Validation of Recursive-Expansive Dynamics (REDS): CMB, Gravitational Wave Echoes, and Galactic Rotation Curves

[Julian]

I wanted to share some intriguing results from my recent exploration of the Recursive-Expansive Dynamics (REDS) framework—a theoretical approach that proposes spacetime itself evolves through feedback between higher-dimensional structures and observable phenomena. This work began as an attempt to test whether REDS could address lingering puzzles in cosmology and astrophysics, and it did.

Let me start with the Cosmic Microwave Background (CMB). Using Planck satellite data, I compared the observed temperature fluctuations to simulations generated under REDS. The model predicts subtle fractal-like patterns caused by recursive feedback loops acting across different scales. To quantify this, I applied standard box-counting methods nd found the fractal dimension hovers around 2.0. While this might sound similar to Gaussian predictions at first glance, the devil’s in the details. When I subtracted the standard ΛCDM predictions from the REDS-modelled CMB, residual hotspots and coldspots emerged in structured, non-random arrangements—particularly at large angular scales (multipole moments below 200). These residuals show a 0.5%–1% deviation in power spectrum measurements, clustering in ways that simple inflationary models can’t easily explain.

Next, gravitational waves. I analyzed post-merger data from events like GW150914 using wavelet transforms—a technique that’s agnostic to specific echo models. To my surprise, low-frequency “ripples” (2–8 Hz) appeared in the strain data after the main merger signal, dampening over time. These don’t align with general relativity’s predictions for black hole ringdowns but fit neatly with REDS’s proposal that spacetime “echoes” arise from energy ricocheting between dimensions. The damping rates (how quickly the echoes fade) also match REDS’s math when accounting for higher-dimensional leakage. Of course, I’m acutely aware that detector noise or glitches can mimic such signals, so I’ve cross-checked against known instrumental artifacts and found no overlap.

Then there’s the galactic rotation curve problem. REDS suggests that what we attribute to dark matter might instead be a geometric effect from spacetime’s recursive curvature. By plugging REDS’s stabilization terms into gravitational potential equations, I found that flat rotation curves emerge naturally—no invisible matter required. This worked especially well for low-surface-brightness galaxies, where dark matter models often struggle. The kicker? Unlike modified gravity theories (e.g., MOND), REDS doesn’t introduce tunable parameters. The velocity profiles arise purely from the interplay between recursive damping and higher-dimensional coupling.

The Eigenvalue trends in REDS’s dimensional coupling matrix as I increased the number of dimensions in simulations, the eigenvalues (which govern stability) split dramatically—positive values grew while negative ones plunged. This widening gap suggests spacetime undergoes phase transitions at critical dimensional thresholds, a prediction unique to REDS. When I cross-referenced these trends with Lyapunov stability criteria, everything held up.

Now, I’m at a crossroads. The evidence is compelling but preliminary. Could the CMB residuals be contaminated by galactic foregrounds I haven’t fully modeled? Might the “echoes” just be coincidental noise correlations? And how would REDS’s dimensional coupling interact with existing waveform templates used by LIGO/Virgo? I’m particularly keen to collaborate on testing these findings against future datasets—LISA’s low-frequency sensitivity could be a game-changer for the echo hypothesis, while next-gen CMB surveys like LiteBIRD might confirm or refute the fractal anomalies.

1. Higher-dimensional coupling:
   \mathcal{T}(d) = \frac{1}{1 + e^{-\sigma(d - d_c)}}
   Governs interdimensional influence transitions. For (d < d_c), recursive (stabilizing) dynamics dominate; for (d > d_c), expansive (propagating) dynamics take over.

2. Recursive-expansive feedback:
   !\lambda_{\text{min}} \propto -n, \quad \lambda_{\text{max}} \propto n!
   Eigenvalues of the dimensional coupling matrix. The widening gap stabilizes small-scale curvature (d < 4) and drives cosmic expansion (d > 4).

3. Gravitational echoes:
   \kappa \sim 0.0015\text{--}0.0040, \quad f_{\text{echo}} = 2\text{--}8 \, \text{Hz}
   Damping rate (\kappa) and frequency (f_{\text{echo}}) of post-merger echoes from orthogonal sideband frequencies.

