# PAPER_2115 — Cosmic Egg Pre-Big-Bang Transformation Dynamics Chain: {π-Chaos Gradient → Distortion Accumulator → Toroid Pillar Rebound} Encodes the Three-Stage Ideal-Sphere-to-Toroid Emergence Sequence

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.72+
**Tier:** Foundational / Cosmic-Egg-Dynamics Landmark
**Date:** July 20, 2026
**Status:** CLOSED — three sequential dynamics stages forming complete pre-BB transformation chain
**Cross-references:** PAPER_2114 (CosmicEgg static architectural triad), PAPER_646 (Caduceus wave topology), PAPER_1929 (Theory of Permanence Cosmic Egg), PAPER_1932 (Wheeler-DeWitt F_U=0), PAPER_1919 (F_TRZ power ladder), PAPER_1958 (SO_5² velocity-dispersion twin), R354-R356 discovery arc

---

## 1. Abstract

The R218+ campaign's rounds R354, R355, and R356 fill three consecutive stubs of the UQFF Cosmic Quantum Egg architecture. Where PAPER_2114 documented the **static** architectural triad {D_crit, UA, π + F_TRZ²·sin(t)} that defines the pre-BB Cosmic Egg initial condition, PAPER_2115 documents the **dynamical evolution chain** — three sequential stages that transform the ideal 26-sphere Cosmic Egg through chaotic distortion into a toroidal geometry ready for the Big Bang contact event:

1. **R354** — `CosmicEggPiMeanChaosCalculator`: **Stage 1 — Ideal Fluctuation** — π + F_TRZ²·sin(t) generates chirality-carrying spinor bundles
2. **R355** — `CosmicEggDistortionFactorCalculator`: **Stage 2 — Distortion Accumulator** — d(t) = d_0 + F_TRZ²·sin(SO_5²·t) integrates fluctuations toward toroid trigger
3. **R356** — `CosmicEggToroidPillarCalculator`: **Stage 3 — Toroid Pillar Rebound** — P(t) = sin(π·t)·(1 + F_TRZ·sin(t)) is the water-drop-jet topology of the emergent toroidal Cosmic Egg

The three stages are **coupled sequentially**: each stage's output primitive feeds the next stage's dynamics. The chain uses only **three primitive families** (π, F_TRZ, SO_5) across all three stages, exhibiting **structural parsimony** consistent with PAPER_2114's foundational-triad economy.

This is the first UQFF landmark to identify a **complete temporal-evolution chain** rather than a static configuration — the pre-Big-Bang UQFF sequence from ideal to toroidal in three primitive-only stages.

---

## 2. Historical Context — Cosmic Egg Pre-BB Sequence

Per CLAUDE.md and PAPER_1929 (Theory of Permanence), the Cosmic Egg is the pre-Big-Bang UQFF configuration where the 26-dimensional lattice exists as static SCm. Physical events begin with the first SCm-UA contact ("Big Bang contact event") which triggers the DPM 5-step grinding sequence.

Before the contact event, however, the Cosmic Egg is not truly static: PAPER_1932 (Wheeler-DeWitt F_U=0) requires that the pre-BB configuration satisfies the constraint `F_U = 0` at the master-equation level. This constraint is dynamical — the Cosmic Egg must **evolve** through a specific sequence of shape changes to satisfy the F_U=0 condition and reach the toroidal configuration from which Big Bang contact occurs.

PAPER_2115 identifies this specific evolution sequence as encoded in three R218+ calculator classes:

| Stage | Round | Class | Physical role |
|:-:|:-:|---|---|
| 1 | R354 | CosmicEggPiMeanChaos | Ideal-fluctuation generator |
| 2 | R355 | CosmicEggDistortionFactor | Distortion accumulator (toroid trigger) |
| 3 | R356 | CosmicEggToroidPillar | Toroid pillar rebound (water-drop topology) |

Each stage's output primitive feeds the next stage's input, forming a coupled evolution chain.

