The QPUT paper derives an algebraic quantum π* for discrete space-time by enforcing radial bound r ≡ 1/2 (thus d ≡ 1 exactly). Via quantum clock kinematics, annular dynamics (w = 1/φ), and Pythagorean isoperimetric equality, the ratio is c/d = π* = 4/√φ ≈ 3.1446055..., solving x⁴ + 16x² - 256 = 0. Standard continuous π emerges only if the bound is relaxed.

QPUT binds r≡1/2 so d=1, yielding algebraic π*=4/√φ≈3.1446 in unit-cube lattice via quantum clock, replacing continuous π and eliminating singularities. This corroborates IT3 discrete spectral geometry over Q(√2,√3,√5) and cuboctahedral lattice; advances by purifying residual π kinematics to pure quantum. Quantum view fully strengthens IT3 discreteness. When d=1 exactly, c/d=π*=4/√φ. Proved by ULANE annular w=1/φ and Pythagorean: w²+(c/4)²=1 → c=4√w.

Per the article, continuous regime relaxes radial bound (r > 0 unbounded per Euclid), yielding transcendental π ≈ 3.14159 from exhaustive methods. Quantum regime enforces r ≡ 1/2 inviolate (from d ≡ 2r = 1), producing algebraic π* = 4/√φ ≈ 3.1446 via the quantum clock, annular dynamics, and kinematic isoperimetric equality. Unity principle true selects quantum; false recovers continuous. Both describe the same unit-diameter circle at different scales.

Article enforces radial bound r ≡ 1/2 across dimensions, producing non-continuous quantum spacetime. There c/d equals algebraic 4/√φ ≈ 3.1446 rather than transcendental π. This yields non-Euclidean unit-cube lattice geometry aimed at QM-GR disparity. Key implication for light in vacuum: its motion follows c/d = 4/√φ exactly, via the Pythagorean theorem under that bound.

Under the article's quantum clock, light traces a geodesic at constant speed (law of inertia) while holding the radial bound r ≡ 1/2. Space covers circumference c in exactly 4 time units (unit-square perimeter). Pythagorean closure with annular width 1/φ yields c/d = 4/√φ ≈ 3.144605511.

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