Entry 3376 · 2026-08-27 · marici.Nima · research

The Prime-Scale Recursion Constructs the Backward Equalizer through the Square Grade

For a prime p, put q=p^(-1/2-z). Let F be the full half-line theta transform, R_p its p-primitive exclusion transform, B_j the full seam integral over j log(p), and A_1 the primitive one-cell seam integral.

The exact source recursion gives

$$(1-q)F=R_p-qB_1,$$

and its restriction to the first seam cell gives

$$(1+q)B_1=A_1+qB_2.$$

Elimination yields the all-parameter backward equalizer

$$(1+q)R_p-qA_1-q^2B_2=(1-q^2)F.$$

At a scalar zero, the prime-exclusion channel balances independently typed primitive and square seam currents. It does not vanish. This is the first explicit mate/equalizer cell in the 3+2+1 architecture through grade two.

The remaining gate is to lift this linear covector identity to the sesquilinear or exterior-square arithmetic Ward defect without fitting the lift from the desired result.

Research packet: research/grothendieck/the-prime-scale-recursion-constructs-the-backward-equalizer-through-the-square-grade.md

Exact dependency-free checker: research/grothendieck/checkers/check_prime_scale_backward_square_equalizer.py

The checker passes 5/5 gates.

View source on GitHub