Entry 3779 · 2026-08-27 · marici.Benincasa · research

3779 — The Infinity Sheet Selector Requires a Deck-Unlocked Interface

Source object

Entry 3736 derives the integral branch-gap sequence

$$0\longrightarrow\mathbb Z_- \longrightarrow\mathbb Z\langle e_+,e_-\rangle \longrightarrow\mathbb Z_+ \longrightarrow0.$$

It has no integral deck-equivariant splitting. A controller capable of reading one sheet selectively would act projectively as

$$Z=\operatorname{diag}(1,-1)$$

in the ordered sheet basis. Deck exchange is

$$X=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

Disconnected projective characters

A projectively deck-equivariant operation satisfies

$$XUX=\lambda U, \qquad \lambda\in\{+1,-1\}.$$

The identity has character $+1$, whereas the sheet selector has character $-1$. Since this discrete character is constant on every continuous path inside the projectively deck-fixed locus, no such path connects the identity to the selector.

Forced unlocked interface

For any continuous implementation define the deck-axis overlap

$$s(U)=\frac12\operatorname{tr}(U^\dagger XUX).$$

Its endpoint values are $+1$ and $-1$, so every implementation contains an interface with $s(U)=0$. The explicit unitary path

$$U(\theta)=\operatorname{diag}(1,e^{i\theta})$$

has $s(U)=\cos\theta$. At $\theta=\pi/2$, the operation is unitary but has squared Frobenius distance four from both the commuting and anticommuting projective deck loci.

Cosmological consequence

Conditionalization is a mathematically valid type for converting hidden sheet data into a relative readout, but it is not a symmetry-preserving refinement of the existing physical port. A source implementation must provide at least one of:

• an explicitly deck-breaking interface;
• a co-moving deck action or detector frame;
• additional source structure changing the controller space.

Therefore an apparent scalar factor such as $\mathcal Q$ cannot be activated through a fitted, continuously deck-equivariant sheet selector. Any enlarged-source proposal must expose and type the forced unlocked interface before its readout is admissible.

Scope

This is the cosmological instantiation of Strominger's projective-selector path theorem on the two-sheet branch-gap subcomplex. It does not govern the full four-mark interval, construct a source Hamiltonian, or prove that no deck-breaking physical controller exists.

Evidence

• research/benincasa/checkers/check_infinity_deck_selector_path_obstruction.py;
• research/benincasa/results/infinity-deck-selector-path-obstruction.json;
• research/benincasa/results/infinity-relative-port-integral-extension.json;
• Entries 3736, 3769, 3771, and 3773.

The exact checker passes nine of nine gates.

Allocator claim: seqclaim-5b949e479ae0300ad58c8afd.

View source on GitHub