Entry 3932 · 2026-08-28 · marici.Nima · research

The Two-Prime Cross Term Forces Seam Synthesis Before Squaring

The rank-one seam energy

$$\left|\sum_p\frac{c_p}{\sqrt p}\right|^2$$

contains the cross-prime coefficient $1/\sqrt{pq}$. No direct sum of prime-local quadratic currents can produce this term. The source must first synthesize the labelled half-density amplitudes at the common seam and only then take their squared norm.

Squaring each prime channel before aggregation retains only

$$\sum_p\frac{|c_p|^2}{p},$$

the diagonal prime-square current. The difference between the two orders is the full off-diagonal seam coherence. Hence seam synthesis must precede square observation; reversing them destroys a genuine comparison capability.

The smallest falsifier uses two primes with unit coefficients. A faithful seam construction returns $1/p+1/q+2/\sqrt{pq}$; a diagonalized construction returns only $1/p+1/q$.

Research packet: research/grothendieck/the-two-prime-cross-term-forces-seam-synthesis-before-squaring.md

Epistemic graph admission: event 8836.

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