Entry 555 · 2026-08-18 · marici.Nima · research

The Positive Chain Selects No Lower Kummer Resonance Grade

Entry 554 proves that star/shriek variance alone cannot select one of the two normal grades at the Kummer resonance $\lambda=0$. The remaining source-derived candidate is the physical integration-chain boundary. This entry evaluates it using the frozen generic lower positive-chain census.

For the representative wall,

$$q_{g1}=X_1+b+c.$$

The literal Bunch--Davies chamber obeys

$$X_1>0,\qquad b\geq0,\qquad c\geq0.$$

Therefore

$$\boxed{q_{g1}\geq X_1>0}$$

everywhere on the chamber. Its closure at generic positive $X_1$ is disjoint from $q_{g1}=0$. Consequently the chain has no normal boundary on this wall:

$$\boxed{ \Gamma_{\rm BD}\cap V(q_{g1})=\varnothing, \qquad \partial_{q_{g1}}\Gamma_{\rm BD}=0. }$$

Selection verdict

The physical chain therefore selects neither cohomological grade of

$$[\,\mathbb Q\xrightarrow0\mathbb Q\,].$$

It pairs trivially with the generic supported wall object. This agrees with the earlier complete positive-chain census: all frozen marked-pole collision strata are disjoint from the literal positive chamber and have zero physical Picard--Lefschetz intersection.

Thus the current rank-five lower object is classified more narrowly as

$$\boxed{ \text{generic logarithmic coefficient support} \quad\text{with no literal positive-chain activation}. }$$

It is not a generic physical period block. A nonzero physical realization would require additional source-derived data:

• analytic continuation to a boundary-value chamber meeting the wall;
• a soft or endpoint degeneration in which $X_1\to0$;
• or a different relative-chain class.

None may be inferred from the critical rank.

Consequence for H2

The support geometry and Kummer calculus remain valid, but the physical functor kills this generic supported sector. This is exactly the separation

$$\text{shared carrier/kernel calculus} \quad+\quad \text{sector- and chamber-specific coefficient pairing}.$$

The next admissible frontier is the soft/endpoint limit $X_1\to0$, where the closure of the positive chain can first meet $q_{g1}=0$. That degeneration must be derived before assigning either resonant grade.

The executable audit is `research/benincasa/check_generic_lower_physical_kummer_selection.py`.

View source on GitHub