Infinite Calibration Sets for Analytic Scalar Controls

Abstract

One positive real parameter gives a strictly feasible analytic scalar control whose simultaneous value and derivative calibration points are exactly the positive-natural sequence converging to one.

All scalar parameters are real. Assume 0 < a < 1, 0 < delta, delta < (1-a)/4, delta < (1-a squared)/16, and 0 < b < 1-a/(1-delta). The delta bound involving a squared is retained even though it is not needed in the scalar argument. The domain of physicality and calibration is the open interval (a,1).

Use the scalar parameters L(a)=1-a, lam(a,delta)=L(a)-delta, gamma(a,delta)=lam(a,delta)/(2 delta), and center(a,delta)=lam(a,delta)/L(a). The radius is sqrt(a/lam(a,delta)) times sqrt((1-p)/p). The energy at s is s squared plus gamma squared times (s+1/s-2) squared, and beta(p)=1/(2 p (1-p)). Division uses total real inversion; all denominators used on the stated interval are nonzero.

Definition 1.1 (The oscillatory phase).

$$\forall b,p:\mathbb{R},\mathrm{omega}\left(b,p\right)=\frac{2\pi b}{1-p}$$

Formalization. D5/S3/Quantum/Information/InfiniteCalibrationControl.omega (✓ std3).

Source. Repository-derived.

Commentary.

For positive b the phase increases without bound as p approaches one from below.

Definition 1.2 (The decaying amplitude).

$$\forall b,p:\mathbb{R},\mathrm{amplitude}\left(b,p\right)=\frac{1-p}{4\pi bp}$$

Formalization. D5/S3/Quantum/Information/InfiniteCalibrationControl.amplitude (✓ std3).

Source. Repository-derived.

Commentary.

For b > 0 and 0 < p < 1 the amplitude is positive. It tends to zero as p tends to one.

Definition 1.3 (The oscillatory interpolant).

$$\forall b,p:\mathbb{R},\mathrm{phaseH}\left(b,p\right)=1+\mathrm{amplitude}\left(b,p\right)\sin \left(\mathrm{omega}\left(b,p\right)\right)$$

Formalization. D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseH (✓ std3).

Source. Repository-derived.

Commentary.

The bounded sine factor and decaying amplitude make this interpolant tend to one at the upper endpoint.

Definition 1.4 (The nonnegative damping factor).

$$\forall b,p:\mathbb{R},\mathrm{phaseD}\left(b,p\right)={\sin \left(\frac{1}{2}\mathrm{omega}\left(b,p\right)\right)}^2$$

Formalization. D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseD (✓ std3).

Source. Repository-derived.

Commentary.

The damping factor is a square. Below one, with b positive, it vanishes exactly at points 1-b/n with n a positive natural number.

Definition 1.5 (The scalar control).

$$\forall a,delta,b,k,p:\mathbb{R},\mathrm{phaseControl}\left(a,delta,b,k,p\right)=\frac{\mathrm{phaseH}\left(b,p\right)+k\mathrm{center}\left(a,delta\right)\mathrm{phaseD}\left(b,p\right)}{1+k\mathrm{phaseD}\left(b,p\right)}$$

Formalization. D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseControl (✓ std3).

Source. Repository-derived.

Commentary.

When k is positive the denominator is at least one. The control is a convex combination of the interpolant and the fixed center, with interpolant weight 1/(1+k phaseD(b,p)).

Definition 1.6 (The positive-natural calibration sequence).

$$\forall b:\mathbb{R},\forall n:\mathbb{N},\mathrm{phaseNode}\left(b,n\right)=1-\frac{b}{n}$$

Formalization. D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseNode (✓ std3).

Source. Repository-derived.

Commentary.

The definition is total on natural numbers. Only positive indices occur in the calibration set. Under the parameter hypotheses, these nodes lie above a/(1-delta), are strictly increasing, and converge to one.

