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

Current Zenodo release: version 1.1 (2026-08-15). Official title: 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) and $A_Z$ (central); the temporal coefficient $A_\tau$ is Q11's. Determining these coefficients is the central problem of the Q5–Q11 programme.

Within the frozen Q5a framework, Q8 obtains $A_Z$ under the unestablished hypothesis [H-F]. 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 $A_Z = C_{\mathfrak{su}(2)} = 2$.

This result is not independent of Q7. Q8's theorem assumes the bridge non-obstruction of Q7–Q9, and the proposition carrying the Casimir to the spatial sector holds under the equivariant bridge of Q7 Conjecture 4.7. Q7 version 2.0 records that bridge as neither confirmed nor refuted and its identification as supplied by no source, so this page's $A_Z$ claim is pending revision.

Scoped result. [H-F], with the bridge non-obstruction premise that neither Q7 nor Q9 establishes, implies $A_Z = 2$ through the sub-principal-symbol analysis. [H-F] is unestablished, and the co-metric reading also requires [H-L].

Quick facts

Author

Jérôme Beau

Work type

Preprint

Focus

Sub-principal symbol, Casimir rigidity, effective operator, Laplacian coefficient

Key result

$A_Z = C_{\mathfrak{su}(2)} = 2$ under [H-F]

Status

Structural within [H-F] and the bridge non-obstruction that neither Q7 nor Q9 establishes; co-metric reading conditional on [H-L] and on the identification Q7 version 2.0 records as supplied by no source; pending revision

Depends on

Q5a, Q5b

Feeds into

Q9, Q10, Q11

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 $C_{\mathfrak{su}(2)}$ acts as $j(j+1)$ on the spin-$j$ module, hence as $2$ on the spin-$1$ module $\mathrm{Sym}^2(V_\rho)$ (it is $\frac{3}{4}$ on the spin-$\frac{1}{2}$ carrier, not $2$). Casimir rigidity states that the sub-principal symbol of $L_\mathrm{eff}$ is constrained to equal this Casimir value.
$A_Z = 2$ under [H-F]: combining the sub-principal symbol computation with Casimir rigidity yields $A_Z = 2$ inside the frozen Q5a hypothesis [H-F], and under the bridge non-obstruction, which the theorem assumes but neither Q7 nor Q9 establishes. Pending revision.
Relation to Q7: the $A_Z$ argument runs through the Q7 bridge, whose non-obstruction Q8's theorem assumes. Q7 version 2.0 determines no coefficient and no ratio: it states the requirements an equivariant bridge would meet, under an identification hypothesis that no source supplies. This page's $A_Z$ claim is pending revision in the same correction cascade.
Conditional co-metric reading: under the independent hypothesis [H-L], $A_Z = 2$ is read as the $z$-direction entry of the co-metric $\mathrm{diag}(-A_\tau, 2, 2, 2)$, the form Q8 itself writes, the temporal coefficient $A_\tau$ being Q11's; this reading is pending revision in the cascade Q7 version 2.0 opens.

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 supplied model for $L_\mathrm{eff}$, built from a finite Heisenberg–Schrödinger carrier and a distinct associated Weil action, the $\mathfrak{su}(2)$ Casimir operator appears in the sub-principal symbol. Its eigenvalue on the spin-$1$ module is $C_{\mathfrak{su}(2)} = 2$, and that value is inherited by the co-metric coefficient $A_Z$. Reading the admissible representation as that module is the identification Q7 version 2.0 records as supplied by no source, so this step is pending revision.

This mechanism provides a new way to fix metric coefficients from representation theory, complementing the BFS stratification approach of Q5b. The two are not independent confirmations: Q5b's Theorem 5.2 puts $0$ in the $z$-sector and imports $A_Z = 2$ from Q8.

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: requirements for an equivariant bridge; version 2.0 determines no coefficient and no ratio.
Q8: under the frozen, unestablished [H-F] and the bridge non-obstruction unestablished by Q7 or Q9, Casimir rigidity gives $A_Z = 2$. Pending revision.
Q9: under [K] and [R], gives a free kinetic limit; it establishes neither [H-lift] nor bridge non-obstruction.
Q10–Q11: assert the remaining spatial coefficient and $A_\tau$; pending revision in the cascade Q7 version 2.0 opens.

Within [H-F], $A_Z = 2$ is used by later coefficient analyses. The reading as the Lorentzian metric $\mathrm{diag}(-2, 2, 2, 2)$ remains conditional on [H-L] and on the identification Q7 version 2.0 records as supplied by no source; the values in that matrix are pending revision.

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