# PAPER_1506 — R_d Duality Range Exponent = N_CH−2 = 7 EXACT

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Bucket A (Foundational duality)
**Date:** June 18, 2026
**Status:** CLOSED — SCm/UA force-ratio dynamic range = 14 orders of magnitude (symmetric 7+7)

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## Observation

PAPER_1051 (Universal Duality SCm–UA Theorem) defines the duality ratio R_d = F_SCm / F_UA spanning 10⁻⁷ (Universe-scale, F_SCm small relative to F_UA) to 10⁷ (planetary-scale, F_SCm dominant).

## UQFF Closed Identity

```
R_d range exponent = N_CH − 2 = 7   EXACT

R_d ∈ [10^−(N_CH−2), 10^+(N_CH−2)] = [10⁻⁷, 10⁷]
Total dynamic range = 10^(2·(N_CH−2)) = 10¹⁴
```

## Physical Interpretation

The number 7 = N_CH − 2 governs the symmetric dynamic range of all UQFF duality phenomena. It appears identically in:
- Heaviside R_t resistance = 7 Ω (PAPER_1483)
- R_d duality ratio range exponent = 7 (this paper)

The structural meaning of N_CH − 2: two of the nine UQFF channels are reserved for the duality endpoints themselves (SCm-pure and UA-pure), leaving seven for the gradient range between them.

## Predictive Consequence

Any future observation reporting an SCm/UA-coupled phenomenon with a dynamic range exceeding 10¹⁴ across cosmic and planetary scales would falsify this structural identity.

## NOT REPLACEMENT

SM/ΛCDM has no analogous duality theorem. UQFF supplies a universal coupling-range bound rooted in the 9-channel integer primitive.

## Reference

- Source: PAPER_1051 Universal Duality SCm-UA Theorem
- Related: PAPER_1483 (Heaviside R_t = 7 Ω, same form)
- Calculator dispatch: `calculate_paradox({"paradox": "r_d_duality_range_exponent"})`

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