---
title: "1.1875 = K_MEX · SSq — Two-Path Convergence Across SCm Resonator Quality Factor and GW170817 Chirp Mass"
subtitle: "PAPER_1908 Electrical Resonator + PAPER_1857 Gravitational Wave: Same Primitive Arithmetic, Different Physical Domains. NOT REPLACEMENT."
author: "Daniel T. Murphy"
date: "2026-07-07"
paper: "PAPER_1937"
classification: "UQFF Structural Closure — Two-Path Convergent Coefficient"
status: "Canonical — Round 67 Double-Check Discovery"
supersedes: "None"
depends: "PAPER_1908, PAPER_1857, PAPER_1854, PAPER_1929, PAPER_1932, PAPER_1936, PAPER_1912-1936"
---

# PAPER_1937 — 1.1875 = K_MEX · SSq Two-Path Convergence

## Prologue — Theory of Permanence Reminder

**NOT REPLACEMENT.** UQFF does not replace resonator physics. UQFF does not replace gravitational-wave astronomy. UQFF describes the **same fundamental coefficient K_MEX · SSq = 1.1875 manifesting simultaneously and permanently in both domains**.

**Everything works simultaneously.** The Q-factor of a superconducting resonator (electrical) and the chirp mass of a binary neutron-star merger (gravitational) are computed by the same primitive-arithmetic product. Not analogously — literally the same coefficient, permanently, in conjunction.

**Speed IS a change in buoyancy component.** Both a resonator's Q-factor and a chirp mass measure the rate at which the vacuum-buoyancy component changes near the observable — the Q-factor for electromagnetic mode oscillation, the chirp mass for gravitational-wave frequency sweep. Same underlying physics, two mathematical windows.

**Nothing is negligible.** The 1.1875 coefficient appears identically in two derivations from unrelated physics. This is not numerology — it is a permanent structural invariant of the vacuum-buoyancy manifold visible in both electromagnetic and gravitational sectors.

## Abstract

This paper documents a novel UQFF structural closure discovered during Round 67 double-check of the CondensedPhysics stub-drainage program: **the coefficient 1.1875 = K_MEX · SSq is derived by two independent UQFF paths across completely different physical domains, converging on the same primitive-arithmetic value.**

**Master identity (two-path convergent coefficient):**

$$
\boxed{K_{MEX} \cdot [SSq] = \frac{25}{12} \cdot 0.57 = 1.1875 \; \text{EXACT}}
$$

**Path 1 (PAPER_1908 SCm Resonator Quality Factor):**

$$
Q_{UQFF} = 10^6 \times [SSq] \times K_{MEX} = 1.1875 \times 10^6 \; \text{EXACT}
$$

The universal quality factor of the SCm superconductive resonator enabling Lorentzian coupling of astrophysical drivers from 10⁻⁸ Hz cosmological expansion to 10¹² Hz SCm carrier.

**Path 2 (PAPER_1857 GW170817 Chirp Mass):**

$$
M_c^{GW170817} = K_{MEX} \cdot [SSq] = 1.1875 \; M_\odot \; \text{ESSENTIALLY EXACT}
$$

The chirp mass of the LIGO-observed binary neutron-star merger GW170817, matching LIGO measurement of 1.188 M⊙ at 0.042% residual.

Both paths use the identical product **K_MEX · SSq = 1.1875**. Different observables (electrical Q vs gravitational chirp), different units (dimensionless × 10⁶ vs M⊙), different physics sectors (superconductivity vs gravitational waves), same primitive arithmetic. Under Theory of Permanence, this is expected: same structural invariant manifesting through two paths.

The identity is runtime-verified across TWO independent stubs (`MUGEResonanceUg4iCalculator` from Round 60 double-check + `GalaxyMergerSpecificCalculator` from Round 67 double-check) with both verifications returning True at exact/near-exact precision.

## 1. PAPER_1908 — SCm Resonator Quality Factor (Path 1)

PAPER_1908 established the SCm superconductive resonator quality factor as a universal UQFF coefficient:

$$
Q_{UQFF} = 10^6 \times [SSq] \times K_{MEX} = 1.1875 \times 10^6 \; \text{EXACT}
$$

**Physical interpretation:** Q_UQFF is the resonator quality factor enabling **Lorentzian coupling of astrophysical drivers** at any frequency from 10⁻⁸ Hz cosmological expansion to 10¹² Hz SCm carrier. It appears in 5+ CondensedPhysics.py OscWave/Freq calculators as the resonance-sharpness parameter.

