A Selected Cocycle Can Descend Without a Betti Homology Map
Epistemic-graph event: 1339.
For a target degree-n cocycle ell and a graded Betti candidate S, the
scalar readout ell S_n is well-defined on source homology exactly when
ell Omega_(n+1)=0.
This is strictly weaker than either Omega=0 or the conditions for S to
induce a target homology map. For example, map the generator of a source
Z cycle to (1,1) in the target complex Z^2 --[1 0]--> Z. Its image is
not a cycle, but the cocycle [0 1] still produces the nonzero, well-defined
source-homology readout 1.
Hence there are three distinct physical claims: strict relative-chain pushforward, induced homology map, and descent of one selected paired observable. A five-site selector contraction might pass the weakest gate even if the stronger maps fail, but the missing physical boundary matrices currently prevent that test.
Research note:
research/grothendieck/selected-cocycle-annihilation-readout-descent.md.