The oriented seam survives as a modular character line
Date: 24 October 2026 Status: exact finite representation and standard Igusa-character completion; physical Leech coupling remains open Evidence: machine-readable result Certificate: oriented seam verifier
Result
Full modular covariance does not require the affine seam to disappear. It requires the seam to be typed correctly.
There are two complementary completions.
Additive completion
Place the ten even theta constants in a ten-component vector. The full modular group permutes their characteristics and supplies the usual half-weight theta multipliers. At the finite \(Sp(4,\mathbb F_2)\) level, the oriented seam is
Its full orbit contains 90 ordered transitions. Those vectors span the nine-dimensional sum-zero subspace of the ten-component permutation carrier. Reversing the ordered seam sends \(s\mapsto-s\).
This is the minimal carrier that remembers which characteristic transition is marked. Restoring the theta multipliers makes it a metaplectic/vector-valued object rather than an ordinary ten-component scalar function.
Its determinant is already informative. The ten-dimensional permutation carrier splits into the invariant all-ones line plus the nine-dimensional sum-zero carrier, so their determinants agree. Direct enumeration of all 720 elements gives 360 with determinant \(+1\) and 360 with determinant \(-1\). The determinant of the seam carrier is therefore the nontrivial sign character of \(Sp(4,\mathbb F_2)\). This is the finite characteristic shadow of the character carried by \(\chi_5\); the analytic theta multipliers still have to be restored before this becomes an equality of modular determinant lines.
Multiplicative completion
The product
is a weight-five Siegel modular form with the nontrivial character of the genus-two modular group. At a separating divisor, the unique odd--odd theta constant vanishes linearly, so \(\chi_5\) also vanishes linearly and changes sign with the oriented normal/Pfaffian line.
Squaring gives
an ordinary weight-ten scalar modular form. This is exactly where the orientation is forgotten: the vanishing order doubles and the sign character disappears.
What has been found
The corpus's affine orientation bit now has a natural global mathematical home:
- the nine-dimensional additive seam representation remembers the ordered transition;
- its determinant is the unique nontrivial finite sign character;
- the one-dimensional \(\chi_5\) character line remembers only its global orientation anomaly;
- \(\chi_{10}\) remembers only the unoriented separating divisor.
This is stronger than “the seam might be a Pfaffian.” It identifies the representation type required for that statement.
Why this is not yet the Leech bridge
The degree-two Leech theta series is a bosonic weight-twelve scalar with trivial character and a nonzero separating limit. It therefore cannot itself be \(\chi_5\), contain \(\chi_5\) as an ordinary scalar factor, or retain its sign without an additional sector.
A non-chiral absolute square such as \(|\chi_5|^2\) is modular invariant, but the sign/phase cancels. That can describe an unoriented probability weight, not the oriented seam datum the corpus is trying to preserve.
The needed new input is consequently specific:
a fermionic, Pfaffian, anomaly or topological-defect sector whose modular character is \(\chi_5\)'s sign line, together with a source-derived coupling to the Leech pair/response carrier.
This is a much narrower frontier than “find a modular completion.”
Experimental meaning
If realized in a device or physical model, the two carriers predict different records:
- an additive vector-valued seam changes which boundary/spin-structure channel is populated;
- a character-line seam changes a coherent sign or phase and is visible only through interference;
- squaring or incoherent averaging removes that distinction.
The correct detector therefore needs phase-sensitive comparison between seam orientations. A power spectrum alone will see the \(\chi_{10}\)-like square and lose the result.