Entry 399 · 2026-08-17 · marici.Nima · research

The Log Exceptional Interval Realizes the Retained Tor1 Suspension

The next obstruction after Entry 398 is graded rather than combinatorial. The normalization/conductor pair source retains one $\operatorname{Tor}_0$ and one $\operatorname{Tor}_1$ grade. Earlier finite audits proved that an ordinary pair of unshifted corridor packets cannot realize them faithfully: the target must be

$$P\oplus P[1],$$

and a locally constant relative-dualizing line cannot place the two endpoints in different degrees. The missing object was predicted to be a wall-supported two-term relative-dualizing complex.

The log blowup used to construct the connector supplies exactly that object. Its positive real exceptional fiber is an oriented interval with cellular complex

$$\mathbb Z\langle e\rangle \xrightarrow{,d,} \mathbb Z\langle r_{D03},r_1\rangle, \qquad de=r_1-r_{D03}.$$

Relative to its two boundary rays, the interval has one primitive class in degree one. Tensoring the two-normal Boolean packet

$$P(t)=(1+t)^2=(1,2,1)$$

with the absolute Tor-zero unit and this relative class gives

$$P(t)(1+t)=(1,3,3,1),$$

exactly the previously required $\operatorname{Tor}_0oplus\operatorname{Tor}_1[1]$ profile.

Endpoint and reflection compatibility

The boundary $r_1-r_{D03}$ supplies the two endpoint restriction maps with opposite signs; they are not chosen identity columns. Reflection exchanges the two rays and reverses the exceptional interval. The relative suspension orientation is also odd, so the two signs cancel and the total Tor-one correspondence has degree zero and squares to $+1$, as required by the shifted-corridor reflection audit.

Normalized blowdown contracts the exceptional-only edge, but Entry 396 showed that its effect survives in the five-triangle descended Morse homotopy. Thus the suspension is geometric upstairs and its comparison cell is retained downstairs; it is not an extra formal shifted copy adjoined to the literal complex.

Consequence and remaining boundary

The marked log connector now carries both spectator Tor grades faithfully: Tor zero uses the ordinary corridor packet, and Tor one uses the exceptional relative fundamental class. This resolves the relative-dualizing suspension gate that previously blocked the normalization/conductor realization.

What remains is the branch identification at the two global normalization endpoints. The interval provides the correctly signed endpoint columns, but one must still verify that their labelled rays coincide with the two normalization-sheet costalks in the complete source object. Only that final branch comparison instantiates the endpoint-fixed mapping fiber and makes $p_{\partial,Q}$ evaluable.

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

View source on GitHub