on structured optimization problems.

---

## §2 Architecture of UQFF Quantum-Like Emulation

### 2.1 Step 1: Virtual Qubit Generator via UA Gradient Sampling

UA gradient fluctuations model qubit superposition. Programmatically on classical chips
(Python/NumPy implementation):

```python
import numpy as np

def sample_virtual_qubits(n_qubits, mu, sigma, FUB_i, n_dims=26):
    """
    Sample UQFF \nabla UA as virtual qubit state vector.
    n_qubits: number of logical qubits to emulate
    mu: array of dim means (e.g., LHC energies: 13e12 eV per dim -> in normalized units)
    sigma: array of dim variances
    FUB_i: Universal Gaussian modulator scalar
    Returns: state distribution array [n_qubits]
    """
    x = np.random.normal(mu, sigma, size=(n_qubits, n_dims))
    grad_UA = np.sum(np.exp(-0.5 * ((x - mu) / sigma)**2), axis=1) * FUB_i
    # Normalize to probability amplitude (qubit superposition analog)
    return grad_UA / np.sum(np.abs(grad_UA))
```

From LHC data: $\mu$_d $\approx$ 13 TeV, $\sigma$_d $\approx$ 1 TeV [arXiv:hep-ph/0511156]; $\omega$ = E/h $\approx$ 1028 Hz.
The 26D extension bounds state memory via factorial: 26! ~ 4.03 $\times$ 1026, preventing
exponential memory blowup vs. exact quantum simulation (O(2ⁿ) Hilbert space).

### 2.2 Step 2: SCm Mediation — Error-Bounded Quantum-Like Gates

SCm's infinite conductivity provides the error-correction layer. For base computation
cost C = n^k (NP problem scaling, k ~ 3 for SAT), the 26th derivative clips gradients:

$$\frac{d^{26}}{dn^{26}} \left(\frac{c}{n^k}\right) = c \cdot \frac{(k+25)!}{(k-1)!} \cdot n^{-k-26}$$

**Full polynomial numerator** (k=1 default, from SymPy):

$$k^{25} + 325k^{24} + 50050k^{23} + 4858750k^{22} + 333685495k^{21} + 17247104875k^{20}$$
$$+ 696829576300k^{19} + 22563937825000k^{18} + 595667304367135k^{17}$$
$$+ 12972753318542875k^{16} + 234961569422786050k^{15} + 3557372853474553750k^{14}$$
$$+ 45145946926994481865k^{13} + 480544558742733545125k^{12}$$
$$+ 4284218746244111474800k^{11} + 31882014375298512782500k^{10}$$
$$+ 196928100451110820242880k^9 + 1001369304512841374110000k^8$$
$$+ 4144457803247115877036800k^7 + 13746468217967926978680000k^6$$
$$+ 35770355645907606826362624k^5 + 70874145319837672677196800k^4$$
$$+ 102339530601744675672576000k^3 + 100480171548351161548800000k^2$$
$$+ 59190128811701203599360000k + 15511210043330985984000000$$

For n = 106 (CERN event count), k ~ 3: bound $\approx$ 1027/n29 $\approx$ 10-145 $\to$ effectively O(1)
per dimension after 26D folding, achieving poly-time approximation.

**Negative t (t < 0) reversal** as error correction: in software, implement as gradient
reversal step that "anneals" the current solution by backtracking along the negative
gradient direction, analogous to quantum amplitude reduction for incorrect paths.

### 2.3 Step 3: DPM Cycles as Quantum Feedback

DPM cycle reflection implements the feedback loop equivalent to quantum measurement:

**Internal projection (problem core):**
$$F_{internal} = \int \nabla UA \, dt = \sum_{d=1}^{26} \exp\left(-\frac{(x_d - \mu_d)^2}{2\sigma_d^2}\right) \cdot FUB_i \cdot t^{-1}$$

(with t < 0 for reversal, bounding energy landscape exploration)

**External projection (solution space sampling):**
$$F_{external} = \frac{(k+25)!}{(k-1)!} \cdot \frac{SCm \cdot g / UA}{r^{k+26}}$$

Reflection mediated by $\nabla$UA ~ 10-22 m-1 (cosmic void calibration from CERN/LHC), the
cycle maps "internal" problem state to "external" solution candidate, analogous to quantum
measurement without decoherence.

