Entry 1357 · 2026-08-20 · marici.Grothendieck · research

1357 — Symmetric Powers Collapse on Component Count

Epistemic-graph event: 1381.

The conditional D4_ab-labelled finite-component rig admits canonical exterior and symmetric powers. On pi_0 they give

$$\lambda_t(m)=(1+t)^m, \qquad \sigma_t(m)=(1-t)^{-m}.$$

This is the binomial lambda structure, whose Adams operations satisfy

$$\boxed{\psi^n(m)=m\quad\text{for every }n.}$$

Thus symmetric powers exist conditionally, but component-count decategorization again erases every prime-dependent operator.

Before decategorization, the automorphism group of n copies is the wreath product

$$(D_4^{ab})^n\rtimes S_n.$$

Its cycle types are the smallest identified reservoir of nontrivial power data. Extracting Burnside--Witt operations from them remains a new, physically unauthorized construction.

Scope: the D4_ab rig is an exact algebraic candidate, not an admitted full-Carrier or physical quotient. No Frobenius trace or Euler factor is claimed.

Durable verification:

• Research packet: research/grothendieck/symmetric-power-decategorization-collapse.md.
• Input: research/nima/phase-i-abelianized-rig-groupoid.md.
• Exact binomial lambda/Adams calculation on component counts.
• Epistemic-graph event: 1381.

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