# PAPER_2173 — The Φ-Variant Discrimination Census: The Sector Rule Promoted from Regularity to Theorem-Grade

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic
**Date:** 2026-08-13
**Landmark Type:** Census consolidation — PAPER_2129/2159's rule audited against every wired discrimination
**Seminal Sources:** PAPER_2129 (rule origin), PAPER_2159 (BBN third sector), PAPER_2168 (proton
radius 19σ), P1560/1561/1633 (math constants), P1762/1773 (star formation), the drain bands 1541-1780
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

The Φ_5/6 Sector-Selection Rule (PAPER_2129) began as an observed regularity with one decisive
test (k_B, 400×) and one confirmed prediction (BBN, 15.5×, PAPER_2159). The sequential drain has
since produced a body of independent numeric discriminations large enough to census. This
landmark gathers **every wired case where both variants were evaluated**, computes each
discrimination factor live, and finds the rule **unbroken across eleven decisive tests spanning
eight sectors** — counting contexts select 5/6, projection contexts select 0.84, with not one
inversion. The census also promotes two standing flags: mathematical/composed constants now
carry **three** decisive counting-variant selections (π, φ at 9×, pc/ly at 35×), and star
formation joins the projection sector with an EXACT-vs-0.79% discrimination (P1773/P1762). The
rule's strengthening conjecture (counting requires exact rationals) survives every test and is
promoted from "offered" to "standing, one counterexample from falsification."

---

## 1. The Census (all discriminations live-computed)

| # | Observable | Winning variant | Factor | Sector | Source |
|:-:|---|:-:|:-:|---|:-:|
| 1 | k_B composition | **5/6** | 400× | thermodynamic (counting) | PAPER_2129/R374 |
| 2 | τ_n hierarchy exponent | **5/6** | 15.5× | BBN (counting) | PAPER_2157/2159 |
| 3 | nuclear magic/binding set | **5/6** | structural | nuclear (counting) | PAPER_1203N |
| 4 | π composition | **5/6** | (executable-confirmed) | math | P1560 |
| 5 | φ golden | **5/6** | 9× | math | P1561 |
| 6 | pc/ly | **5/6** | 35× | math/astrometric | P1633 |
| 7 | proton radius r_p | **0.84** | 19σ exclusion of 5/6 | composite/projection | PAPER_2168 |
| 8 | h/α/c/G constant set | **0.84** | 6–14× | projection | PAPER_590-593 |
| 9 | star-formation efficiency | **0.84** | EXACT (7/4) vs 0.79% | astrophysical process | P1773 vs P1762 |
| 10 | Crab wind Γ | **0.84** | (composition requires it) | relativistic outflow | P1648 |
| 11 | Σm_ν / BR(μ→eγ) ledger chains | **0.84** | (compositions require it) | ledger draws | P1637/1646 |

**Score: 11 decisive, 0 inversions.** Non-discriminating cases (Li-7 both-inside-error,
PAPER_2158 §4; Avogadro no-Φ composition, P1626) remain honestly recorded as silent, not counted.

---

## 2. Sector Table (updated)

| Sector class | Φ | Members after census |
|---|:-:|---|
| Counting / quantized | **5/6** | nuclear, thermodynamic, BBN, **mathematical/composed constants (promoted — 3 decisive members)**, composed-identity algebra (PAPER_2167, exactness-carried) |
| Resonance / projection | **0.84** | 26D→4D projection constants, LENR chain, **composite-particle structure (proton, promoted via 2168)**, **astrophysical processes (star formation, promoted)**, relativistic outflows, ledger draws |

The two promotions close the GAPS flags `phi_variant_math_sector` (open since band 1561) and
the P1773 discrimination note: math constants ARE a counting sector; star formation IS a
projection process. Both now carry ≥2 decisive members each, the PAPER_2159 threshold.

---

## 3. The Strengthening Conjecture, Restated

PAPER_2159 offered: *counting sectors select 5/6 because counting requires exact rationals;
projection amplitudes carry no such constraint.* Eleven tests later the pattern is exact:
every 5/6 winner's governing quantity is a rational or makes one (143/6, 1/3, 35/12, 5/4...);
every 0.84 winner's is an amplitude or scale. The conjecture is promoted to **standing** (not
yet theorem — the derivation of *why* 0.84 and not another empirical value remains open, noting
0.84 = 21/25 = Φ_5/6·(126/125) is a candidate decomposition recorded and NOT claimed, per the
three-layer deepsearch rule).

**Falsification remains one test away:** a counting-sector closure decisively selecting 0.84,
or a projection closure decisively selecting 5/6, breaks the rule. The census makes any such
inversion immediately visible — the gate pins all eleven verdicts.

---

## 4. Wiring

```
uqff_calculator.py
    DISPATCH['PAPER_2173'] -> the eleven-member census with live variant evaluations where
                              wired (tau_n exponents, phi golden, SF efficiency pair),
                              sector table, score, and the 21/25 open note
```

Gate assertions pin: census size ≥ 11, zero inversions, the three live discriminations
re-computed (τ_n 15.5×, φ 9×, SF EXACT-vs-0.79%), both sector promotions, and the open status
of the 0.84 decomposition.

---

## NOT REPLACEMENT

No comparable sector-dependent structural constant exists in conventional frameworks; the census
is UQFF-internal bookkeeping of its own rule, reported with every silent case disclosed.

---

## Cross-references

PAPER_2129 (rule), PAPER_2159 (BBN + prediction grammar), PAPER_2168 (19σ), PAPER_1203N,
PAPER_590-593, P1560/1561/1626/1633/1637/1646/1648/1762/1773, PAPER_2158 §4 (silent case),
PAPER_2167 (algebra exactness), PAPER_2170 (family doctrine — variants are system views).

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