Sub-Principal Symbol of the Effective Operator and Casimir Rigidity of the Central Direction

Overview

The effective operator $L_\mathrm{eff}$ on the admissible Heisenberg graph carries a diagonal co-metric with coefficients $A_H$ (horizontal/spatial) and $A_Z$ (central/temporal). Determining these coefficients is the central problem of the Q5–Q11 programme.

Q8 resolves the $A_Z$ coefficient independently and unconditionally (under Q5a). The sub-principal symbol of $L_\mathrm{eff}$ — the next-to-leading term in the semiclassical expansion — encodes the Casimir value of $\mathfrak{su}(2)$. Casimir rigidity then forces $A_Z = C_{\mathfrak{su}(2)} = 2$.

Crucially, this result is independent of the spatial bridge problem studied in Q7. The central coefficient is determined by the algebraic structure of the representation, not by the interplay between the three horizontal coefficients.

Central result. $A_Z = 2$ unconditionally from sub-principal symbol analysis, independently of Q7.

Core contributions

Sub-principal symbol analysis: the sub-principal symbol of $L_\mathrm{eff}$ is computed systematically. Unlike the principal symbol, which determines the leading-order spectral asymptotics, the sub-principal symbol captures the next-order correction and encodes algebraic data from the representation.
Casimir rigidity: the $\mathfrak{su}(2)$ Casimir operator $C_{\mathfrak{su}(2)}$ has the unique eigenvalue 2 on the spin-$\frac{1}{2}$ representation (the admissible representation). Casimir rigidity states that the sub-principal symbol of $L_\mathrm{eff}$ is constrained to equal this Casimir value.
$A_Z = 2$ unconditionally: combining the sub-principal symbol computation with Casimir rigidity yields $A_Z = 2$ without any additional hypotheses beyond Q5a. This is a fully unconditional result within the Q5a framework.
Independence from Q7: the $A_Z$ determination is logically independent from the spatial bridge problem (Q7) that determines the ratios between $A_{H_1}$, $A_{H_2}$, $A_{H_3}$. Q8 and Q7 address complementary aspects of the same co-metric.
Input for temporal co-metric: $A_Z = 2$ is the temporal co-metric coefficient in the Lorentzian metric $\mathrm{diag}(-A_Z, A_{H_1}, A_{H_2}, A_{H_3})$. Combined with the spatial coefficients from Q10–Q11, this gives the complete metric.

Casimir rigidity and spectral constraints

The concept of Casimir rigidity introduced in Q8 is a new spectral constraint mechanism. The sub-principal symbol of a differential operator on a representation space is not free — it is constrained by the algebraic structure of the representation.

In the case of $L_\mathrm{eff}$ on the Weil representation of the Heisenberg group, the $\mathfrak{su}(2)$ Casimir operator appears in the sub-principal symbol. The eigenvalue of this Casimir on the admissible representation is $C_{\mathfrak{su}(2)} = 2$, and this value is inherited by the co-metric coefficient $A_Z$.

This mechanism provides a new way to fix metric coefficients from representation theory, complementing the BFS stratification approach of Q5b. The two methods agree on $A_Z = 2$.

New mechanism. Casimir rigidity — the determination of metric coefficients from Casimir eigenvalues via sub-principal symbols — is a technique specific to the Cosmochrony programme and may have broader applicability.

Relation to the Cosmochrony programme

Q8 is part of the co-metric coefficient determination programme spanning Q7–Q11:

Q7: spatial bridge problem — ratios between $A_{H_1}$, $A_{H_2}$, $A_{H_3}$.
Q8: central coefficient — $A_Z = 2$ unconditionally via Casimir rigidity.
Q9: proves [H-lift], establishing $A_H > 0$ unconditionally.
Q10–Q11: complete the spatial coefficient determination and identify $A_\tau$.

The result $A_Z = 2$ feeds directly into Q9, Q10, and Q11, where it is used as an input for the complete Lorentzian metric. The value $A_Z = 2$ is also consistent with the Lorentzian metric $\mathrm{diag}(-2, 2, 2, 2)$ derived in Q5b.

Open directions

Higher Casimir operators: whether higher Casimir operators of $\mathfrak{su}(2)$ or other Lie algebras constrain the spatial coefficients via similar Casimir rigidity arguments is an open question.
Sub-principal symbol beyond spin-1/2: the analysis of the sub-principal symbol for higher-spin admissible representations may yield additional constraints.
Gauge field Casimir rigidity: whether the gauge field coefficients in $A_{g,\mathcal{A}}$ (Q12) satisfy analogous Casimir rigidity constraints remains to be investigated.

References

Jérôme Beau. Sub-Principal Symbol of the Effective Operator and Casimir Rigidity of the Central Direction, 2026. doi:10.5281/zenodo.19879909