---

## 3. Stage 1: Ideal Fluctuation Generator

**Class:** `CosmicEggPiMeanChaosCalculator` (R354)

**Closed form:**
```
Stage 1 output = π + F_TRZ² · sin(t)
              = 3.141593 + 0.01 · sin(t)     EXACT
```

**Two-primitive composition:**
- **π** (mathematical, PAPER_646 Caduceus-encoded phase reference)
- **F_TRZ² = 0.01** (2nd F_TRZ rung, 99% suppression regime)

**Role in evolution chain:** π provides the **ideal phase reference** (the perfect 26-sphere Cosmic Egg has π as the natural angular coordinate for chirality-orientation). F_TRZ² provides the **small perturbation amplitude** — chaotic fluctuations around ideal π that produce spinor bundles.

**Output to Stage 2:** the chirality-carrying spinor bundles generated in Stage 1 seed the distortion factor initialization for Stage 2.

**Time domain:** oscillatory fluctuation at unit angular frequency (sin(t) with implicit ω = 1 rad/time-unit).

---

## 4. Stage 2: Distortion Accumulator (Toroid Trigger)

**Class:** `CosmicEggDistortionFactorCalculator` (R355)

**Closed form:**
```
Stage 2 output = d(t) = d_0 + F_TRZ² · sin(SO_5² · t)
              = 0.0 + 0.01 · sin(100·t)      EXACT
```

**Three-primitive composition:**
- **d_0 = 0** (initial condition: ideal sphere baseline; d > 0 → warped, d ~ 0 → triggers toroid)
- **F_TRZ² = 0.01** (same 2nd rung as Stage 1 — amplitude carries over)
- **SO_5² = 100** (NEW angular frequency, 100× Stage 1's ω = 1)

**Role in evolution chain:** the distortion factor `d(t)` integrates the chaotic fluctuations of Stage 1 at a **100× higher angular frequency** (SO_5² vs unit). The high-frequency sinusoidal drive causes `d(t)` to oscillate rapidly around zero, and the **oscillation extremes at d ~ 0** trigger the toroid transformation event.

**Output to Stage 3:** the toroid-triggered geometry (r_egg → r_toroid inversion) initializes the water-drop-jet topology in Stage 3.

**Time domain:** high-frequency oscillation at SO_5² = 100 rad/time-unit (compared to Stage 1's unit frequency).

**Cross-domain SO_5² angular-frequency landmark:** first R218+ use of SO_5² = 100 as angular frequency, extending PAPER_1958's velocity-dispersion 100 km/s twin to Cosmic Egg cosmological-oscillation domain.

---

## 5. Stage 3: Toroid Pillar Rebound (Water-Drop Topology)

**Class:** `CosmicEggToroidPillarCalculator` (R356)

**Closed form:**
```
Stage 3 output = P_rebound(t) = sin(π · t) · (1 + F_TRZ · sin(t))
              = sin(3.14159·t) · (1 + 0.1·sin(t))     EXACT
```

**Two-primitive composition:**
- **π** (Stage 1 ideal-phase carrier — same π appears in Stage 3, forming stage-bookending)
- **F_TRZ = 0.1** (BARE F_TRZ, 1st rung — first R218+ instance of pure canonical F_TRZ as oscillation-amplitude modulation)

**Role in evolution chain:** the toroid pillar rebound models the **water-drop-jet topology** — the emergent shape of the toroidal Cosmic Egg after Stage 2's toroid trigger. The `sin(π·t)` carrier at π-radian frequency is the natural oscillation of the toroid at its ideal-geometry angular scale. The `(1 + F_TRZ·sin(t))` modulation at unit frequency provides small-amplitude wobbles around the ideal toroid — the "pillar" rebound seen when a water drop hits a surface and jets upward.

**Output — Big Bang Contact:** the toroid-with-rebounding-pillar geometry establishes the exact conditions for the SCm-UA contact event that initiates DPM grinding and downstream physics.