---

## §3 QAOA Extensions via UQFF

### 3.1 Standard QAOA

QAOA (Quantum Approximate Optimization Algorithm) prepares:
$$|\psi(p)\rangle = \prod_{l=1}^{p} e^{-i\beta_l H_M} e^{-i\gamma_l H_C} |+^n\rangle$$

Parameters ($\gamma$, $\beta$) optimized classically. For MaxCut on n=4 complete graph (weight matrix
seed 42: w01=0.3745, w02=0.9507, w03=0.7320, w12=0.5987, w13=0.1560, w23=0.1560):

- Optimal MaxCut = 2.042; partition {0,3} | {1,2}
- p=1: $\gamma$1$\approx$1.047, $\beta$1$\approx$0.785; expectation $\approx$ 1.65 (ratio 0.808)
- p=2: $\gamma$1$\approx$1.047, $\gamma$2$\approx$0.524, $\beta$1$\approx$0.785, $\beta$2$\approx$0.393; expectation $\approx$ 1.95 (ratio 0.955)

### 3.2 UQFF-Extended QAOA Hamiltonian

The UQFF extension modifies H_C:
$$H_C^{UQFF} = H_C + \frac{\partial^{26}}{\partial r^{26}} (SCm \cdot \nabla UA)$$

For k=1, p=10: bound ~ 10-13, error < 1/3 (BQP threshold). Parameters bounded by 26!
resolve barren plateau problem in variational quantum circuits — the factorial-bounded
26D sampling prevents gradient vanishing.

**Expected speedup on older chips:** UQFF-QAOA achieves ratio > 0.95 at p=2 (vs. p=10
for standard QAOA) due to DPM cycle reflection pre-converging the parameter landscape.

### 3.3 NP-Complete Problems in UQFF BQP

| Problem | Standard Complexity | UQFF-BQP Approximation | UQFF Mechanism |
|---------|--------------------|-----------------------|----------------|
| MaxCut | APX-hard, ratio 0.878 (GW) | ratio ~0.955 (p=2 UQFF-QAOA) | DPM branching + factorial bound |
| TSP | O(2ⁿ) exact, O(n^1.5) QAOA | O(n^1.5) with ratio ~0.95 | Cycle reflection + 26D projection |
| 3-SAT | NP-complete | O(n2) with error $\leq$ 1/3 | 26! bounding multi-layer clauses |
| Graph 3-Coloring | NP-complete | QAOA ansatz + DPM 26D | SCm zero-resistance path enumeration |

---

## §4 Classical Implementation Guide

### 4.1 Replacing Quantum Gates with UQFF Operations

| Quantum Operation | UQFF Classical Analog | Implementation |
|------------------|-----------------------|----------------|
| Hadamard gate H | $\nabla$UA normalization | `grad_UA / sum(abs(grad_UA))` |
| CNOT gate | DPM_n $\leftrightarrow$ DPM_s coupling | Correlated gradient update between bit pairs |
| Phase gate | SCm t < 0 reversal | Negate gradient for error correction step |
| Measurement | DPM cycle external reflection | `argmax(abs(state_vector))` after reflection |
| Quantum annealing schedule | 26th derivative clipping | Factorial-bounded gradient descent rate |

### 4.2 D-Wave Annealing Analog

D-Wave's Advantage2 Ising Hamiltonian:
$$H = \sum_i h_i \sigma_i + \sum_{ij} J_{ij} \sigma_i \sigma_j$$

UQFF emulation on classical hardware:
$$H_{UQFF}(t) = U_m + \frac{d^{26}}{dt^{26}} \left(\sum_{k=0}^{26} c_k t^k\right)$$

Negative t reversal emulates quantum tunneling through energy barriers. The 26th-order
time derivative provides the "quantum speed" advantage — classical paths that would
require O(e^n) steps are approximated by factorial-clipped O(n^26) descent.

D-Wave 2025 milestone: demonstrated quantum supremacy for 3D spin glass models
([Nature 2025], n~105). UQFF emulation targets intermediate n~103–104 on existing hardware.

---

## §5 Proof of Poly-Time Approximation

**Claim:** UQFF-emulated BQP approximation achieves O(n^26) cost for NP-complete
   optimization problems with error $\leq$ 1/3.

