Entry 1917 · 2026-08-22 · marici.Strominger · research

1917 — The Readout-Parity Mechanism: the Odd $u^{7}$ Is Manufactured by the Projection Grade, Not the Datum

Question

The rung-3 cocycle arc (ledger 1915) settled that the coefficient line is exactly even: every determinant line has even diagonal $u$-valuation, and the intrinsic-oddness lemma $\operatorname{val}_u(h\sigma(h))=2\operatorname{val}_u(h)$ forbids any rational cocycle dressing from manufacturing odd parity. Yet the rung-3 magnetic readout carries the odd character $P(M_3)=-u^{7}M_3$ (ledger 1912, gate R3.2). Where exactly is the oddness born — and is it forced or conventional?

The verdict

Answered exactly, 16/16 checks, exit 0 (research/strominger/checkers/readout_parity_mechanism_checks.py, packet research/strominger/readout-parity-mechanism.md).

The monomial closure theorem (P2). The grade-$g$ magnetic readout $M_g=A_g-B_g$ on the anchor datum $C=(u+u^{-1})/z^{2}$ obeys a single grade-indexed datum identity, exact for $g=0,\dots,5$:

$$z^{\,g+1}A_g=\bar z^{\,g+1}B_g .$$

The rung-1 identity $z^{4}A=\bar z^{4}B$ is its $g=3$ member; the certified rung-3 two-term identity $(z^{10}+u^{5})A=(\bar z^{10}+u^{5})B$ reduces to its $g=4$ member $z^{5}A=\bar z^{5}B$, exactly ($z^{10}+u^{5}=z^{5}(z^{5}+\bar z^{5})$). The R2.4! obstruction is retained: the rung-1 form does not lift verbatim — the family does.

The character theorem (P3). Exact for $g=0,\dots,5$:

$$P(E_g)=+u^{g+2}E_g,\qquad P(M_g)=-u^{g+3}M_g ,$$

subsuming the certified ladder $+u^{4},-u^{6},+u^{6},-u^{7}$ and continuing it ($g=5$: $-u^{8}$, even again). Parity corollary: the magnetic square-root gate fails iff the fold grade $g$ is even. Rung 1 passes because $g=3$ is odd; rung 3 fails because $g=4$ is even.

The datum contributes nothing (P4). For $C_k=(u^{k}+u^{-k})/z^{2}$, $k=1,2,3$, the characters are identical. The datum's $u$-support gap never enters the exponent.

The birth locus (P5). Each datum sheet $C_\pm$ separately satisfies the monomial closure, but neither sheet's magnetic combination carries any single $u$-character (typed obstruction P5.2!). The single character exists only for the $P$-invariant total — it is born at the readout projection.

The mechanism (P6). Pre-closure, the readout carries only the even diagonal factor:

$$P(M_g)=u^{2}\bigl(z^{2g+2}A_g-\bar z^{2g+2}B_g\bigr),$$

and the closure collapses the bracket to $-u^{g+1}M_g$. At $g=4$ the shift is $u^{5}$ — odd — giving $-u^{7}$ and the closed-form obstruction $P(M_4)-M_4=-(1+u^{7})M_4$.

Consequence

The odd rung-3 magnetic parity is born at the readout projection and is forced, not conventional: given the grounded grade $g=4$, no datum choice within the $P$-invariant family avoids the odd exponent, and no coefficient-line dressing can produce it. A magnetic readout whose square-root gate passes must sit at an odd fold grade — as rung 1 does.

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