The canonical period-matrix map does not send separation to the Jordan boundary
Date: 24 October 2026 Status: exact no-go for \(Q=\operatorname{Im}\Omega\); noncanonical response maps remain open Evidence: machine-readable gate Certificate: gate verifier
Result
The most economical proposed base-space bridge fails.
A genus-two period matrix belongs to the Siegel upper half-space:
Because \(Y\) is a positive real symmetric \(2\times2\) matrix, it looks exactly like the corpus's rank-two Jordan response carrier \(H_2(\mathbb R)_+\). This makes
the canonical first map to test.
But the separating divisor is
Therefore
The surface separates while the Jordan carrier remains in the interior of its positive cone. The \(\chi_5\) zero at separation is therefore not the response-root sheet meeting at \(\det Q=0\).
Two different meanings of “boundary”
The distinction is structural:
- \(\Omega_{12}=0\) is a complex codimension-one locus where a genus-two surface factorizes into two genus-one components;
- \(\det Q=0\) is a real codimension-one boundary of the positive-semidefinite Jordan cone where a two-channel response loses rank.
For \(Q=\operatorname{Im}\Omega\), the first locus lies over diagonal positive-definite matrices. The second is not contained in the Siegel upper half-space at all.
The Siegel modular action does not repair this. If
then
which is positive definite whenever \(\operatorname{Im}\Omega\) is. No ordinary modular transport of an interior period matrix reaches \(\det Q=0\).
Implication for the Pfaffian bridge
The two sign lines can still be representation-compatible, but their zero sets are not identified by the canonical positive-cone projection. The proposed commuting square
needs additional structure.
A surviving map must do at least one of the following:
- use a derived response matrix rather than \(\operatorname{Im}\Omega\);
- include sewing/defect amplitudes whose vanishing is transferred into a physical response determinant;
- approach a genuine cusp or degeneration where the positive metric itself becomes singular after a declared normalization;
- introduce a detector map that converts the off-diagonal sewing channel into a rank-losing two-port response.
Each option is new physics or new model data, not a consequence of the shared \(2\times2\) matrix shape.
The local sewing-response gate sharpens the alternatives. A phase-blind positive scalar built from the complex sewing normal necessarily begins with \(|\epsilon|^2\), so it squares the seam and loses orientation. A linear signed response requires either two coherent quadratures or a separately sourced phase reference.
What remains useful
The identification \(Y\in H_2(\mathbb R)_+\) is still mathematically exact. It says that genus-two complex structures carry a natural positive rank-two Jordan datum. What fails is the stronger claim that the modular seam zero is automatically the Jordan rank-one boundary.
This is a good example of the programme's central discipline:
the carriers match, but the physically important strata do not.
The next bridge should therefore be tested on strata and zeros before comparing symmetry groups.
Reopening test
Supply an explicit observable response map
with:
- a declared transformation law;
- positivity on the physical domain;
- \(\det Q_{\rm phys}=0\) exactly on the intended separating/defect locus;
- a microscopic or experimental interpretation for both channels;
- no arbitrary multiplication by \(\chi_5\) inserted merely to force the zero.
Without those conditions, the period-matrix/Jordan resemblance remains a carrier analogy rather than a physical seam map.
The later Xi genus-two audit does not alter this no-go. It supplies a new genus-two curve whose branch geometry is constructed from the Riemann \(\Xi\)-function, so its normalized period matrix gives another authentic point of \(\mathfrak H_2\). No reported result sends its arithmetic nonresonance, Kummer specialization or heat motion to \(\det\operatorname{Im}\Omega=0\). The new comparison must preserve marked cycles and strata rather than identify the carriers by genus alone.