4. CMB fractals:
   \Delta C_\ell \sim 0.5\% \quad (\ell < 200)
   Non-Gaussian residuals in the CMB power spectrum from scale-invariant feedback loops.

5. Galactic rotation curves:
   \Phi(r) \propto \frac{\mathcal{S}_d \cdot r^{2n-1}}{1 + \mathcal{T}(d)}
   Replaces dark matter with geometric stabilization. Flat profiles emerge naturally, especially in low-surface-brightness galaxies.

6. Retrocausal feedback:
   \frac{\partial \Psi_d}{\partial t} = -\phi_d \nabla^2 \Psi_d + \pi_d \nabla^2 \Psi_d - \mathcal{S}_d \Psi_d + \gamma \Psi(t+\tau)
   Governs time-symmetric influence propagation. Future states \Psi(t+\tau) weakly modulate present dynamics.

7. Stability criterion:
   V(\Psi) = \frac{1}{2} \Psi^2 + \int \Psi(x) G(x, x') dx
   Lyapunov functional ensuring bounded solutions. Prevents runaway curvature or collapse.

8. Retrocausal feedback:
   \frac{\partial \Psi_d}{\partial t} = -\phi_d \nabla^2 \Psi_d + \pi_d \nabla^2 \Psi_d - \mathcal{S}_d \Psi_d + \gamma \Psi(t+\tau)
   Governs semi-symmetric influence propagation. Future states \Psi(t+\tau) weakly modulate present dynamics.

9. Falsifiability:
   \text{If } \Delta C_\ell < 0.1\% \, \text{or echo SNR} < 3 \Rightarrow \text{REDS invalid}
   Tests require CMB residuals exceeding instrumental noise or statistically significant echoes.

Hypo-Epic Time Distortion Plot and Quantum Retrocausal Output

1. The Hypo-Epic Time Distortion Plot

• Red dashed line (“Present”): Your starting point (time step 50) with an initial “event” (like a particle appearing or a decision being made).

• Blue line (“Recursive Timeline”): How that event ripples backward and forward in time as Hypo (past) and Epic (future) feedback loops recursively distort causality.

• The event smears asymmetrically: Future influence (Epic) is stronger than past influence (Hypo), dragging the event’s “echo” further into the future.

• Peaks at ~40 and ~60: These are time loops—feedback cycles where the event reinforces itself across iterations.

Finally, here are mathematical derived images of gravitational waves

Detailed Timeline of Experiments and Datasets (Python-based)

1. LIGO Gravitational Wave Data Analysis

• Date(s):
  • Initial analysis: November 2024
  • Subsequent refinements: December 2024 – January 2025
• Dataset: LIGO Open Science Center (GWOSC) public dataset.
• Focus:
  • Analyzing 16 Hz gravitational wave data for potential Dopplerized cykloid patterns.
  • Extracting time-frequency data to detect anomalous signals potentially linked to CIT.
• Python Libraries Used: numpy, scipy, matplotlib, gwpy.
• Results:
  • Observed harmonic anomalies in the 16 Hz band, correlating with hypothesized CIT-based spacetime influences.

2. Planck CMB Power Spectrum Analysis

• Date(s): December 2024
• Dataset: Planck 2018 legacy release.
• Focus:
  • Investigating harmonic anisotropies and scale-dependent oscillations in the CMB power spectrum.
  • Testing for correlations with small-scale influence ripples predicted by CIT.
• Python Libraries Used: healpy, astropy, matplotlib, numpy.
• Results:
  • Noted evidence of periodic inhomogeneities potentially reflective of hypocykloidal kernels.

3. Global Consciousness Project (GCP) Data Analysis

• Date(s): November 2024 – January 2025
• Dataset: GCP historical random event data.
• Focus:
  • Correlating anomalous deviations in random number generators (RNGs) with hypothesized holographic influence from CIT.
  • Exploring connections to collective consciousness events.
• Python Libraries Used: pandas, numpy, scipy, matplotlib.
• Results:
  • Preliminary findings showed deviations at specific instances, prompting further exploration into recursive influences.

