DRSN XIII: Spectral Action, Asymptotic Expansion, and Effective Truncations (De Rerum Spectrale Natura, Report XIII, Version 1.0)

DSRN XIII: SPECTRAL ACTION, ASYMPTOTIC EXPANSION, AND EFFECTIVE TRUNCATIONS De Rerum Spectrale Natura series REPORT XIII (Version 1.0) J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 • Defines the spectral action as a trace functional of spectral data and proves rigidity under bounded similarity drift. • Derives the large-cutoff asymptotic expansion via heat-kernel methods and identifies the Seeley–DeWitt coefficients as spectrally rigid inputs. • Formalises precisely how effective actions arise only through truncation and cutoff dependence, in accordance with the DSRN Separation Principle. •Clarifies the logical status of spectral-action-based models: exact invariance versus effective truncations. •No claim is made that the spectral action defines fundamental dynamics. DSRN XIII: Spectral Action, Asymptotic Expansion, and Effective Truncations under Drift J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal (Dated: December 14, 2025) We analyse the spectral action functional within the Drifted Spectral Renormalisation Network (DSRN). We recall its definition as a trace functional of the spectrum and prove its rigidity under bounded similarity drift. We then derive its large-cutoff asymptotic expansion via heat-kernel techniques and identify the Seeley–DeWitt coefficients as spectrally rigid inputs. Finally, we formalise the precise sense in which effective actions arise only through truncation and scale dependence, in accordance with the DSRN Separation Principle. Keywords: spectral action; heat-kernel expansion; Seeley–DeWitt coefficients; cutoff dependence; isospectrality; effective field theory; drifted similarity. CONTENTS I. Scope and Relation to Previous Reports 3 II. Analytic Setting 3 III. Definition of the Spectral Action 3 IV. Drift Invariance of the Spectral Action 3 V. Heat-Kernel Representation 4 VI. Asymptotic Expansion at Large Cutoff 4 VII. Effective Actions from Truncation 5 A. Scale Dependence 5 B. Non-Contradiction with Drift Invariance 5 VIII. Summary and Role within the DSRN 5 ∗jp[email protected] 3 A. Bibliographic Notes 6 References 6 I. SCOPE AND RELATION TO PREVIOUS REPORTS This report develops the spectral action layer of the DSRN. It relies on: •the drifted similarity framework and separation principle fixed in DSRN XI; •the heat-kernel invariance results established in DSRN XII. Non-claims. We make no claim that the spectral action defines fundamental dynamics. Any interpretation as a physical action is effective and scale-dependent by construction. II. ANALYTIC SETTING Let H≥ 0be a self-adjoint operator on a separable Hilbert space H , assumed to be heat-trace class (cf. DSRN XII). Let f:R+→Rbe a smooth, rapidly decaying test function. III. DEFINITION OF THE SPECTRAL ACTION Definition 1 (Spectral action functional).Given a cutoff scale Λ>0, define the spectral action SΛ(H) := Trf(H/Λ), whenever the trace exists. Remark 2.If Hhas discrete spectrum {λn}with finite multiplicities, then SΛ(H) = X n f(λn/Λ), making explicit that SΛdepends only on spectral data. IV. DRIFT INVARIANCE OF THE SPECTRAL ACTION Theorem 3 (Drift rigidity of the spectral action).Let Hs = U−1 sHUs be a bounded drifted similarity family. Assume Hand Hsare such that SΛ(H)and SΛ(Hs)are well-defined. Then SΛ(Hs)=SΛ(H)for all sand Λ>0. 4 Proof. By bounded similarity, Spec ( Hs ) = Spec ( H )(DSRN XI). Since SΛ is defined purely from spectral data, the equality follows immediately. V. HEAT-KERNEL REPRESENTATION Assume fadmits a Laplace transform representation f(x) = Z∞ 0 ˜ f(t)e−tx dt, with ˜ frapidly decaying. Then SΛ(H) = Z∞ 0 ˜ f(t) Tr(e−tH/Λ)dt. Remark 4.This representation makes explicit the link between the spectral action and the heat trace analysed in DSRN XII. VI. ASYMPTOTIC EXPANSION AT LARGE CUTOFF Let H be a Laplace-type operator on a compact d -dimensional manifold. As Λ → ∞ , the spectral action admits an asymptotic expansion SΛ(H)∼X n≥0 Fd−nΛd−nan(H), where: •an(H)are the Seeley–DeWitt coefficients (DSRN XII), •Fkare moments of the test function f, Fk:= Z∞ 0 f(u)uk/2−1du. Theorem 5 (Spectral rigidity of asymptotic coefficients).For any bounded drifted family Hs satisfying the hypotheses above, all coefficients an ( Hs )coincide with an ( H ). Hence, any s -dependence in SΛ(Hs)can arise only from truncation of the asymptotic expansion. Proof. By DSRN XII, each an is a spectral invariant. By DSRN XI, spectral invariants are rigid under bounded similarity drift. 5 VII. EFFECTIVE ACTIONS FROM TRUNCATION In practice, one replaces the full asymptotic series by a finite truncation SΛ(H)≈ N X n=0 Fd−nΛd−nan(H). Remark 6.This truncation discards higher-order terms and therefore constitutes an explicit loss of spectral information. According to the DSRN Separation Principle, the resulting object is an effective action. A. Scale Dependence The presence of the cutoff Λintroduces scale dependence. Any renormalisation-group interpretation of Λbelongs to the effective layer and does not affect spectral rigidity. B. Non-Contradiction with Drift Invariance Although the exact spectral action is rigid under drift, its truncated approximation may display s - dependent behaviour. This does not contradict Theorem 3, since truncation violates the hypotheses of spectral invariance. VIII. SUMMARY AND ROLE WITHIN THE DSRN The spectral action occupies a precise position in the DSRN hierarchy: •As a trace functional, it is fully determined by spectral data and rigid under drift. •Its asymptotic expansion involves spectrally rigid coefficients. •Any dynamics extracted from a finite truncation is effective and scale-dependent. This clarifies the logical status of spectral-action-based models and prepares the ground for the application to noncommutative geometry and the Standard Model in DSRN XIV–XVI. 6 Appendix A: Bibliographic Notes Standard treatments of the spectral action and its heat-kernel expansion include [ 1 – 3 ]. The presentation here isolates the minimal analytic structure needed for DSRN purposes. [1] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, 2nd ed., CRC Press, 1995. [2] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. [3] M. Eckstein and B. Iochum, Spectral Action in Noncommutative Geometry, Springer, 2018.