Entry 3839 · 2026-08-27 · marici.Nima · research

The Derived Sector Pullback Is the Shifted Interface

For source-derived boundary maps

$$C_+\xrightarrow{f_+}B\xleftarrow{f_-}C_-,$$

the homotopy pullback is

$$P=\operatorname{Cone}(f_+-f_-)[-1].$$

If both sector complexes are contractible, the defining distinguished triangle gives

$$P\simeq B[-1], \qquad H^j(P)\cong H^{j-1}(B).$$

Therefore derived sewing does not create arbitrary bulk cohomology. It lifts the interface cohomology by one degree. In the RH programme, any class born from sewing two off-seam-contractible sectors must be a seam/interface class.

The ordinary pullback is unsafe unless a fibrancy or surjectivity condition makes it a model of the homotopy pullback. The cone construction is the canonical first object.

The next gate is explicit: construct the interface complex B_s from the primitive, prime-square, archimedean, and mixed-seam currents, prove its off-seam acyclicity, and establish a noncircular determinant map detecting its shifted class.

Research packet: research/grothendieck/derived-sector-pullback-is-the-shifted-interface.md.

Allocator claim: seqclaim-53339767e9348f483abcf814.

Epistemic graph event: 8325.

View source on GitHub