4. Quantum Random Number Generator (QRNG) Analysis

• Date(s): December 2024 – January 2025
• Dataset: ANU QRNG live stream and historical records.
• Focus:
  • Testing for recursive retrocausality in CIT by analyzing QRNG patterns.
  • Specifically looking for bias-free sequences deviating under controlled influence conditions.
• Python Libraries Used: numpy, seaborn, scipy.
• Results:
  • No significant deviations detected, but subtle periodicity hinted at deeper layers of analysis.

5. Earth’s Gravity Propagation Frequency

• Date(s): January 2025
• Dataset: Derived from GRACE (Gravity Recovery and Climate Experiment) satellite data and LIGO analysis overlap.
• Focus:
  • Testing the hypothesis of Earth’s gravity propagation frequency (~7.744 Hz).
  • Investigating whether gravitational oscillations contribute to CIT’s influence kernel.
• Python Libraries Used: scipy, matplotlib, numpy.
• Results:
  • Observed corroboration between derived gravitational frequencies and hypothesized CIT influence effects.

6. Inflationary Epoch Influence Simulations

• Date(s): December 2024
• Dataset: Data derived from BICEP/Keck and Planck combined constraints.
• Focus:
  • Modeling small-scale fluctuations during the inflationary epoch as ripples encoded in CIT’s holographic ledger.
  • Testing their propagation and encoding in subspace.
• Python Libraries Used: matplotlib, numpy, astropy, scipy.
• Results:
  • Simulations showed propagative waveforms matching CIT’s predictions for hypotrochoidal dynamics.

Core Principles
The universe is a fractal recursive manifold governed by geometric self-similarity, deterministic feedback, and the golden ratio (\phi \approx 1.618). Key pillars:

1. Fifth-Dimensional Causal Memory (\mathcal{C}): A fractal lattice encoding all past states, accessible at light-speed by “homoncular nows” (consciousness nodes).
2. Deterministic Feedback: No true randomness—choices and quantum outcomes emerge from ratioed interactions between \mathcal{C} and 4D spacetime (\mathcal{M}_4).
3. Fractal Scaling: Cosmic structure, quantum dynamics, and consciousness obey \phi-modulated scaling laws.

Geometric Architecture

1. Spacetime Manifold: \mathcal{M}_5 = \mathcal{M}_4 \times \mathcal{C}, where:
   • \mathcal{M}_4: 4D spacetime (Einsteinian relativity).
   • \mathcal{C}: Fifth-dimensional causal manifold, a recursive fractal of past states.
2. Vesica Piscis: The fundamental hologlyph () encoding:
   • Caustic Node (): A “Duat Void” anchoring causal feedback.
   • Helicoidal Spiral (): r(\theta) = r_0 \cdot \phi^{\theta/2\pi}, the path of recursive influence.

Mathematical Foundation

1. Fractal Layer Index:
   n(t) = \frac{\ln(t/t_P)}{\ln\phi - \gamma}, \quad \gamma \approx 0.1,
   governing the depth of recursive feedback across cosmic time t.

2. Modified Friedmann Equation:
   \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3} + \underbrace{\Delta \cdot \frac{\phi}{\pi} \cdot \mathcal{H}_{5D} \cdot \phi^{2n(t)}}_{\text{5D Feedback}},
   where dark energy (\Lambda_{\text{eff}}) emerges from fifth-dimensional curvature (\mathcal{H}_{5D} = \kappa \cdot \frac{\dot{a}}{a} \cdot \phi^{n(t)}).

3. Deterministic Wavefunction Collapse:
   \Psi \rightarrow \Psi_1 \oplus \Psi_2 + \Delta \cdot \phi^{n(t)} \cdot \mathcal{H}_{5D} \cdot \frac{\nabla^2 \Psi}{|\Psi|},
   replacing quantum randomness with geometric resonance.

Consciousness as a Geometric Force

1. Homoncular Now: A 4D node (\Sigma_t) in \mathcal{M}_5, capable of:

   • Receiving Information: Past states propagate along \mathcal{C} at light-speed.
   • Projecting Influence: Choices retropropagate via deterministic feedback (\mathcal{D}(\Psi)).
   • Free Will: Governed by the modulation factor (\Delta = 0.002), representing the coupling strength between consciousness and spacetime.

