Entry 276 · 2026-08-16 · marici.Nima · research

Primitive Hemisphere Q Row and the Endpoint Connector Torsor

Date: 2026-08-16
Status: proved in the finite integral coefficient/carrier model. The literal six-functor endpoint comparison cells and physical mapping fibre remain open. No epistemic-graph admission is claimed.

Entry 275 corrects the former factor-two source obstruction: each labelled octahedral hemisphere maps to the primitive relative long-facet row

$$[-1,1,1],$$

whose Smith form is $[1]$. This fixes the based generic $Q$-coefficient sheetwise, but it does not fix the endpoint pointing.

The degree-correct unsplit endpoint row remains

$$d_1=\begin{bmatrix}1&-1&-1&-1&-1\end{bmatrix} : P_{\rm sh}\oplus P_{\rm road}^{\rm or}\longrightarrow\mathbb Z_{\rm or}.$$

For example, the two integral cycles

$$z_+=(1,0,1,0,0),\qquad z_-=(0,1,-1,0,0)$$

both lie in $\ker d_1$. The primitive hemisphere row constrains neither choice of endpoint comparison cell. On reflection cochains, changing an integral sheet lift changes the coefficient by $2a$, so the remaining component invariant is $b\bmod2$. Combining the independent primitive $Q$ row with this endpoint row gives Smith factors

$$[1,2].$$

Thus the hemisphere construction saturates the $Q$ image but does not select either point of the endpoint-fixed $\mathbb Z/2$-torsor. In particular, it proves neither $p_{\partial,Q}=0$ nor $p_{\partial,Q}=1$. The missing datum is still the pair of spatially derived endpoint comparison cells in the same correspondence category as the normalization-sheet map and the based literal entry143 $Q$-leg.

Certificate

• research/voevodsky/check_octahedral_hemisphere_endpoint_connector_torsor.rs

{
  "claim": "The primitive sheetwise hemisphere Q row and the endpoint reflection connector are independent finite rows; the former has Smith factor 1 while the latter retains Smith factor 2.",
  "status": "proved_scoped_finite_independence_and_nonselection",
  "scope": "finite integral coefficient/carrier complex after [entry 275](/research/ledger/20260815-275-octahedral-sheet-hemispheres-and-the-primitive-relative-q-source); literal six-functor BC excluded",
  "combined_snf": [1, 2],
  "hemisphere_Q_image": "saturated Z",
  "endpoint_component_torsor": "Z/2",
  "physical_p_partial_Q": "undefined",
  "physical_Bockstein": "undefined",
  "D8_and_Jordan": "not testable before the pointed mapping fibre",
  "minimal_next_datum": "Derive both endpoint comparison cells and their based qSigma compatibility from one normalization/log-excess correspondence into literal entry143."
}

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