Quantum-knot modular dynamics investigation

An exhaustive 310-profile prime-field campaign on the reduced zero-framed coloured Jones invariant of the figure-eight knot found every observed period equal to the multiplicative order of q. The Habiro cyclotomic expansion supplies an exact proof for the whole expansion class, an independent 5 sub 2 implementation confirms it, and the corresponding root-of-unity symmetry is established prior art. No distinct dynamical modular channel was identified and no composite factorisation experiment was performed.

A bounded investigation of coloured Jones modular dynamics: 310 exhaustive prime-field profiles, an exact cyclotomic periodicity proof, a trace-zero classification and an honest prior-art boundary.

Quantum-knot modular dynamics.

We proved a real theorem. Someone else proved it first.

What is this study actually trying to do?

Make the sequence.

Measure the cycle.

Anything left over?

From a hopeful hypothesis to an exact boundary.

Every claim here maps to code, data, a proof, or a citation.

Quantum knot · Frozen bounded study

A knot invariant produces long, intricate sequences of numbers. Do those sequences repeat in a genuinely new way, or does ordinary arithmetic already explain every repetition?

The periodicity result is exact and independently reconstructed here, but the corresponding root-of-unity symmetry is already established in the coloured-Jones literature. We claim the reconstruction, the finite-field audit and the bounded negative conclusion, not the theorem.

A knot invariant is a way of turning a tangled loop into numbers. The one studied here, the coloured Jones invariant, gives you not a single number but a whole sequence, one value for each "colour", which you can think of as a dial you keep turning up.

Fix the arithmetic, then read off one value of the knot invariant for each colour. You get a long sequence of numbers to inspect.

The sequence loops. We measured how long every loop was, for every case in the range, not a sample of them.

• Quantum-Knot Modular Dynamics: A Bounded Investigation
• Could a knot invariant hide a richer kind of repetition?
• Classes · O: order / q-orbit · Z: coefficient zeros · T: finite-field algebra · D: genuine q-difference dynamics
• The verified model
• We rebuilt the figure-eight invariant two independent ways.
• Jₙ(q) = Σ_{k<n} Π_{j≤k} (qⁿ + q⁻ⁿ − qʲ − q⁻ʲ)
• The exhaustive pilot
• The exact proof
• The computation made its own follow-up computation unnecessary.
• The theorem we proved was already in the literature.