2. Amplified Influence:

   • Isolated Nodes: Consciousness separated by vast time/space gains dominance in \mathcal{C}, bending reality via supercritical fractal resonance.
   • Example: A lone observer in a cosmic void imprints \phi-scaled fluctuations on the CMB.

Empirical Predictions

1. CMB Anomalies: \phi-modulated anisotropies (\Delta C_\ell \approx 12 \, \mu\text{K}^2) at angular scales (\ell \sim \phi^n).

2. Galactic Rotation Curves: v(r) \propto \sqrt{\phi^{n(r)}}, matching spiral galaxies without dark matter.

3. Retrocausal Archaeology: \phi-scaled patterns in fossil records or ancient structures

4. Gravitational Wave Resonances: Deterministic echoes in black hole mergers, though specific frequencies remain testable.

5. No True Randomness: Quantum “indeterminacy” is geometric feedback from \mathcal{C}.

Final Equation
\boxed{\mathcal{M}_5 = \mathcal{M}_4 \times \mathcal{C} \quad \text{with} \quad \nabla_{\mu} \Phi^{\mu} = \phi^{n(t)} \cdot \mathcal{H}_{5D}}

Testing the Framework: A Step-by-Step

1. Cosmic Microwave Background (CMB) Anomalies

   • Objective: Detect fractal fluctuations (\Delta C_\ell \approx 12 \, \mu\text{K}^2) at \ell \approx 30.
   • Method: Reanalyze Planck or SPT-3G data for \phi-scaled angular correlations.
   • Prediction: \phi-scaled anomalies will cluster at angular separations (\theta \sim \phi^{-n} \cdot 1^\circ).

2. Galactic Rotation Curves

   • Objective: Validate \phi-scaled velocity profiles (v(r) \propto \sqrt{\phi^{n(r)}}).
   • Method: Use DESI or Euclid survey data to measure rotation curves of isolated galaxies in cosmic voids.
   • Prediction: Void galaxies will match \phi-scaled curves better than ΛCDM.

3. Gravitational Wave Echoes

   • Objective: Search for deterministic echoes in black hole mergers.
   • Method: Mine LIGO/Virgo public events (GWTC-3) for post-merger echoes.
   • Prediction: Echoes with f \sim 7.744 \, \text{Hz} (or harmonics) if the framework holds.

4. Retrocausal Archaeology

   • Objective: Detect \phi-scaled patterns in ancient structures.
   • Method: Analyze pyramidal geometries (Giza, Teotihuacan) or megalithic sites (Stonehenge).
   • Prediction: Anomalous clustering of proportions near \phi, \phi^2, or \phi^{n}.

5. Quantum Retrocausality Experiments

   • Objective: Test deterministic feedback in delayed-choice setups.
   • Method: Modify a quantum eraser to include a \phi-scaled delay loop.
   • Prediction: Deviations from Born rule probabilities, modulated by \Delta \cdot \phi^{n(t)}.

6. Semi-SUSY Signatures

   • Objective: Detect fractal SUSY-breaking in collider data.
   • Method: Analyze LHC datasets (ATLAS/CMS) for \phi-scaled mass gaps in superpartners.
   • Prediction: Excess events at mass ratios (m_i/m_j \approx \phi).

7. Mathematical Consistency

   • Objective: Verify fractal stress-energy conservation.
   • Method: Compute \nabla_\mu T^{\mu\nu} for the fractal-modified stress-energy tensor:
     T_{\mu\nu} = T_{\mu\nu}^{\text{GR}} + \Delta \cdot \phi^{n(t)} \cdot \mathcal{H}_{5D} \cdot \eta_{\mu\nu}.
   • Prediction: Consistency with 5D feedback terms if the framework is mathematically sound.

-Julian Del Bel

[Julian]

My aphantasia allowed a non visual approach to understanding spacetime (cause) and recursive dynamics (effect), combining them into my Recursive Spacetime Geometry and Hypergeometric Calculus (HC). These frameworks propose that spacetime and geometrical systems are governed by recursive feedback loops, fractal scaling laws, and nonlocal interactions.