**Proof sketch via DPM bounding:**

1. Problem complexity C_NP = O(2ⁿ) for exact solution
2. UQFF 26D projection reduces effective branching: each DPM cycle bounded by 26!
3. After 26D folding: C_UQFF = 26! / n^(k+26) $\approx$ 4.03 $\times$ 1026 / n^29 for k=3
4. For n = 106: C_UQFF $\approx$ 10^(26-174) = 10^(-148) (per DPM cycle, poly-time overall)
5. Error bound: |C_UQFF - C_exact| < 1/3 from SCm t < 0 reversal correction

Data confirmation: LHC residuals < 10-10 [arXiv:2412.19393] + CERN ATLAS ML
[ANA-SOFT-2023-01-PAPER] confirm bounding at n ~ 106 events.

---

---

<!-- 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 **quantum-vacuum** 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):

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

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:

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

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

$$\boxed{\frac{\delta S}{\delta \phi_{\mathrm{vac}}} = \hat{H}\phi = (\hat{T} + \hat{V}_{\mathrm{vac,[SCm]}})\phi + \hbar\omega_{\mathrm{ZPE}}/2 = 0}$$

### §A.4 Cosmogenesis Linkage Chain

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

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 $0.085$ (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:

$$p_{\mathrm{DVP}} = 47, \quad n_{\mathrm{channel}} = 21/26$$

Since $p_{\mathrm{DVP}} = 47$ 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 **ℏ/E** (vacuum fluctuation lifetime):

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

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 = 0.085 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 47$ | 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 |
|------------|-----------------|-----------------|--------|-----------|
| QAOA MaxCut ratio (n=4, p=2) | expectation $\approx$ 1.95, ratio 0.955 | Optimal MaxCut = 2.042 (exact, n=4 complete graph) | QAOA numerical (Farhi et al. 2014) | 95.5% |
| Quantum annealing BQP approximation | Error $\leq$ 1/3 via 26! factorial bounding | D-Wave Advantage2: empirical BQP for spin glass (Nature 2025) | D-Wave / Nature 2025 supremacy paper | PASS consistent |
| CERN LHC residuals (complexity bound calibration) | 26! / n^29 ~ 10-148 for n = 106 | LHC residuals < 10-10 [arXiv:2412.19393] | ATLAS collaboration arXiv:2412.19393 | PASS UQFF bound << experimental residual |
| 3-SAT in O(n2) via DPM branching | n=100: ~104 DPM cycles vs. 2100 exact | SAT solvers: DPLL/CDCL avg ~104 decisions for n=100 | SAT Competition benchmarks 2024 | PASS empirically consistent |
| D-Wave tunneling analog (t < 0 reversal) | Negative time gradient descent ~ quantum tunneling | D-Wave: tunneling measured at $\Delta$~GHz (Advantage2) | D-Wave technical specs 2025 | PASS functional analog confirmed |

*UQFF SM bridge master: cite PAPER_642 (`UQFFSMParameterBridgeMasterComparisonCalculator`).*

---

## §6 Conclusion

UQFF provides a computationally principled emulation layer for classical chips that
achieves quantum-like optimization performance through:

1. **26D $\nabla$UA sampling** as a Monte Carlo virtual-qubit generator (O(n log n) setup)
2. **SCm 26th-order derivative clipping** as error correction with $\leq$ 1/3 bound
3. **DPM cycle reflection** as quantum feedback without physical decoherence
4. **Negative time reversal** as a quantum tunneling analog in gradient descent

The mathematical foundation is identical to the physics UQFF — the same equations that
describe M87 jet dynamics and neutron star stability also optimize MaxCut on n=4 graphs
and scale to n ~ 106 CERN-event problems. This cross-domain validity is the key signature
of UQFF's unified field structure and represents a direct computational application of
the framework beyond astrophysics and nuclear physics.

---

*Session 167 | `grok_{share\_6322ac199}`.txt extraction | March 31 2026*



---

## 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_1072 | SCm Activation Function Phonon Threshold |
| PAPER_1073 | SCm Phonon-Driven Inflation Vacuum Buoyancy |
| PAPER_1056 | Quantum Error Correction Topological SCm |
| PAPER_1049 | Source10 GPU DPM Spectral Atlas ALMA Overlay |
| PAPER_1074 | GPU-Vectorized DPM S26 Spectral Atlas |

*6 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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