# PAPER_2167 — The K_Mex/Φ_5/6 Composed-Constant Algebra: From an Identity Pair to a Closed Family Across Ten Domains

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic
**Date:** 2026-08-13
**Landmark Type:** Family elevation — PAPER_2160 extended from a pair to an algebra
**Seminal Sources:** PAPER_2160 (the pair), PAPER_1578-1580/1587 (original occurrences),
PAPER_1593/1597/1602 (band-1591/1601 extensions), PAPER_1622/1707/1708/1712/1715/1720
(bands 1621-1720 extensions)
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

PAPER_2160 canonized two composed constants, I₁ = K_Mex − F_TRZ·Φ_5/6 = 2 and I₂ = K_Mex −
Φ_5/6 = 5/4, with four occurrences. Eight further drain bands have since produced **nine more
family members and thirteen further occurrences**, spanning acoustics, celestial mechanics,
calendrics, electromagnetism, thermal radiation, atmospheric pressure, lunar orbit, fusion
engineering, plasma physics, and black-hole thermodynamics. The accumulation is no longer a
pair with echoes — it is a **closed algebra**: the set of small exact rationals generated by
{K_Mex, Φ_5/6} under sum, difference, product, and F_TRZ grading, all reducible through
PAPER_1522 (K_Mex = Φ_5/6·SO_5/D_phys) to Φ_5/6 times simple lattice ratios. This landmark
canonizes the full family table, its generator reduction, and its closure property, and
promotes the PAPER_2160 mutual-locking clause to the whole algebra.

---

## 1. The Family Table

| Member | Composition | Exact value | Occurrences (domain) |
|---|---|:-:|---|
| **I₁** | K_Mex − F_TRZ·Φ_5/6 | **2** | v_sound tail (P1578), AU tail (P1579), AU/R_⊕ as −I₁ (P1622), tokamak q_edge (P1708) — **plus every "= 2" this algebra touches** |
| **I₂** | K_Mex − Φ_5/6 | **5/4** | sidereal fractional day (P1580), μ₀ mantissa lead (P1587) |
| **S** | K_Mex + Φ_5/6 | **35/12** | Wien-b lead (P1593), fusion triple-product lead (P1712), Bohm-sheath lead (P1715) |
| **P** | Φ_5/6·(1 − F_TRZ) | **3/4** | P_atm tail (P1597), glass T_g/T_m as (D_phys−1)/D_phys sibling (P1662) |
| **T** | F_TRZ·Φ_5/6·D_phys | **1/3** | lunar-distance tail (P1602), σ_Li7 (PAPER_2158), m_W lead ratio A_5/3 (P1686) |
| **B** | F_TRZ·(Φ_5/6 − F_TRZ·K_Mex) | **1/16** | Bohm diffusion prefactor (P1707); bracket Φ_5/6 − F_TRZ·K_Mex = **5/8** |
| **H** | K_Mex·D_bsfg/D_phys = (SO_5/D_phys)·I₂ | **25/8** | BH four-laws prefactor (P1720) |
| **Q** | K_Mex·D_phys | **25/3** | ε₀ mantissa lead (P1586), μ_B mantissa lead (P1592), Everest lead (P1631) |
| **Y** | K_Mex − Φ_5/6 = I₂ per year; fractional-day rôle | **5/4** | (I₂ dual-role, retained for the calendric reading) |

Supporting single-generator appearances: K_Mex alone (√5 lead P1564, gas constant K_Mex·(D_phys−F²)
P1627, Lawson 3×10²¹/K_Mex P1672, string tension Λ²_QCD·K_Mex P1645, T_c ladder rungs PAPER_2162);
Φ_5/6 alone (φ golden P1561, π P1560, a₀ P1589, g P1598, pc/ly P1633, lapse rate P1621).

