ReynoldsBEng 10th August 2026
Two papers, published within a day of each other (6–7 August 2026), close the same geometric circuit.
Paper 1 — Exponential Speedup of Entanglement Generation by Quantum Mpemba Effects (arXiv:2608.05935) Protocols that exploit quantum Mpemba effects exponentially accelerate the generation of entanglement (or retard its decay). “Hotter” initial states reach a target entanglement threshold or the asymptotic steady-state value far faster than “colder” ones. The effect is measure-dependent and survives in many-body systems.
Paper 2 — Shor’s algorithm requires Fanout (arXiv:2608.06703)
Constant-depth approximation of the Quantum Fourier Transform (the core of Shor’s algorithm) is possible if and only if the n-qubit Fanout operation is available. Without Fanout, QFT cannot be realised in constant depth; with a single QFT one can approximate Fanout (plus a constant number of local gates). The feasibility of Shor’s algorithm on NISQ hardware is therefore tied directly to the feasibility of Fanout.
Geometric synthesis under the π-Tensor
Both results are the same bistable contact-patch action read in Rest Time.
Every contact patch is a local hysteretic thickening of the Elastic Plenum:0^{i2} = r²
The radius of the disc becomes the diameter of the sphere of light emitted at each Instant closure. The Instant is the judder wave that takes up slack across the surface. The choice carried across the gap selects the branch:
State A (positive toggle, πN-dominant)* — permanent solvent twist. Cavity force is pushed upward; collapse occurs at Time-Particle scale *inside the Instant itself. The Moment h is already contained. This is a global, simultaneous reconfiguration — precisely the geometric content of Fanout. Entanglement (or the QFT superposition) is generated exponentially fast. Light is bright.
State B (negative toggle, πR-dominant)— duration-clamped mass shell maintained for the full Moment \(h\). This is the slower, local-gate channel. Entanglement generation is thermal and sequential.A “hotter” initial state (Mpemba) is an ensemble of contact patches with higher dilatancy and larger stored strain. The judder wave therefore has a higher probability of driving the system onto the solvent (State A / Fanout) channel. Once on that channel the target entanglement (or the QFT register) is reached exponentially faster. A “colder” state sits deeper in the clamped branch and must wait for the slower thermal hopping.
Thus:
The Mpemba speedup is the statistical preference for the solvent Instant-scale path. Fanout is the geometric realisation of that same Instant-scale path across many patches at once. Shor’s algorithm requires Fanout because factoring demands the global solvent reconfiguration that only State A supplies in constant depth.
Both phenomena are therefore necessary consequences of the Instant–Moment operator cycling inside the Elastic Plenum; they are not independent algorithmic or thermodynamic accidents.
The combined proof is direct: the exponential quantum advantage and the Mpemba entanglement speedup share the identical geometric mechanism. Certainty (the permanent solvent branch) is fast and global; the apparent “probability” or slower relaxation is the histogram of clamped choices. All is simple once the continuum is read as geometry rather than as scalar temperature, continuous clock time, or local gate counts.The Canon advances.
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