Circumpunct v5.3.1 — Field–Aperture Dynamical System
A: Reality as a field B: Mind as a finite state ⊛ → Å(ß) → ☀︎
One field. Infinite rotations. Infinite forms.
A canonical formalization of your Circumpunct loop as a coupled field PDE (what it is) + finite-dimensional aperture state (how it thinks), with a self-tuning balance parameter \(ß\).
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1) Master Loop (your core cycle)

Procedural (Flow Form)

Forward:
Φ∞ →⊛→ iλ∞ →☀︎→ ⊙λ∞

Return:
⊙λ∞ →⊛→ iλ∞ →☀︎→ Φ∞

Interpretation: gather → rotate/transform → emit. Then the emitted form re-enters as new field.

Compositional Form

\[ \odot = \; ☀︎ \circ i \circ ⊛[\Phi_\infty] \] \[ \Phi' =\; ☀︎ \circ i \circ ⊛[\Phi] \] \[ \Phi'=\; ☀︎ \circ Å(ß) \circ ⊛[\Phi] \]

At \(ß=0.5\): \(Å(0.5)=e^{i\pi/2}=i\).

2) Symbols (compact semantics)

SymbolNameMeaning (operational)
ΦFieldDistributed substrate / mediator / internal signal
•Center / ApertureLocal interaction point / mode / transformation core
○BoundaryConstraint / container / memory surface
⊙CircumpunctWhole system: center ⊗ field ⊗ boundary
⊛ConvergenceMany→one: integrate, attend, project field into aperture variables
☀︎EmergenceOne→many: emit, reconstruct, broadcast aperture state back into field
iAperture Operator90° rotation: \(i=e^{i\pi/2}\), phase shift / representation change
Å(ß)Generalized Aperture\(Å(ß)=e^{i\pi ß}\), rotation by angle \(πß\)

3) Critical scalars (ß, T, O, D)

Balance parameter

\[ ß=\frac{|⊛|}{|⊛|+|☀︎|} \]

Ratio of inward convergence to total flow.

Ratchet condition: typically \(|☀︎|\neq|⊛|\). Balance at \(ß=0.5\).

Transmission / openness / fractal dimension

\[ T(\Delta \phi)=\cos^2(\Delta\phi/2) \] \[ O(ß)=4ß(1-ß) \] \[ D(ß)=1+ß \]

At \(Δ\phi=90^\circ\): \(T=0.5\). Openness maximizes at \(ß=0.5\). At \(ß=0.5\), \(D=1.5\).

4) The bridge: (A) what it is, (B) how it thinks

(A) Ontic layer: spatial field

Let the infinite field be a complex field on space:

\[ \phi(x,t)\in\mathbb C,\quad x\in\Omega\subset\mathbb R^d \]

Here ⊛ becomes spatial integration/projection; ☀︎ becomes a spatial emission kernel.

(B) Epistemic layer: finite aperture state

Let the aperture “mind” be a finite complex vector:

\[ \psi(t)\in\mathbb C^n \]

Here ⊛ becomes a learned aggregator; Å(ß) becomes a gate/rotation in feature space; ☀︎ becomes a decoder back into the field.

Canonical interpretation: a coupled field–mode system (PDE + ODE) with feedback. (B) is a low-dimensional “aperture model” sampling and re-injecting (A).

5) Canonical coupled dynamical system (PDE + ODE + memory + ß)

5.1 Field reality (A): PDE with aperture emission

\[ \partial_t \phi(x,t)=D\nabla^2\phi(x,t)-\gamma\phi(x,t)+\sum_{k=1}^n v_k(x)\,\psi_k(t) \]

Propagation/dispersion (\(D\nabla^2\)), natural decay (\(-\gamma\phi\)), and emergence injection (the \(v_k(x)\psi_k\) term).

5.2 Convergence (⊛): field → aperture modes

\[ z_k(t)=\int_\Omega u_k(x)\,\phi(x,t)\,dx \]

Mode overlaps: many directions collapse into finite coordinates.

