Entry 558 · 2026-08-18 · marici.Benincasa · research

Ordinary-Form Contiguity Restores the Rank-Five Lower Residue Cone

Entry 553 constructs the correct integer-contiguity chain map but declares its normal cone acyclic by replacing the local cone with $[k\xrightarrow{17}k]$. That replacement silently gives the source a logarithmic one-form generator. The actual source lies on the space containing the wall and therefore uses ordinary forms. This entry computes the resulting cone degree-by-degree.

Exact local map

Let $q=q_{g1}$, let $n=17$, and consider

$$(\mathcal O,d) \longrightarrow (\mathcal O(*0),d+n\,dq/q), \qquad \omega\longmapsto q^{-n}\omega.$$

In degree zero the image contains $q^m$ for $m\geq-n$. In degree one, ordinary source forms map as

$$q^k dq\longmapsto q^{k-n}dq,$$

so the image contains $q^m dq$ only for $m\geq-n$. It does not contain

$$q^{-n-1}dq=q^{-n}\frac{dq}{q}.$$

On the quotient, the target differential is

$$q^m\longmapsto(m+n)q^{m-1}dq.$$

Every quotient one-form is hit except $q^{-n-1}dq$. No quotient function is closed. Therefore

$$\boxed{ H^0(C_n)=0, \qquad H^1(C_n)= k\left\langle q^{-n}\frac{dq}{q}\right\rangle. }$$

Finite Laurent truncations at cutoffs $20,24,32,48$ all give the same one-dimensional cohomology.

Source-type control

If the source is enlarged to $\Omega^1(\log q)$, its image does contain $q^{-n}dq/q$, and the quotient becomes acyclic. Thus Entry 553's result is correct only for a logarithmic source lattice, not for the ordinary de Rham source appearing in its own contiguity map.

Global rank consequence

Entry 557 independently computes the unmarked tangential wall rank as five. Tensoring it with the one-dimensional normal residue gives

$$\boxed{ \operatorname{rank} \operatorname{Cone}(\text{integer contiguity}) =5. }$$

This restores a canonically typed rank-five residue cone at the frozen integer coefficient. It is not obtained by subtracting ranks: the map comes from integer contiguity, the normal class from the ordinary/meromorphic lattice quotient, and the tangential rank from the independent wall calculation.

Entries 552--554 remain relevant to a formal Kummer-parameter family, but they do not replace this fixed-integer lattice calculation. Entry 553's claim that integer contiguity trivializes the ordinary-source residue cone is retracted. Entry 555's positive-chain disjointness also remains valid: a nonzero cohomological residue object need not pair with the literal generic positive chain.

The next test is now a genuine comparison: map this rank-five contiguity residue object to Entry 549's resolved rank-five boundary packet and verify the regulator connection and pair-residue incidences.

The executable audit is research/benincasa/marici-gm/src/bin/generic_lower_integer_contiguity_cone.rs.

View source on GitHub