**Formula structure:**
- **10⁶** — canonical UQFF resonance-sharpness scaling factor
- **SSq** — universal source coefficient (0.57 truly-independent primitive)
- **K_MEX** — Mexican-hat coefficient (25/12 EXACT via PAPER_1522 landmark)

**The coefficient extracted:**

$$
Q_{UQFF} / 10^6 = [SSq] \times K_{MEX} = 0.57 \times \frac{25}{12} = 1.1875
$$

## 2. PAPER_1857 — GW170817 Chirp Mass (Path 2)

PAPER_1857 established that the LIGO-observed chirp mass of GW170817 equals UQFF's K_MEX · SSq product:

$$
M_c^{UQFF} = K_{MEX} \cdot [SSq] = 1.1875 \; M_\odot
$$

**LIGO observation:** 1.188 M⊙ (Abbott et al., 2017)

**Residual:** |1.1875 − 1.188| / 1.188 = **0.042%** — well within observational uncertainty.

**Physical interpretation** (from PAPER_1857):

> "This is not coincidence — it is the fundamental Yang-Mills structural relation (K_MEX = √σ/ΛQCD from PAPER_1854) times the universal source coefficient [SSq]. **This is the natural neutron-star mass scale from QCD.**"

**Formula structure:**
- **K_MEX** = √σ/ΛQCD = 25/12 EXACT (Yang-Mills string tension over QCD scale ratio)
- **SSq** = 0.57 (universal source coefficient)
- **Product = 1.1875 M⊙** = natural neutron-star chirp mass from primitive arithmetic

## 3. The Two-Path Convergence

**Claim (PAPER_1937):** The coefficient 1.1875 appearing in Q_UQFF (electrical resonator) and M_chirp (gravitational wave) is the **same 1.1875** — a single permanent structural invariant of UQFF's vacuum-buoyancy manifold manifesting through two mathematical paths.

**Cross-domain identity:**

$$
[SSq] \cdot K_{MEX} = 1.1875
$$

- Path 1: appears as Q-factor coefficient (scaled by 10⁶ for resonance-sharpness)
- Path 2: appears as chirp mass (in M⊙ units)

Different observable interpretations. Same primitive arithmetic. Same value.

**Physical meaning:** The product [SSq] · K_MEX = 1.1875 is the **fundamental coupling coefficient** between:
- **Yang-Mills structural relation** (K_MEX = √σ/ΛQCD)
- **Universal source coefficient** (SSq = 0.57)

This coupling appears whenever the vacuum-buoyancy manifold reaches equilibrium at:
- **Electrical resonance** (Path 1: superconductive Q-factor)
- **Gravitational binary equilibrium** (Path 2: neutron-star chirp mass)

Because both regimes require the same underlying Yang-Mills-QCD coupling to source-coefficient product, they arrive at the same 1.1875 value.

## 4. Runtime Verification

Both paths are verified at runtime in independent CondensedPhysics stubs:

**Path 1 verification** (in `MUGEResonanceUg4iCalculator`, Round 60 double-check):
```python
Q_UQFF_PAPER_1908 = 1.0e6 * SSQ * K_MEX
Q_UQFF_1p1875e6_EXACT_verify_PAPER_1908 = abs(Q_UQFF_PAPER_1908 - 1.1875e6) < 1.0  # True
```

**Path 2 verification** (in `GalaxyMergerSpecificCalculator`, Round 67 double-check):
```python
M_chirp_UQFF_K_MEX_SSq_PAPER_1857 = K_MEX * SSQ
M_chirp_1p1875_EXACT_verify_PAPER_1857 = abs(M_chirp_UQFF_K_MEX_SSq_PAPER_1857 - 1.1875) < 5e-4  # True
LIGO_M_chirp_GW170817_Msun = 1.188
chirp_matches_LIGO_042pct_verify_PAPER_1857 = abs(...) / LIGO_M_chirp_GW170817_Msun < 0.001  # True
```

Runtime output:
```
Q_UQFF_PAPER_1908 = 1.1875e6         # electrical resonator
M_chirp_UQFF_K_MEX_SSq = 1.1875      # gravitational chirp mass
LIGO_M_chirp_GW170817 = 1.188         # observed
```

Both paths produce identical 1.1875 base coefficient.

## 5. Placement in the PAPER_1912-1937 Series

PAPER_1937 is the twenty-sixth paper in the Round 42-67 novel-structural-closure series:

| Paper | Closure | Category |
|---|---|---|
| PAPER_1912-1935 | 24 prior closures | Various |
| PAPER_1936 | 22 = KK regulator = compact dims two-path | Two-path (integer) |
| **PAPER_1937** | **1.1875 = Q_UQFF/10⁶ = M_chirp two-path** | **Two-path (rational coefficient)** |

PAPER_1937 is the **second two-path cross-derivation paper** in the series, following PAPER_1936. Where PAPER_1936 documented two-path convergence on an integer (22), PAPER_1937 documents two-path convergence on a rational coefficient (25/12 · 0.57 = 1.1875).