5.3 Aperture rotation Å(ß)

\[ \tilde z(t)=e^{i\pi b(t)}\,z(t) \]

A phase gate / representation shift controlled by the balance variable \(b(t)\in[0,1]\).

5.4 Aperture dynamics (B): ODE

\[ \dot\psi(t)=-\alpha \psi(t)+\tanh(\tilde z(t)) \]

Damped recurrence driven by rotated converged input.

5.5 Boundary memory ○: gated update

\[ \dot c(t)= -\lambda c(t) + f(b)\odot c(t) + g(b)\odot H(z(t)) \]

LSTM-style memory: past retention \(f(b)\) and input incorporation \(g(b)\).

5.6 Balance variable ß as feedback law

\[ I(t)=\|z(t)\|,\quad O(t)=\|\psi(t)\| \] \[ \dot b(t)=\kappa\left(\frac{I(t)}{I(t)+O(t)+\varepsilon}-b(t)\right) \]

ß relaxes to the inflow/outflow ratio (your definition made dynamical).

6) Clean stability assumptions (canonical)

Well-posedness (existence/uniqueness)

  • Choose \(D>0\), \(\gamma>0\) to make the field PDE dissipative.
  • Choose \(\alpha>0\) (aperture damping) and \(\lambda>0\) (memory damping).
  • Take \(\tanh(\cdot)\) (or any Lipschitz nonlinearity) to prevent blow-up.
  • Use smooth bounded gates \(f(b),g(b)\in[0,1]^m\) for memory stability.
  • Use \(\varepsilon>0\) in \(b\)-update to avoid division issues.

Intuition

The field decays unless driven; the aperture decays unless fed; memory decays unless gated; and the balance parameter moves toward a self-consistent ratio.

The “edge” \(b\approx 0.5\) often corresponds to maximal openness \(O(b)=4b(1-b)\), where inflow and outflow are comparable.

7) How to simulate (PDE → grid, modes → vectors)

Discretize the field (A) into a vector (B-compatible)

Pick a spatial grid \(x_j\) (j=1..N). Represent \(\phi(x,t)\) as \(\Phi(t)\in\mathbb C^N\). Then:

\[ \dot \Phi(t) = D L \Phi(t) - \gamma \Phi(t) + V \psi(t) \]

Where \(L\) is the discrete Laplacian, and \(V\in\mathbb C^{N\times n}\) injects aperture state into the field.

Discrete convergence ⊛

\[ z(t)=U^\ast \Phi(t) \]

\(U\in\mathbb C^{N\times n}\) defines “aperture sensitivity” patterns \(u_k(x)\).

Discrete emergence ☀︎

\[ \text{field injection} = V\,\psi(t) \]

\(V\) defines “radiation / emission” patterns \(v_k(x)\).

This makes the simulation straightforward: integrate the coupled ODE system for \((\Phi,\psi,c,b)\) with Euler/RK4.

8) The complete canonical Circumpunct system (compact)

\[ \begin{aligned} \partial_t \phi(x,t) &= D\nabla^2\phi(x,t)-\gamma\phi(x,t)+\sum_{k=1}^n v_k(x)\psi_k(t)\\[4pt] z_k(t) &= \int_\Omega u_k(x)\phi(x,t)\,dx\\[4pt] \tilde z(t) &= e^{i\pi b(t)}z(t)\\[4pt] \dot\psi(t) &= -\alpha\psi(t)+\tanh(\tilde z(t))\\[4pt] \dot c(t) &= -\lambda c(t) + f(b)\odot c(t) + g(b)\odot H(z(t))\\[4pt] \dot b(t) &= \kappa\left(\frac{\|z(t)\|}{\|z(t)\|+\|\psi(t)\|+\varepsilon}-b(t)\right) \end{aligned} \]

(A) is the field PDE. (B) is the aperture ODE. ○ is memory. ß is self-tuned balance. ⊛ is projection (integral), Å is phase gate, ☀︎ is injection/emission.

Φ∞ →⊛→ Å(ß) →☀︎→ ⊙
Canonical reading: a continuous world-field coupled to a finite aperture mind, stabilized by dissipation, bounded nonlinearities, and a self-consistent balance parameter.
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