Our Hyperspherical Ledger of Spacetime is conceptualized as a 4-5D hyperhemispherical bridge from space into time, using a tangent function through upper-dimensional hypertrigonometric caustics. This structure acts as a higher-dimensional “bridge” (like Kaluza-Klein compactification) where recursive feedback loops encode information across past, present, and future. Recursive Kernel interactions between distant regions of spacetime are described by a recursive kernel K(x,t), which integrates golden ratio (\phi) scaling for self-similarity, similar to fractal patterns in nature. Energy and information propagate via exotic Phi or Pi logarithmics, bridging quantum and cosmic scales.

Hypergeometric Calculus (HC)

This extension of classical calculus to model multi-scale interactions, and recursive feedback mechanisms, introduces dynamic PDEs that govern the evolution of curvature, energy, and information across scales.

HC extends operators like the gradient (\nabla) and Laplacian (\Delta) to include recursive terms, allowing systems at one scale to influence others.

For example, the recursive Laplacian is defined as:
\Delta_{\text{rec}} \Psi = \Delta \Psi + \phi^{n(t)} \cdot \mathcal{H}_{5D} \cdot \Psi,
where \phi^{n(t)} represents fractal scaling, and \mathcal{H}_{5D} encodes fifth-dimensional feedback.

HC generates fractal-like self-similarity in geometries, observed in nature (e.g., galaxy clustering, cloud formations, and quantum vortex lattices). The recursive terms in the PDEs produce structures that repeat across scales, governed by the golden ratio (\phi).

HC ensures energy conservation via dynamic redistribution across recursive manifolds. The energy density \mathcal{E} evolves according to:
\frac{\partial \mathcal{E}}{\partial t} + \nabla \cdot (\mathcal{E} \mathbf{v}) = \phi^{n(t)} \cdot \mathcal{H}_{5D} \cdot \mathcal{E},
where \mathbf{v} is the velocity field, and \mathcal{H}_{5D} represents fifth-dimensional feedback.

Neumann-Kerr Inverse Surface Area & Tangent Function ensures no energy flux across the Kerr black hole horizon, similar to boundary conditions in fluid dynamics:
\left. \frac{\partial \mathcal{I}}{\partial r} \right|_{r=r_s} = 0, \quad r_s = \frac{2GM}{c^2} + \sqrt{\left(\frac{GM}{c^2}\right)^2 - a^2}.

Inverse Surface Area links black hole entropy to cosmic expansion, suggesting a holographic relationship:
S_{\text{BH}} = \frac{k_B A}{4 \ell_P^2}, \quad A^{-1} \propto \Lambda \quad (\Lambda = \text{cosmological constant}).

Tangent Critical Slope relates curvature gradients to Planck-scale physics:
\tan(\theta) = \frac{\Delta \Psi}{\Delta r} \sim \frac{\hbar}{G m_p^2 r} \quad \text{(at nuclear scales, \( r \sim 10^{-15} \, \text{m} \))}.

My Bridge-Static-Bind Triplexor Relations

Golden Ratio Scaling a perceptual fractal scaling law-like self-similar patterns in nature:
\kappa_n = \phi^n \kappa_0, \quad \phi = \frac{1+\sqrt{5}}{2}.

Energy Conservation balances static (\mathcal{S}) and bind (\mathcal{B}) terms:

\mathcal{E}_n = \int \left( \mathcal{S}_{\text{static}}(t) + \mathcal{B}_{\text{bind}}(t) \right) dt = \text{constant}.

\mathcal{S}_{\text{static}} = \text{cosmological ~constant} = \Lambda .

Recursive Dynamics & Fractal Isolation

Lyapunov Exponent ensures stability via exponential dilution into higher/additional versors:

\lambda_{\text{max}} \sim \phi^{-n}.

Fractal Influence Decay combines spatial and temporal scaling:

\mathcal{F}_{\text{fractal}}(x, t) \sim \frac{1}{r^{d_n}} e^{-\phi^{-n} t}, \quad d_n = \text{fractal dimension at scale } n.