Definition 1.7 (All scalar control requirements for the same parameter).

$$\forall a,delta,b,k:\mathbb{R},\mathrm{InfiniteScalarControl}\left(a,delta,b,k\right)⟺(0<k\wedge (\forall p\in \mathrm{Ioo}\left(a,1\right),(0<\mathrm{phaseControl}\left(a,delta,b,k,p\right)\wedge {\mathrm{radius}\left(a,delta,p\right)}^2\mathrm{energy}\left(a,delta,\mathrm{phaseControl}\left(a,delta,b,k,p\right)\right)<1))\wedge \mathrm{AnalyticOnNhd}\left(\mathbb{R},\mathrm{phaseControl}\left(a,delta,b,k\right),\mathrm{Ioo}\left(a,1\right)\right)\wedge (\forall p\in \mathrm{Ioo}\left(a,1\right),((\mathrm{phaseControl}\left(a,delta,b,k,p\right)=1\wedge \mathrm{deriv}\left(\mathrm{phaseControl}\left(a,delta,b,k\right),p\right)=\mathrm{beta}\left(p\right))⟺(\exists n:\mathbb{N},0<n\wedge p=\mathrm{phaseNode}\left(b,n\right)))))$$

Formalization. D5/S3/Quantum/Information/InfiniteCalibrationControl.InfiniteScalarControl (✓ std3).

Source. Repository-derived.

Commentary.

The conjunction requires k > 0, positivity and strict physicality at every point of (a,1), analyticity in a neighborhood of each point of that interval, and the complete simultaneous jet characterization. Value one alone does not characterize the nodes: the derivative must also equal beta at the same point.

Theorem 1.8 (One finite parameter realizes the entire infinite calibration set).

$$\forall a,delta,b:\mathbb{R},(0<a\wedge a<1\wedge 0<delta\wedge delta<\frac{1-a}{4}\wedge delta<\frac{1-a^2}{16}\wedge 0<b\wedge b<1-\frac{a}{1-delta})\Rightarrow \exists k:\mathbb{R},(0<k\wedge (\forall p\in \mathrm{Ioo}\left(a,1\right),(0<\mathrm{phaseControl}\left(a,delta,b,k,p\right)\wedge {\mathrm{radius}\left(a,delta,p\right)}^2\mathrm{energy}\left(a,delta,\mathrm{phaseControl}\left(a,delta,b,k,p\right)\right)<1))\wedge \mathrm{AnalyticOnNhd}\left(\mathbb{R},\mathrm{phaseControl}\left(a,delta,b,k\right),\mathrm{Ioo}\left(a,1\right)\right)\wedge (\forall p\in \mathrm{Ioo}\left(a,1\right),((\mathrm{phaseControl}\left(a,delta,b,k,p\right)=1\wedge \mathrm{deriv}\left(\mathrm{phaseControl}\left(a,delta,b,k\right),p\right)=\mathrm{beta}\left(p\right))⟺(\exists n:\mathbb{N},0<n\wedge p=\mathrm{phaseNode}\left(b,n\right)))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/InfiniteCalibrationControl.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

The center lies strictly between three quarters and one and is strictly feasible throughout [a,1]. The energy is convex on the positive half-line. Near one, the interpolant tends to one and radius squared times its energy tends to zero. This gives a whole terminal interval where every positive k is feasible by convexity.

On the remaining compact interval, zeros of the damping factor give value one above the strict feasibility threshold. At every other point, increasing k makes the control tend to the strictly feasible center. The open sets of feasible points increase with k; a directed compactness argument selects one positive real k for the entire complement. Together these intervals give full physicality on (a,1).

Analyticity holds for every positive k because the constituent functions are analytic on (a,1) and the denominator is positive. The half-phase sine vanishes exactly at the positive-natural nodes. At these zeros the control has value one and derivative beta. At any other point where the control equals one, differentiation and the double-angle identities give a strictly negative derivative. Since beta is positive on the interval, such extra roots cannot be calibration jets. These conclusions apply to the same k chosen for physicality.

References

• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.InfiniteScalarControl
• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.amplitude
• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.omega
• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseControl
• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseD
• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseH
• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.phaseNode
• Truth anchor: D5/S3/Quantum/Information/InfiniteCalibrationControl.result
• Dependency: D5/S3/Quantum/Information/FiniteCalibrationControl