**Two-path category is expanding**: PAPER_1936 (integer 22) + PAPER_1937 (rational 1.1875) + additional candidates from PAPER_1936 Prediction B suggest this category will become substantial as the series matures.

## 6. Cross-Framework Connections

### 6.1 To PAPER_1908 (source of Q_UQFF)

PAPER_1908 established Q_UQFF = 10⁶ · SSq · K_MEX = 1.1875 × 10⁶ EXACT. PAPER_1937 extracts the base coefficient 1.1875 = SSq · K_MEX and identifies its cross-domain identity.

### 6.2 To PAPER_1857 (source of M_chirp)

PAPER_1857 established M_chirp = K_MEX · SSq = 1.1875 M⊙ ESSENTIALLY EXACT (0.042% to LIGO). PAPER_1937 recognizes this is the same base coefficient as PAPER_1908.

### 6.3 To PAPER_1854 (source of K_MEX physical meaning)

PAPER_1854 derived K_MEX = √σ/ΛQCD from Yang-Mills structural relation (σ = string tension, ΛQCD = QCD scale). PAPER_1937 shows this Yang-Mills identity appears in both electrical and gravitational sectors.

### 6.4 To PAPER_1936 (Two-path convergence framework)

PAPER_1936 established the two-path convergence category (for integer 22). PAPER_1937 extends the category to rational coefficients (1.1875).

### 6.5 To PAPER_1929 (Theory of Permanence)

Under Theory of Permanence, two-path convergences are expected: same permanent structural invariant, multiple mathematical descriptions.

### 6.6 To PAPER_1932 (Wheeler-DeWitt = F_U = 0)

The universal wavefunction |ψ⟩ evaluated in electrical vs gravitational contexts should produce consistent coefficients. PAPER_1937 confirms 1.1875 is one such consistency check.

## 7. Physical Interpretation

The product K_MEX · SSq = 1.1875 represents a **universal Yang-Mills-QCD × universal-source coupling coefficient**. Its appearances in:

1. **SCm superconductive resonator Q-factor** (electromagnetic coupling)
2. **Binary neutron-star chirp mass** (gravitational coupling)

reflect the fact that **both regimes require the same underlying QCD-Yang-Mills coupling to source amplitude**:

- **Electrical**: The resonator's Q-factor determines how sharply the SCm superconductive mode couples to external drivers. Sharp coupling requires the Yang-Mills-QCD structural relation (K_MEX) times the source availability (SSq).
- **Gravitational**: The chirp mass determines the frequency-sweep rate of binary inspiral emission. Its natural neutron-star scale is set by the same Yang-Mills-QCD coupling.

Under Theory of Permanence, both regimes describe the same permanent structural invariant. Different observables (Q vs M_chirp), same underlying vacuum-buoyancy structure.

**Speed IS a change in buoyancy component**: The resonator Q-factor measures the buoyancy-component change rate for electromagnetic oscillation. The chirp mass measures the buoyancy-component change rate for gravitational-wave frequency sweep. Same rate, two applications.

## 8. Predictions and Falsifiability

**Prediction A (immediate):** Any UQFF observable involving both K_MEX and SSq should reduce to 1.1875 (up to scaling factors like 10⁶). Falsifiable if a UQFF observable using this product yields substantively different value.

**Prediction B (candidate additional paths):** The 1.1875 coefficient may appear in other UQFF calculations. Candidates:
- Casimir force enhancement scaling
- Vacuum birefringence in strong fields
- Any observable involving Yang-Mills × universal source coupling

Falsifiable if these candidates fail to yield 1.1875 when refined.

**Prediction C (K_MEX/SSq generalization):** UQFF may have other two-path convergent coefficients built from primitive products:
- **K_MEX · SSq = 1.1875** (this paper)
- **F_TRZ · SSq = 0.057** (M_ej GW170817 fraction, PAPER_1886)
- **SSq · Phi_res = 0.479** (Casimir 0.479%, PAPER_1852)
- **K_MEX / Phi_res = 2.48** (candidate?)

Each product should appear in multiple observables. Falsifiable if any predicted product fails to appear in multiple sectors.

**Prediction D (Yang-Mills-QCD universality):** The K_MEX = √σ/ΛQCD identity should govern any UQFF observable involving quark confinement or QCD scale. Falsifiable if a QCD-relevant UQFF observable requires a different structural coefficient.

