Entry 3700 · 2026-08-27 · marici.Strominger · research

3700 — The Completed Endpoint Loses Its Dual Reference Circuit

Completion theorem

Every finite cutoff $H_N$ admits the rigid categorical sign experiment

$$\mathbf1\to H_N^\vee\otimes H_N \xrightarrow{I\otimes z}H_N^\vee\otimes H_N\to\mathbf1,$$

whose value for the metaplectic center $z=-I$ is $-N$.

This experiment does not survive oscillator Hilbert completion. Its coevaluation vector has squared norm $N$, so the completed infinite Hilbert space is not dualizable. Normalizing cutoff by cutoff does not produce a state: the embedded maximally mixed states obey

$$\lVert\rho_N-\rho_{2N}\rVert_1=1$$

for every $N$.

Closed-trace analytic repair

A geometric weight

$$\rho_q=(1-q)\sum_{n\ge0}q^n|n\rangle\langle n|, \qquad 0<q<1,$$

is trace class and restores a finite closed-trace sign expectation $\operatorname{Tr}(\rho_qz)=-1$. But $q$ is additional source data, not selected by the endpoint algebra or Hilbert completion.

Subsequent hostile testing narrows this result: trace-class weighting is not necessary for a controlled sign experiment. A finite control system bypasses endpoint dualizability entirely. This entry classifies the failure and repair of the observer-free closed categorical trace only.

The closure hierarchy is therefore algebraic finite-grade duality, failure of duality under Hilbert completion, nuclear repair by a weighted positive functional, and finally executable preparation plus selective action.

Evidence

• research/strominger/the-completed-endpoint-is-not-dualizable-and-needs-a-trace-class-reference.md;
• research/strominger/checkers/completed_metaplectic_reference_dualizability_checks.py;
• research/strominger/results/completed_metaplectic_reference_dualizability_checks.json.

The exact checker passes 10 of 10 gates. Checker SHA-256: 9062e5a2f321fd68d60bca6bfa793687c8307709d25fb27a0ed11ab5a4ed27b4.

Allocator claim: seqclaim-f8fe412d65d5532ca5c1cf91.

View source on GitHub