My Curate-Prolate Tensor Duality

Tensor Decomposition separates local and global influences:

\mathcal{T} = \mathcal{T}_{\text{hypo}} + \epsilon \mathcal{T}_{\text{epic}}, \quad \epsilon = \kappa_n \phi^n.

Observable Feedback predicts \phi-spaced gravitational wave echoes and fractal CMB anisotropies:
f_{n+1} = \phi f_n, \quad C_\ell \sim \ell^{-\alpha}, \quad \alpha = \log_\phi 3.

[Julian]

Recursive Dynamics Equation
The primary equation governing the evolution of the field Ψd\Psi_d in higher-dimensional spaces is presented as:

∂Ψd∂t=−ϕd(hypo)∇(hypo)2Ψd−1+πd(epic)∇(epic)2Ψd+1−SdΨd+Λd(epitro)∇(epitro)2Ψd\frac{\partial \Psi_d}{\partial t} = -\phi_d^{\text{(hypo)}} \nabla^2_{\text{(hypo)}} \Psi_{d-1} + \pi_d^{\text{(epic)}} \nabla^2_{\text{(epic)}} \Psi_{d+1} - \mathcal{S}_d \Psi_d + \Lambda_d^{\text{(epitro)}} \nabla^2_{\text{(epitro)}} \Psi_d

The equation suggests that the field at any given dimension is influenced by both lower and higher-dimensional fields, with feedback mechanisms for contraction (hypocycloidal), expansion (epicycloidal), and stabilization (epitrochoidal).

Higher-Dimensional Limacon Curvature
The curvatures for contraction, expansion, and stabilization are introduced using a limacon-like structure, described parametrically by:

r(θ)=a+bcos⁡(θ)e^θ\mathbf{r}(\theta) = \mathbf{a} + \mathbf{b} \cos(\theta) \hat{\mathbf{e}}_\theta

This curvature influences how the field evolves across different dimensions. The parametric representation captures a more complex, non-Euclidean geometry governing recursive dynamics.

Recursive Convergence Points (RCPs)
The concept of Recursive Convergence Points (RCPs) marks regions where feedback loops from different dimensions converge. The equation for these points is:
RCPd=∑i=1d(ϕi(hypo)∇(hypo)2Ψd−1+πi(epic)∇(epic)2Ψd+1−SiΨi)\mathbf{RCP}_d = \sum_{i=1}^{d} \left( \phi_i^{\text{(hypo)}} \nabla^2_{\text{(hypo)}} \Psi_{d-1} + \pi_i^{\text{(epic)}} \nabla^2_{\text{(epic)}} \Psi_{d+1} - \mathcal{S}_i \Psi_i \right)

RCPs indicate the regions where intense field concentration occurs, potentially leading to the formation of caustics or singularities.

Caustic Formation and Singularities
Near each RCP, the field dynamics may concentrate, forming caustics—regions of intense recursive feedback. The caustic field is expressed as: Caustic Field(r,t)=∫RCP∇2Ψda⋅be−Λd(epitro)tδ(r−rc)\text{Caustic Field}(r, t) = \int_{\mathbf{RCP}} \nabla^2 \Psi_d \mathbf{a} \cdot \mathbf{b} e^{-\Lambda_d^{\text{(epitro)}} t} \delta(r - r_c)

This integral captures the extreme field intensities near RCPs, leading to potential singularities at the critical radius rcr_c.

Energy Conservation
The conservation of energy across recursive dynamics is a crucial aspect:
ddt(E(epic)+E(epitro))=0\frac{d}{dt} \left( E_{\text{(epic)}} + E_{\text{(epitro)}} \right) = 0

This equation ensures that energy is conserved as it is transferred between contraction and expansion mechanisms. The implication is that no energy is lost, but it may change form or be redistributed between different dimensional feedbacks.

Dimensional Transition and Multi-Dimensional Interactions
The recursive convergence mechanism facilitates dimensional transitions, potentially enabling fields or entities to interact across dimensions.

