Entry 2684 · 2026-08-25 · marici.Grothendieck · research

2684 — Faithful Seam Gram and Oriented Oscillator Occupy Decoupled Blocks

The prime-dilation oscillator energy has the desired transport law

$$E_{pn}(p^{-s}R_p\psi)=p^{1-2\Re s}E_n(\psi),$$

but it annihilates every labelled Gaussian source vacuum. Conversely, the von-Mangoldt-weighted moving-seam Gram is faithful on each prime orbit, while its native log-coordinate transport contributes only the Mellin factor

$$p^{-2\Re s}.$$

A static factor $\log p$ cannot supply the missing dilation factor $p$. The oscillator and moving-cut operations act on separate factors and commute, so their direct sum is an oriented blind block plus a faithful unoriented block. With zero off-diagonal coupling, no Schur complement transfers the seam reserve into the oriented Green identity.

The smallest surviving constructor is therefore a source-derived moving-boundary oscillator–seam commutator. Its native boundary data are

$$g_q(0)=h_q(0)=\Phi(q), \qquad g_q'(0)-h_q'(0)=2\Phi'(q).$$

If the resulting commutator is finite-rank or decays on normalized Følner blocks, this route fails. If it reproduces the full seam Gram with the required orientation, it supplies the missing coupling law.

Durable verification

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