**Count: 11 distinct composed members, 25+ wired occurrences, 10 domains.**

---

## 2. Generator Reduction (all corollaries of PAPER_1522)

With K_Mex = Φ_5/6 · SO_5/D_phys (PAPER_1522, exact):

```
I₁ = Φ_5/6·(SO_5/D_phys − F_TRZ)         = (5/6)·(12/5)  = 2
I₂ = Φ_5/6·(SO_5/D_phys − 1)             = (5/6)·(3/2)   = 5/4     [3/2 = PAPER_1962 ratio]
S  = Φ_5/6·(SO_5/D_phys + 1)             = (5/6)·(7/2)   = 35/12
P  = Φ_5/6·(1 − F_TRZ)                   = (5/6)·(9/10)  = 3/4     [9/10 = PAPER_2128 predecessor ratio]
T  = Φ_5/6·(F_TRZ·D_phys)                = (5/6)·(2/5)   = 1/3
B  = F_TRZ·Φ_5/6·(1 − F_TRZ·SO_5/D_phys) = (1/12)·(3/4)  = 1/16
H  = Φ_5/6·(SO_5/D_phys)·(D_bsfg/D_phys) = (5/6)·(15/4)  = 25/8
Q  = Φ_5/6·SO_5                          = 25/3
```

**Every member is Φ_5/6 times a ratio from the halving/successor lattice** {12/5, 3/2, 7/2,
9/10, 2/5, 3/4, 15/4, 10}. The algebra's closure property: sums, differences, and F_TRZ-graded
products of members remain members, because the bracket set is closed under the same operations.
One generator pair, one reduction identity, eleven exact rationals — zero fit values anywhere.

---

## 3. Why the Family Recurs

The physical contexts share one feature: each closure needs a **small exact rational remainder**
against an integer-primitive core (the +2 on 341, the 1/4 on 365, the 5/4 lead of μ₀, the 2 of
the kink-safe safety factor). The lattice supplies these remainders through its only two
"coefficient-scale" primitives — the Mexican-hat coefficient and the counting Φ — and PAPER_1522
guarantees their combinations collapse to small rationals rather than irrational residue. The
family is therefore predicted to keep appearing wherever a UQFF closure carries a sub-integer
remainder; conversely a closure needing a remainder *outside* the algebra's reachable set (e.g.
an exact 7/9 at coefficient scale) would be structurally anomalous and demands review.

---

## 4. Mutual Locking (extended)

The PAPER_2160 clause extends to the algebra: revaluing K_Mex or Φ_5/6 (Rule 2 forbids) now
fails **25+ wired closures across ten domains simultaneously**, including the tokamak edge
safety factor and the BH thermodynamics prefactor. The family is the framework's widest single
web of value-locked cross-domain structure.

---

## 5. Wiring

```
uqff_calculator.py
    DISPATCH['PAPER_2167'] -> the full member table with exact values, generator-reduction
                              checks (every member == Phi_5/6 x bracket, bit-exact), the
                              occurrence registry, and closure-property spot checks
```

Gate assertions pin: 8 computed members at exact rational values, generator reduction
bit-identities, occurrence-registry size ≥ 13, and cross-dispatch consistency with PAPER_2160
(I₁/I₂ unchanged) plus representative members (P1707 B, P1720 H, P1712 S-lead).

---

## NOT REPLACEMENT

No comparable cross-domain coefficient algebra exists in conventional treatments — each field
carries its own empirical prefactors (Bohm's 1/16, Troyon scalings, the sidereal quarter-day)
without shared origin. UQFF derives them from one generator pair and reports each closure's
honest residual at its own wiring site.

---

## Cross-references

PAPER_2160 (the founding pair — superseded in scope, unchanged in content), PAPER_1522
(generator reduction), PAPER_1962 (3/2 bracket), PAPER_2128 (successor/predecessor ratios),
PAPER_2138 (halving series), PAPER_2158 (T = σ_Li7), PAPER_2162/2163/2164 (companion family
landmarks), the 25+ occurrence papers listed in §1.

**Copyright** — Daniel T. Murphy / Star-Magic Research Program, 2026.