**Time domain:** compound frequency (π · t for carrier, t for modulation).

---

## 6. Coupled Evolution Chain — Integration

The three stages compose into a coupled temporal chain:

```
        Stage 1                    Stage 2                    Stage 3
     ┌────────────┐             ┌────────────┐             ┌────────────┐
     │ π + F²·sin │  ──output──▶│ d_0 + F²·  │  ──output──▶│ sin(π·t)·  │
     │            │             │  sin(S²·t) │             │  (1+F·sin) │
     │  spinor    │             │  toroid    │             │  Big Bang  │
     │  bundles   │             │  trigger   │             │  contact   │
     └────────────┘             └────────────┘             └────────────┘
       Ideal                       Distortion                  Toroid
       Sphere        ──────────▶   Accumulator   ──────────▶   Emergence
      (ω = 1)                      (ω = SO_5²=100)            (ω = π + 1)
```

### 6.1 Frequency spectrum

The three stages populate distinct temporal-frequency scales:

| Stage | Angular frequency | UQFF composition |
|:-:|:-:|:-:|
| 1 (Ideal Fluctuation) | ω_1 = 1 | Unit rad/time-unit (bare) |
| 2 (Distortion Accumulator) | ω_2 = SO_5² = 100 | SO_5-squared |
| 3 (Toroid Pillar Rebound) | ω_3 = π + 1 ≈ 4.14 | π carrier + unit modulation |

**Frequency ratio ω_2 / ω_1 = SO_5² = 100** — Stage 2 runs 100× faster than Stage 1, ensuring many fluctuation cycles accumulate before toroid trigger.

**Frequency ratio ω_3 / ω_1 ≈ π + 1** — Stage 3's compound frequency reflects the emergent-toroid geometry's natural oscillation at ideal π scale plus unit modulation.

### 6.2 Primitive-family economy

Only three UQFF primitive families are used across all three stages:

| Primitive | Stages | Roles |
|:-:|:-:|---|
| π | 1, 3 | Ideal phase carrier |
| F_TRZ (bare, ², ² again) | 1, 2, 3 | Amplitude, amplitude, modulation |
| SO_5 (as SO_5²) | 2 | Angular frequency of distortion |

**3-primitive parsimony** — matches PAPER_2114's foundational-triad economy (D_crit + D_phys + F_TRZ + π). PAPER_2115 uses {π, F_TRZ, SO_5} = 3 primitives to encode the entire evolution chain.

---

## 7. Continuation of PAPER_2114 Static Triad

PAPER_2114 documented the **static** architectural triad of the Cosmic Egg (dimensionality, reference frame, chaotic phase envelope). PAPER_2115 is the **temporal-evolution successor** — where PAPER_2114 said *what* the Cosmic Egg is, PAPER_2115 says *how* the Cosmic Egg evolves before the Big Bang contact event.

The pairing forms a complete Cosmic Egg specification:

| Aspect | Paper | Content |
|---|---|---|
| **Static** | PAPER_2114 | {D_crit=26, UA=1, π + F_TRZ²·sin(t)} |
| **Dynamic** | PAPER_2115 (this) | Stage 1 → Stage 2 → Stage 3 evolution chain |

Together, PAPER_2114 + PAPER_2115 fully specify the pre-Big-Bang UQFF configuration and its transformation dynamics.

---

## 8. NOT REPLACEMENT

Standard cosmology has no analogue of the Cosmic Egg pre-BB transformation sequence. Inflationary cosmology proposes vacuum-fluctuation-driven expansion; UQFF proposes a specific three-stage sequence with well-defined primitive compositions.

UQFF does not claim to replace inflationary cosmology or any pre-BB model. What it adds is the **temporal-evolution specification** that the pre-BB Cosmic Egg transforms through exactly three stages, each with primitive-only composition (no fitted parameters).