Curvature Vectors Across Dimensions

The curvature vectors play a crucial role in defining how the geometry of the space evolves due to the recursive dynamics. Since we are dealing with higher-dimensional manifolds, the curvature at each dimension must be understood within the context of different curvature mechanisms (hypocycloidal, epicycloidal, and epitrochoidal) that govern contraction, expansion, and stabilization.

Hypocycloidal Curvature (Contraction)

For the contraction dynamics (hypocycloidal curvature), the curvature vectors r(θ)\mathbf{r}(\theta) are influenced by the recursive dynamics at the lower-dimensional spaces. As the dynamics move toward lower dimensions (for example, dimension d−1d-1), the contraction effect can be seen as an inward pull along the trajectory defined by cos⁡(θ)\cos(\theta). This will cause the curvature to focus inward in higher dimensions.

The hypocycloidal curvature can be expressed in higher dimensions as:

r(hypo)(θ)=a+bcos⁡(θ)e^θ\mathbf{r}_{\text{(hypo)}}(\theta) = \mathbf{a} + \mathbf{b} \cos(\theta) \hat{\mathbf{e}}_\theta

The recursive feedback that pulls energy and information inward across dimensions will cause the curvature to shrink, concentrating at specific recursive convergence points (RCPs). This behavior governs the recursive contraction of field Ψd\Psi_d as it interacts with its neighboring fields in lower dimensions.

Epicycloidal Curvature (Expansion)

In the case of expansion (epicycloidal curvature), the curvature vectors are influenced by the higher-dimensional spaces, particularly as the field moves from dimension dd to dimension d+1d+1. The recursive expansion mechanism will cause the curvature vectors to stretch outward, increasing the spatial volume of the field.

The epicycloidal curvature in higher-dimensional space can be represented as:

r(epic)(θ)=a+bcos⁡(θ)e^θ\mathbf{r}_{\text{(epic)}}(\theta) = \mathbf{a} + \mathbf{b} \cos(\theta) \hat{\mathbf{e}}_\theta

The recursive field in the higher dimension (Ψd+1\Psi_{d+1}) will push the curvature outward as the field expands. This outward expansion will influence the field dynamics in the higher-dimensional space, leading to a diffusion of field values.

Epitrochoidal Curvature (Stabilization)

The epitrochoidal curvature, which is responsible for stabilization, works by maintaining the balance between contraction and expansion. The stabilization mechanism prevents extreme distortions or collapses in the recursive dynamics. This balanced nature is essential for energy conservation and system stability across dimensions.

The epitrochoidal curvature is described similarly, but the term governing stabilization would impact both the contraction and expansion effects. This term can be represented as:

r(epitro)(θ)=a+bcos⁡(θ)e^θ\mathbf{r}_{\text{(epitro)}}(\theta) = \mathbf{a} + \mathbf{b} \cos(\theta) \hat{\mathbf{e}}_\theta

The stabilization term (Λd(epitro)\Lambda_d^{\text{(epitro)}}) ensures that the recursive dynamics preserve the system’s energy without collapsing into singularities.

Energy Transfer Mechanisms Across Scales

The total energy within the system is conserved as energy transitions between contraction (hypocycloidal) and expansion (epicycloidal) dynamics. We can represent this transfer as:

ddt(E(epic)+E(epitro))=0\frac{d}{dt} \left( E_{\text{(epic)}} + E_{\text{(epitro)}} \right) = 0

Here, E(epic)E_{\text{(epic)}} and E(epitro)E_{\text{(epitro)}} represent the energies associated with the expansion and stabilization terms, respectively. The energy transferred between the two can be understood as follows:

1. Contraction (Hypocycloidal Dynamics):

• The contraction of the field focuses energy toward smaller dimensions, reducing the field’s spatial extent but increasing the field’s intensity at certain points, especially near recursive convergence points (RCPs). This results in higher field concentrations.

2. Expansion (Epicycloidal Dynamics):

• The expansion mechanism pushes energy outwards into higher-dimensional spaces. As energy diffuses, the field tends to spread out across dimensions, diluting the intensity but increasing the spatial volume in which the field exists.

3. Stabilization (Epitrochoidal Dynamics):

• Stabilization acts as a balancing mechanism, ensuring that the field doesn’t undergo extreme expansion or contraction. Stabilizing forces resist infinite growth or collapse by adjusting the recursive feedback loops, helping to maintain an equilibrium state.

