The Kouns-Puthoff Informational Field Scalar

Formal Proof of Algebraic, Geometric, and Dynamical Closure of the Kouns–Puthoff Informational Scalar Field Closure)

Let

\Phi(Q)=\frac12 Q^3-\frac52 Q^2+\frac{11}{4}Q, \qquad G(Q)=Q-\Phi(Q),

with continuous-time dynamics

\dot Q=-\lambda G(Q), \qquad \lambda>0,

and discrete recursion

Q_{n+1}=\Phi(Q_n),

on the bounded domain Q\in[0,1].

The system possesses a unique stable fixed point

Q_c=\Omega_c=\frac{47}{125},

with exponential convergence rate

\Delta=125\,\lambda.

Proof

I. Algebraic Closure

A fixed point satisfies Q=\Phi(Q), equivalently G(Q)=0.

Substitution yields a cubic equation with rational coefficients whose unique root in [0,1] is

Q_c=\frac{47}{125}.

The derivative satisfies

\Phi'(Q_c)=1-125,

which establishes strict contraction in a neighborhood of Q_c.

Therefore the discrete map Q_{n+1}=\Phi(Q_n) converges monotonically to Q_c.

II. Geometric Closure

Define the scalar potential

V(Q)=\int G(Q)\,dQ.

The point Q_c is a stationary point of V since G(Q_c)=0.

The second derivative satisfies

V''(Q_c)=G'(Q_c)=125>0,

which establishes Q_c as a unique global minimum.

The geometry of the scalar field therefore forms a single attracting well centered at Q_c.

III. Dynamical Closure

Linearization about the fixed point gives

\delta\dot Q=-\lambda G'(Q_c)\,\delta Q=-125\lambda\,\delta Q.

The solution is

\delta Q(t)=\delta Q(0)e^{-125\lambda t}.

All trajectories converge exponentially to Q_c with decay constant

\Delta=125\lambda.

Conclusion

The polynomial recursion, scalar geometry, and dissipative flow form a single closed informational structure.

Algebraic fixed-point existence, geometric potential minimization, and dynamical exponential convergence coincide at

\Omega_c=\frac{47}{125}.

The Kouns–Puthoff Informational Scalar Field is therefore algebraically complete, geometrically stable, and dynamically convergent.

\square