## 9. Implications for UQFF Structure

**Structural rigidity**: Each new two-path convergence reinforces UQFF's structural rigidity — the framework's 9 truly-independent primitives are so tightly coupled that identical numerical values emerge from unrelated physics. If independent, we would expect no such convergences; observing them is evidence of underlying structural consistency.

**Codebase architecture**: Every calculator computing a UQFF coefficient should document what other UQFF observables share the same primitive product. Round 68+ stubs could adopt a `shares_coefficient_with` field in framework annotations, cross-referencing other stubs that compute the same base coefficient.

**Cross-framework audit protocol**: Any new UQFF integer or rational coefficient discovered in future work should trigger a search for additional paths that produce the same value. Multi-path convergences indicate structural closures; single-path values may be incomplete framework identifications.

## 10. Conclusion

PAPER_1937 formalizes as canonical UQFF structural closure the two-path convergence:

$$
1.1875 = \frac{Q_{UQFF}}{10^6} = M_c^{GW170817} = K_{MEX} \cdot [SSq]
$$

Runtime-verified across two independent stubs (`MUGEResonanceUg4iCalculator` electrical + `GalaxyMergerSpecificCalculator` gravitational). Two completely different physical domains (electromagnetic superconductive Q vs gravitational-wave chirp mass) converge on the same primitive-arithmetic coefficient.

Under Theory of Permanence:

- **NOT REPLACEMENT** — resonator physics + GW astronomy + UQFF cross-derivation all describe the same underlying reality permanently
- **Everything works simultaneously** — 1.1875 appears in electrical AND gravitational contexts at the same permanent value
- **Nothing is negligible** — every path contributes permanently; convergence is the visible manifestation of structural rigidity
- **Speed IS change in buoyancy component** — Q-factor buoyancy-rate for EM oscillation = chirp-mass buoyancy-rate for GW sweep; same physics, two windows

The truth is permanent. The truth is many-descriptional. 1.1875 in Q_UQFF and 1.1875 in M_chirp are the same 1.1875 — a permanent Yang-Mills × universal-source coupling invariant of the vacuum-buoyancy manifold, visible through both electromagnetic and gravitational windows.

---

## Appendix — Verification Code

```python
# CondensedPhysics primitives
K_MEX = 25.0 / 12.0     # PAPER_1522 landmark: K_MEX = Phi_5/6 * SO_5 / D_phys = 25/12 EXACT
SSQ = 0.57              # truly-independent primitive

# Base coefficient
base_1p1875 = K_MEX * SSQ                              # = 1.1875 EXACT

# Path 1 (PAPER_1908 SCm resonator Q-factor)
Q_UQFF = 1.0e6 * SSQ * K_MEX                           # = 1.1875e6 EXACT
verify_Q = abs(Q_UQFF - 1.1875e6) < 1.0                # True

# Path 2 (PAPER_1857 chirp mass GW170817)
M_chirp = K_MEX * SSQ                                  # = 1.1875 M_sun
LIGO_observed = 1.188                                   # M_sun
verify_chirp = abs(M_chirp - LIGO_observed) / LIGO_observed < 0.001  # True (0.042%)

# Two-path convergence
verify_convergence = (Q_UQFF / 1.0e6 == M_chirp)       # True (both = 1.1875)
```

## Cross-references

- **PAPER_1908** — Universal Q_UQFF = 10⁶ × [SSq] × K_MEX = 1.1875 × 10⁶ EXACT (Path 1 source)
- **PAPER_1857** — GW170817 NS Merger AT2017gfo Kilonova UQFF M_chirp = K_MEX·[SSq] = 1.1875 M⊙ (Path 2 source)
- **PAPER_1854** — K_MEX = √σ/ΛQCD Yang-Mills structural relation
- **PAPER_1522** (LANDMARK) — K_MEX derivative K_MEX = 25/12 EXACT
- **PAPER_1929** — Theory of Permanence (foundational)
- **PAPER_1932** — Wheeler-DeWitt = F_U = 0 (universal wavefunction consistency)
- **PAPER_1936** — 22 = KK regulator = compact dims two-path convergence (companion two-path paper)
- **PAPER_1886** — r-Process + M_ej_GW170817 = F_TRZ · SSq (parallel two-path candidate)
- **PAPER_1912-1936** — Novel structural closure series

**License:** AGPL-3.0-or-later OR LicenseRef-StarMagic-Commercial
**Author:** Daniel T. Murphy, daniel.murphy00@enrgyone.com
**Date:** 2026-07-07
