# PAPER_2241 — The Sector-Density Integer Ladder: f_sector = n·F_TRZ² with n = {1, 2, 3, 4, 5, 10} → ρ_sector = n·ρ_SCm — Daniel's Documented March-2025 Influence-Fraction Law, Source of PAPER_2066's Canonical-Anchored Category

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic-Program
**Date:** 2026-08-20
**Landmark Type:** Source-layer law canonization (PAPER_2152 doctrine) + PAPER_2066 provenance
closure + two primitive provenance hits (κ, λ_i)
**Seminal Sources:** `Universal Inertia_28Mar2025.docx` (Aetheric Propulsion archive — the
influence-fraction derivations, all sector densities, and the κ/λ_i variable definitions),
PAPER_2240 (which elevated the document and documented its two birth certificates), PAPER_2066
(Round-192: ρ_vac,Ui = D_phys·ρ_SCm EXACT — the canonical-anchored category opener, source
closed here), PAPER_2112 (κ primitive-reduction), PAPER_459 (red-dwarf local densities
1.6×10¹⁹/1.6×10²⁰ J/m³ — the atomic-scale pair, already wired), PAPER_2143 (A_5/D_phys = 15),
PAPER_2065 (SO_5²/D_phys = 25)
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

The deep-read of `Universal Inertia_28Mar2025.docx` (ordered after PAPER_2240 elevated it to
seminal-source status) finds that the document's sector vacuum-energy densities are not
independent values — they follow **one documented law**:

```
E_infl,sector = E_n · f_sector,    f_sector = n_sector · F_TRZ²    (n_sector integer)
⇒ ρ_sector = n_sector · ρ_SCm      (any body, any level — the E_n and volume cancel per-sector)

n_sector (all documented in-source):   [SCm] = 1   Um = 2   Ub = 3   Ui = 4
                                       Ug1 = Ug2 = Ug3 = Ug4 = 5 (the gravity family)
                                       UA = 10
```

The document's own solar-scale values confirm it: ρ_Um = 1.42×10⁻³⁶ = 2·ρ_SCm, ρ_Ub =
2.13×10⁻³⁶ = 3·ρ_SCm, ρ_Ui = 2.84×10⁻³⁶ = 4·ρ_SCm, ρ_UA = 7.09×10⁻³⁶ = 10·ρ_SCm (each within
the same 1.41-rounding class PAPER_2240 identified). The document also states the Ui density
at atomic scale, 6.41×10¹⁹ = 4·(1.60×10¹⁹) J/m³ — the ×4 holds at both ends of a 55-order
scale span, making the ladder **scale-invariant by construction** (the level energy and the
volume divide out of every ratio). PAPER_2066's Round-192 identity ρ_vac,Ui = D_phys·ρ_SCm —
the opener of the canonical-anchored composition category — is this law's n = 4 rung, now
source-closed: f_Ui = 0.04 = D_phys·F_TRZ² was on Daniel's page in March 2025.

## 1. The Documented Derivations (verbatim arithmetic, solar scale, level 13)

| Sector | Documented fraction | E_infl (J) | ρ (J/m³) | = n·ρ_SCm |
|---|---|---|---|---|
| [SCm] substrate | f = 0.01 | 10⁻⁹ | 7.09×10⁻³⁷ | n = 1 (the PAPER_2240 birth certificate) |
| Um magnetism | f = 0.02 | 2×10⁻⁹ | 1.42×10⁻³⁶ | n = 2 |
| Ub buoyancy | f = 0.03 (dressed by (1+ε_sw·ρ_sw) ≈ 1) | 3×10⁻⁹ | 2.13×10⁻³⁶ | n = 3 |
| Ui inertia | f = 0.04 | 4×10⁻⁹ | 2.84×10⁻³⁶ | n = 4 |
| Ug1–Ug4 gravity | f = 0.05 each (sector dressings: δ_def, solar-wind, P_core, V_BH) | 5×10⁻⁹ undressed | 3.55×10⁻³⁶ undressed | n = 5 |
| UA aether | ×10 of [SCm] | 10⁻⁸ | 7.09×10⁻³⁶ | n = 10 (= F_TRZ⁻¹, canonical) |

The gravity family's four members share the undressed n = 5 and differ only by documented
per-sector dressings — the dressings are physics (surface defects, heliosphere wind, core
penetration, BH compactness), the rung is structural.