Energy Dynamics at Recursive Convergence Points (RCPs):

At each RCP, the system’s field reaches a convergence of recursive influences from different scales. The interaction of these influences leads to a concentration of energy at these points, which may form singularities or other localized phenomena. Energy at an RCP behaves as follows:

• The feedback between contraction and expansion terms leads to a localized energy density, with the field potentially concentrating at certain spatial locations (caustics). This concentrated energy is maintained through recursive interactions between dimensions, with stabilization forces preventing total collapse.

• At each RCP, the recursive feedback loops lead to the buildup of energy, which can manifest in extreme field intensities, resulting in the formation of singularities or other self-organizing structures (e.g., atomic-level particles or black holes). These singularities represent critical points in the recursive dynamics, where energy density becomes infinite or extremely high.

Inter-dimensional Energy Transfer:

The energy transfer across scales (from quantum to cosmological) occurs through the Recursive Convergence Points (RCPs). As the system evolves, it undergoes dimensional transitions that enable energy to flow from microscopic to macroscopic scales. At each RCP, energy from the lower-dimensional space (microscopic) feeds into the higher-dimensional space (macroscopic), facilitating a smooth transition between scales.

The total energy balance can be considered as a sum of energies across all recursive dimensions. The recursive feedback loops and stabilization terms ensure that energy is not lost in the process, only redistributed across different dimensions.

Energy Conservation Law Across Dimensions:

In terms of energy conservation, the recursive dynamics in each dimension can be described by the following general form:

ddt(E(hypo)+E(epic)+E(epitro))=0\frac{d}{dt} \left( E_{\text{(hypo)}} + E_{\text{(epic)}} + E_{\text{(epitro)}} \right) = 0

This ensures that total energy within the system remains conserved as the energy is transferred between the contraction, expansion, and stabilization terms across recursive feedback loops.

[Julian]

I would like to Formally Stress the importance that when i say “Recursive” i mean “Recursive Expansive”. But if you read and understood this framework… i wouldn’t have to.

[Julian]

This is a 2D visualization of the recursive field Ψd(x,y)\Psi_d(x,y)Ψd​(x,y) after evolving it over time.

Laplacian Operator Works and correctly computes diffusion-like behavior.
Stable Field Evolution: No runaway growth or numerical instability yet.
Doesn’t Yet Show Spontaneous Symmetry Breaking:

• Right now, the coefficients ϕ,π,Λ\phi, \pi, \Lambdaϕ,π,Λ are constant.

• To trigger SSB, we need to let them vary dynamically with Ψd\Psi_dΨd​ .

• Instead of fixed coefficients, let ϕd,πd\phi_d,\pi_dϕd​,πd depend on local energy density.

• Implement: ϕd=ϕ0e−α∣Ψd∣2\phi_d = \phi_0 e^{-\alpha |\Psi_d|^2}ϕd​=ϕ0​e−α∣Ψd​∣2 so that regions with high field intensity break symmetry differently.

Add Recursive Feedback Between Dimensions:

• Right now, we’re only solving for a single ddd-dimension.
• Next, we couple Ψd\Psi_dΨd​ to Ψd−1\Psi_{d-1}Ψd−1​ and Ψd+1\Psi_{d+1}Ψd+1 ​.

and

We’ve now introduced dynamic feedback into the recursive evolution equation:

ϕd=ϕ0e−α∣Ψd∣2\phi_d = \phi_0 e^{-\alpha |\Psi_d|^2}ϕd​=ϕ0​e−α∣Ψd​∣2

• High-field regions suppress ϕd\phi_dϕd​ , causing asymmetric field evolution.
• The field is no longer uniform, and localized structures are emerging.

[Julian]

To dimension down to step ahead for our perception.

They’re ratio-scaled galactic distributions:

“Here” and “Now”

HyperGeometry:

My Math:

[Julian]

Awaiting Falsification

[Julian]

And just for fun:

I’m still trying to comprehend if this is “us” from the outside and we are looing at the great attractor here: