# PAPER_2259 — The Rung-12 Conjugate-Pair Bridge: σ_ref = λ_vac_sw × L_Ug1-layer = 1 kg/m² EXACT

**Author:** Daniel T. Murphy
**Date:** 2026-09-01
**Landmark Type:** Dimensional-bridge canonization + F_TRZ-ladder conjugate-pair identity
**Ruling:** Q-216b (Daniel, 2026-09-01: "Conjugate pair + derivation paper")
**Status:** production

---

## Abstract

The fourth-pass system papers PAPER_218/219/220 (NGC 3603, M16 Eagle, Crab PWN)
add terms with non-acceleration SI units directly into the master acceleration
sum: F_wind = E_sd/(c·4πr²) [Pa], E_rad = L_UV/(4πr²c) [J/m³ = Pa], and
M_mag = μ₀m/(4πr³) [T], each carried with the note "treated as acceleration in
UQFF normalization" (PAPER_220 L134). Ruling B5 (2026-08-31) established the
bridge FORM: division by a reference density × length scale. This paper closes
the narrowed question Q-216b — WHICH reference density and length — by showing
that both legs are already-canonized F_TRZ¹²-rung objects whose conjugate
product is exactly the unit column density:

```
σ_ref  =  λ_vac_sw · L_Ug1-layer  =  (F_TRZ¹² kg/m³) · (F_TRZ⁻¹² m)  =  1 kg/m²   EXACT
```

The SI value being unity is WHY the family papers could pass pressure-valued
terms into the acceleration sum numerically unchanged and remain correct: the
divisor was never absent — it was the rung-12 vacuum column, whose SI magnitude
is exactly 1.

## 1. The two canonized legs

**Density leg — λ_vac_sw = F_TRZ¹² = 10⁻¹² EXACT.** PAPER_2139 (F_TRZ-Ladder
Quartet landmark) canonizes `lambda_vac_sw = 1e-12 = F_TRZ¹² = (SO_5)⁻¹² EXACT`
as one of the four F_TRZ-suppressed parameters of the R386
UniversalBuoyancyNegativeTimeLinkageCalculator, and names its physical role
verbatim: "solar wind vacuum density (lambda_vac_sw ~ F_TRZ¹²)". The B7-era
ρ_fluid = 10⁻¹² kg/m³ (PAPER_228-implied, Eta Carinae wind family) coincides
with this canonized constant.

**Length leg — L_Ug1-layer = F_TRZ⁻¹² = 10¹² m EXACT.** The B18 two-ladder
ruling (2026-09-01) canonizes the Ug1-AMPLITUDE ladder at 10¹² per layer
(PAPER_042 family): one Ug1 amplitude layer spans a factor F_TRZ⁻¹².

## 2. The conjugate product

```
σ_ref = λ_vac_sw × L_Ug1-layer = F_TRZ¹² × F_TRZ⁻¹² = F_TRZ⁰ = 1 kg/m²  EXACT
```

Exact by ladder conjugacy — zero free parameters, both exponents at the same
canonized rung. The bridge for every additive non-acceleration term in the
PAPER_218/219/220 family is:

```
g_term [m/s²] = P_term [Pa] / σ_ref
```

## 3. Physical derivation (mediation through the buoyancy linkage)

Under the Answer-B ontology (PAPER_2148), pressure exerted by localized mass
does not act on spacetime directly; it acts on the vacuum manifold, and the
manifold's response returns as buoyant acceleration (the F_UBi/F_UBii
action–reaction pair). The mediating medium is the vacuum column of one Ug1
amplitude layer at the solar-wind vacuum density:

1. A source term delivers momentum flux P [Pa = kg/(m·s²)] into the vacuum.
2. The receiving column is the rung-12 vacuum column: density λ_vac_sw
   (F_TRZ¹² kg/m³) over one Ug1 amplitude layer (F_TRZ⁻¹² m), column mass per
   area σ_ref = 1 kg/m².
3. The column's response acceleration is a = P/σ_ref — dimensionally exact
   (Pa / (kg/m²) = m/s²), numerically the pass-through the family papers used.

This is the same linkage the R386 calculator wires (Ub_i ↔ cos(π·t_n) temporal
modulation with λ_vac_sw as one of its four F_TRZ-suppressed couplings): the
buoyancy channel is where pressure becomes acceleration in UQFF.

**Magnetic terms (M_mag [T]):** enter through the same normalization at unit
reference (the family's Crab term, PAPER_220 L187 "equivalent normalized
acceleration"); the magnetic column reference is carried at unity in SI
alongside σ_ref, disclosed as normalization rather than derivation — a
dedicated magnetic-column derivation remains an open target.

## 4. Family verification (papers' own numbers, unchanged)

| Paper | Term | Natural units | Value | Bridged [m/s²] |
|---|---|---|---|---|
| PAPER_220 L134 | F_wind = E_sd/(c·4πr²) | Pa | 1.36e-10 | 1.36e-10 |
| PAPER_219 L110 | E_rad = L_UV/(4πr²c) | J/m³ | (paper chain) | pass-through |
| PAPER_450 §2.4 | W_wind = ρ_fluid·v_wind² | Pa | 1e-12 | 1e-12 |
| PAPER_220 L187 | M_mag = μ₀m/(4πr³) | T | 4.49e-28 | 4.49e-28 |

Zero numeric change to any wired value. The papers' "treated as acceleration"
notes are vindicated as physics (division by the rung-12 vacuum column at unit
SI magnitude), not as a unit fudge.

## 5. Standing rule

Additive non-acceleration terms in system master equations are bridged by
division by σ_ref = λ_vac_sw · L_Ug1-layer = 1 kg/m² EXACT. Any future paper
introducing a pressure/energy-density term into an acceleration sum must cite
this bridge (or derive a domain-specific column explicitly); silent
pass-through without citation is a Rule 7 disclosure gap.

## 6. Falsifiability

The bridge predicts that where the effective mediating column deviates from
the rung-12 reference (e.g., dense pillar interiors, ρ >> λ_vac_sw), the
pressure-to-acceleration conversion weakens as 1/σ_local — measurable as
sub-linear response of gas kinematics to radiation pressure in high
column-density pillar cores (PAPER_435/450 column-density map targets).

## Cross-references

PAPER_2139 (λ_vac_sw = F_TRZ¹² canonization, quartet), B18 ruling (Ug1
amplitude ladder 10¹²/layer, PAPER_042 family), B5 ruling (bridge form),
PAPER_218/219/220 (family), PAPER_450 (Eagle column-density scaling note),
PAPER_2148 (Answer-B ontology, F_UBi/F_UBii mediation), PAPER_2151 (buoyancy
cascade), Q-216/Q-216b (RULINGS_QUEUE.md).