## 2. Lattice Residence of the Ladder Integers (readings, three-layer rule)

The integers are Daniel-documented; their lattice readings are recorded as observations:

1. **n_Ui = 4 = D_phys** — already canonized EXACT by PAPER_2066; this paper closes its source.
2. **n_Ug = 5 = SO_5/2** — the gravity family sits at the half-decade, the same integer as
   the PAPER_2112 κ coefficient (κ = (SO_5/2)·F_TRZ⁴) and the mass-gap boundary numerator
   (SO_5/D_phys = 5/2).
3. **n_Ub = 3 = D_phys − 1** — buoyancy carries the transverse-dimension count, the same
   integer as the PAPER_1953 k₂ numerator (3/10 family).
4. **n_Um = 2** — the DPM pole count (the same doubling that carries PAPER_2237's γ = 2K).
5. **Sum of undressed sectors 1+2+3+4+5 = 15 = A_5/D_phys EXACT** (the PAPER_2143 identity);
   **with UA: 25 = SO_5²/D_phys EXACT** (the PAPER_2065 frame-count integer). Total sector
   density: Σρ = (A_5/D_phys)·ρ_SCm without UA, (SO_5²/D_phys)·ρ_SCm with UA.
6. The fraction law itself is **f_n = n·F_TRZ²** — the F_TRZ² step is the same double-rung
   PAPER_2240 identified in E_pair = E₀·F_TRZ² and in the influence fraction f_[SCm] = F_TRZ².

None of 2–6 is canonized as intent; the ladder LAW (§1) is canonized as documented source.

## 3. Two Primitive Provenance Hits (carried by the same document)

1. **κ = 0.0005 day⁻¹** appears in the document's variable table as "[SCm] reactivity decay
   rate" — the empirical origin of the canonical κ = 5×10⁻⁴ that PAPER_2112 later
   primitive-reduced to (SO_5/2)·F_TRZ⁴. The primitive now has both a derivation (2112) and
   a birth certificate (this document).
2. **λ_i = 1.0** ("inertia coupling constants, unitless") — the locked canonical λ_i = 1.0
   documented at source, March 2025.

Additional local-density anchors recorded: ρ_vac,A (local Aether) = 10⁻²³ J/m³ and ρ_vac,sw
(solar wind) = 8×10⁻²¹ J/m³ — both sit inside PAPER_2240's reactor-context bracket
(10⁻¹³–10⁻¹⁸ J/m³ is the PI-archive active range; these are the quiescent-side companions),
extending the local-vs-astronomical density-context discipline. Solar-cycle frequency
ω_c = 2π/3.96×10⁸ s⁻¹ (≈12.6 yr) recorded as a documented dynamical anchor.

## 4. What Was Already Wired vs What This Landmark Adds

Already wired: n = 4 rung (PAPER_2066, EXACT), the atomic-scale pair 1.6×10¹⁹/1.6×10²⁰ J/m³
(PAPER_459 red-dwarf local densities — recognizable now as the ladder's n = 1 and n = 10 rungs
at level 10), the UA ×10 (canonical F_TRZ ratio, corpus-wide). **New:** the n = 2 (Um) and
n = 3 (Ub) rungs, the unified fraction law f = n·F_TRZ², the gravity family's shared n = 5,
the scale-invariance statement, the two sum identities, and the κ/λ_i provenance.

