The Dedekind eta line evaluates the congruence defect

Date: 25 October 2026

Status: genuine scalar metaplectic line constructed; finite \(C_4\) identification remains ruled out; defect-line comparison open

Evidence: machine-readable eta-phase ledger

Certificate: exact Dedekind-sum and phase verifier

Result

The scalar \(C_4\) proposal failed because a genuine metaplectic quarter-turn must have a primitive eighth-root value. That does not eliminate every scalar metaplectic line. The Dedekind eta multiplier supplies the canonical survivor.

For the explicit integral generators

use the standard eta multiplier relative to the square-root automorphy factor. Its values are

Consequently,

The first phase belongs to a lift of the base dicyclic center \(-I\); the second is the genuine metaplectic kernel. They are retained as distinct operations. The congruence relation defect

has the exact phase

This completes the previously open scalar cocycle calculation on the selected relation rail. It does not restore the finite character \(S\mapsto i,\ S^2\mapsto-1\). Instead it replaces that inadmissible identification with the anomaly-shifted chain

1. Exact Dedekind-sum calculation

For \(R=(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})\), one has

The multiplier exponent relative to \((c\tau+d)^{1/2}\) is

Hence \(v_\eta(R)=1\) exactly, not just numerically. For the chosen inverse quarter-turn \(S\), the standard transformation \(\eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau)\) gives \(v_\eta(S)=e^{\pi i/4}\).

In \(\mu_{24}\) exponent notation,

2. What survives and what does not

Three claims must now be separated:

1. Finite claim: \(\mathrm{Dic}_{11}^{ab}=C_4\) has a character \(S\mapsto i\), sending its central \(-I\) to \(-1\). This remains true.
2. No-go: that finite character cannot simultaneously be a genuine character of the one-block metaplectic cover. This remains true.
3. Survivor: the eta line is genuine, but it lifts the base center to the phase \(i\), not the finite sign \(-1\). It also detects the relation rail \(K\) by the same phase \(i\).

Thus a scalar route survives only after accepting that the lift changes the order and phase semantics. The base involution is represented by an order-four phase whose square is the metaplectic sign. Calling both objects “the same \(-1\)” would erase exactly the extension data under investigation.

3. Revised decisive comparison

The scalar mathematical target is concrete:

Restrict the sewn order-23 defect/Pfaffian anomaly line to the selected relation rail and compare its phase on \(K\) with \(i\).

The outcomes are sharp:

• phase \(i\): the canonical eta line supplies a compatible genuine scalar seam;
• phase \(-i\): compatibility differs by the alternate lift convention and must be checked globally;
• phase \(\pm1\) or a non-scalar operator: the eta line does not provide the coupling, and the route moves to a vector-valued Weil kernel;
• no definable phase: the required genus-two defect line has not yet been constructed.

No further finite-group counting can decide this comparison.

The subsequent seam-permutation audit shows why a scalar match is only necessary. Although \(K\) is identity modulo 23, it has order three modulo 2 and permutes the ten even characteristics with cycle type \(3^3\,1\). The full target is therefore the monomial operator \(\rho(K)=D_KP_K\), not the phase \(i\) alone.

Boundary

The eta multiplier is a canonical metaplectic character of the modular carrier. It is not evidence that the Monster defect realizes that character, that the phase is a physical Wick rotation, or that the arithmetic carrier has acquired time, energy or a detector.

Sources and corpus links

• NIST DLMF, Dedekind eta modular transformations
• Kong and Teo, An Elementary Proof of the Transformation Formula for the Dedekind Eta Function
• Rao, The Maslov index on the simply connected covering group and the metaplectic representation
• Integral relation rail and scalar 0 no-go
• Metaplectic congruence-lift gate
• Exhaustion checkpoint
• Relation-defect seam permutation