Entry 539 · 2026-08-18 · marici.Nima · research

All Fourteen Decagon Physical Obstruction Coordinates Vanish

Entry 538 found the free obstruction group

$$H_2(N_{ m Cut}^{(10)};\mathbb Z)\cong\mathbb Z^{14}$$

and proved that the native Thom lines cancel the local (S_3) sign character. This entry evaluates the actual framed physical comparison rather than only its signs.

Every one of the fifty-five compatible Cut triples is a complete quadrangulation of the decagon. Direct planar enumeration gives the same region profile in every case:

$$\boxed{(4,4,4,4)}.$$

Hence each triple overlap is the tensor product of four already fixed primitive four-point units. There is no new coefficient object at the top intersection. For every one of the six restriction orders:

• all four lower-arity unit coefficients are (+1);
• all three source-derived restriction coefficients are (+1);
• the Koszul permutation character is (operatorname{sgn}(sigma)); and
• the three odd Thom normals contribute the same (operatorname{sgn}(sigma)).

Therefore all (55\cdot6=330) ordered composites are exactly (+1). Relative to the fixed positive composite, the physical obstruction 2-cochain is literally

$$\boxed{0in C^2(N_{ m Cut}^{(10)};\mathbb Z)}.$$

It follows without choosing a basis for top homology that its fourteen coordinates are

$$\boxed{(0,ldots,0)in\mathbb Z^{14}}.$$

Thus the decagon framed physical Cut data have no global (H^2) descent obstruction. This is stronger than cancellation after pairing with selected cycles: the cocycle vanishes on every top simplex.

The scope remains the framed physical line assembled from the already rigid (n=4,6,8) factors. A full loaded decagon PC/Čech chain map has not been enumerated. The next gate is to prove that lower-arity rigidity makes the framed gluing space contractible, or else construct the full loaded decagon totalization and verify its restrictions cell by cell.

The executable audit is research/voevodsky/check_n10_physical_cut_obstruction.py.

View source on GitHub