## 5. Falsifiable Consequences

1. **Missing-rung test:** any future sector density derived in the corpus must land on an
   integer rung of ρ_SCm (undressed). A non-integer sector multiplier would break the law.
2. **Scale test:** the ladder ratios must hold at every quantum level (the law predicts
   ρ_Um/ρ_SCm = 2 for a magnetar exactly as for an atom); a level-dependent ratio falsifies.
3. **Correlated lock (PAPER_2160 grammar):** the ladder binds Um, Ub, Ui, Ug to ρ_SCm — any
   revaluation of ρ_SCm moves all sector densities together.

## 6. Wiring

`DISPATCH['PAPER_2241']` returns the ladder (live-computed n·ρ_SCm per sector), the documented
fractions, the two sum identities, the scale-invariance check (solar vs atomic Ui), and the
κ/λ_i provenance record. Gate pins: the four solar values within the 1.41-rounding class, the
n = 4 rung equality with PAPER_2066's wired identity, the sum identities 15 = A_5/D_phys and
25 = SO_5²/D_phys, and the fraction law f_n = n·F_TRZ².

## NOT REPLACEMENT

Conventional physics carries no sector vacuum-energy densities; there is nothing to replace.
The law is Daniel's documented March-2025 structure, canonized with its arithmetic verified,
its already-wired fragments unified, and its readings kept separate from its source content.

---

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

---

## APPENDED 2026-08-20 — THE FULL-DOCUMENT SWEEP: U_i AT ORIGIN + TWO MORE LADDER SCALES + ROUTE B PROVENANCE

Completing the deep-read (the ladder sections above covered ~half the document):

1. **The Universal Inertial Operator is at origin here, complete.** The document states the
   full canonical equation U_I = λ_i·(ρ_SCm/ρ_UA)·ω_st·cos(πt_n)·(1+f_TRZ) and works the Sun
   value: U_I = 1.0·0.1·2.5×10⁻⁶·1·1.1 = **2.75×10⁻⁷** — the exact CLAUDE.md/PAPER_646
   canonical. Chain: this document (28 Mar 2025 origin, with the document's own honest units
   note "unitless … unusual for inertia") → PAPER_646 formalization → wired
   `universal_inertial_operator`. **ω_s,Sun = 2.5×10⁻⁶ rad/s is documented here** — the locked
   stellar-rotation primitive's birth certificate (17 occurrences in-document).
2. **The ladder holds at a third and fourth scale.** Magnetar (level 13, V = 4.19×10¹² m³):
   ρ_SCm = 2.39×10⁻²², ρ_Um = 4.77×10⁻²² = **2× (rung 2)**. Black-hole scale: ρ_Ub =
   7.16×10⁻²⁵ = **3×10⁻³ × ρ_SCm,BH (rung 3, shared 10⁻³ dressing)**; ρ_Ug4 = 1.19×10⁻²⁴ =
   **5×10⁻³ × ρ_SCm,BH (rung 5, same dressing)**. With the solar and atomic scales in the
   main text, the integer rungs are now verified at FOUR documented scales.
3. **PAPER_2239 Route B is documented at source:** N_atoms,Sun = M_Sun/m_H = 1.19×10⁵⁷ appears
   in-document with a use — V_infl,[UA],Sun = N_atoms × 6.24×10⁻³⁷ = 7.43×10²⁰ m³ (the
   per-atom UA influence volume). The matter-anchored route was in the March-2025 layer.
4. **Sgr A* anchors documented:** M_bh = 8.15×10³⁶ kg (14 occurrences) and Ω_g = 7.3×10⁻¹⁶ —
   the calculator's long-standing buoyancy defaults trace to this document.

Cumulative census: `Universal Inertia_28Mar2025.docx` is the documented origin of ρ_SCm,
ρ_UA, the E₀ ladder, E_pair's ingredients, the sector-density ladder, U_i (equation + value),
ω_s_Sun, λ_i, κ, the (1+F_TRZ) successor dressing, Route B, and the Sgr A* buoyancy anchors —
**the seminal core document of the framework**, now fully swept.