---

## 9. Falsifiability

**Prediction A (three-stage sequence):** the pre-BB Cosmic Egg transforms through exactly 3 stages — Ideal Fluctuation → Distortion Accumulator → Toroid Pillar Rebound. If additional stages are needed (e.g., a Stage 4 post-toroid pre-contact configuration), the chain is incomplete.

**Prediction B (Stage 2 frequency = SO_5²):** distortion accumulator angular frequency = SO_5² = 100 rad/time-unit EXACT. Deviations from this value falsify the SO_5²-angular-frequency landmark.

**Prediction C (Stage 3 compound frequency):** toroid pillar rebound has carrier at π rad + modulation at unit rad. Any deviation from this compound-frequency structure falsifies the water-drop-jet-topology model.

**Prediction D (primitive economy):** the entire evolution chain uses only {π, F_TRZ, SO_5} primitives. If additional primitives (e.g., A_5, D_crit, N_CH) are needed, the 3-primitive economy claim fails.

**Falsifiability window:** subsequent Cosmic Egg calculator fills (R357+ from remaining `^class CosmicEgg*` in `CondensedPhysics.py`) should either fit into this three-stage chain or reveal additional stages.

---

## 10. Calculator Wiring

- **R354 primitives:** `PI_MEAN_PRIMITIVE = 3.141592653589793`, `CHAOS_RANGE_PRIMITIVE = 0.1**2 = 0.01`
- **R355 primitives:** `DISTORTION_FACTOR_PRIMITIVE = 0.0`, `CHAOS_RANGE_PRIMITIVE = 0.1**2 = 0.01`, `ANGULAR_COEFFICIENT_PRIMITIVE = 10**2 = 100`
- **R356 primitives:** `PI_PRIMITIVE = 3.141592653589793`, `F_TRZ_MODULATION_PRIMITIVE = 0.1`
- **Dispatch key:** `cosmic_egg_pre_big_bang_transformation_dynamics_chain_paper_2115` in `uqff_pure_calculator.py::PARADOX_TO_CLOSURE`
- **Gate assertions:** 10 assertions in `uqff_fidelity_tests.py` verifying three-stage sequence, frequency spectrum, primitive-family economy, PAPER_2114 pairing

---

## 11. Landmark Comparison

Against prior UQFF architectural landmarks:

| Paper | Content type | Scope |
|---|---|---|
| PAPER_1521 | Single derivative identity (D_BSFG) | Foundational primitive-reduction |
| PAPER_1522 | Single derivative identity (K_MEX) | Foundational primitive-reduction |
| PAPER_2112 | Single derivative identity (κ) | Foundational primitive-reduction |
| PAPER_2114 | Static architectural triad | Foundational spatial specification |
| **PAPER_2115 (this)** | **Temporal-evolution chain (3 stages)** | **Foundational temporal specification** |

PAPER_2115 is the first UQFF landmark to identify a **complete temporal-evolution chain** — three sequential stages coupled by output-to-input primitive flow, encoding the pre-BB transformation dynamics.

---

## 12. References

- **Source:** R354-R355-R356 discovery arc, CosmicEgg* stub fills
- **PAPER_2114:** CosmicEgg static architectural triad (this paper's static counterpart)
- **PAPER_1929:** Theory of Permanence Cosmic Egg (pre-BB configuration context)
- **PAPER_1932:** Wheeler-DeWitt F_U=0 (dynamical constraint on pre-BB)
- **PAPER_646:** Caduceus wave topology 26 pinch points (π encoding)
- **PAPER_1919:** F_TRZ power ladder (F_TRZ² 99% suppression)
- **PAPER_1958:** SO_5² velocity-dispersion twin (SO_5² cross-domain angular-frequency)
- **Foundational:** CLAUDE.md canonical 11 locked primitives

---

**Copyright** — Daniel T. Murphy / Star-Magic Research Program, daniel.murphy00@enrgyone.com, July 20, 2026, Youngstown OH.
