DRSN XI: Foundations and Canonical Methodology of Drifted Spectral Renormalisation (De Rerum Spectrale Natura, Report XI, Version 1.0)

DSRN XI: FOUNDATIONS AND CANONICAL METHODOLOGY OF DRIFTED SPECTRAL RENORMALISATION De Rerum Spectrale Natura series REPORT XI (Version 1.0) J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Canonical definition of drifted similarity families for Dirac-type operators. •Proof of isospectrality under bounded similarity and rigidity of all global spectral invariants. • Explicit Separation Principle: spectral data are rigid, while any dynamics must arise from truncation, projection, or other information-loss operations. •Establishes the non-negotiable methodological base for DSRN XII–XVIII and all subsequent applications. •No physical interpretation, quantisation, or ontological assumptions are introduced at this stage. Foundations and Canonical Methodology of Drifted Spectral Renormalisation J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal (Dated: December 14, 2025) This report establishes the canonical operator–theoretic foundations of drifted spectral renormalisation. We define drift as a bounded similarity deformation of Dirac–type operators and prove that such deformations preserve domains, self–adjointness, spectra, and all global spectral invariants. These results isolate a spectrally rigid operatorial core and formalise the Separation Principle that governs the entire DSRN programme: spectral invariants are exact, while any effective dynamics can arise only after truncation, projection, or other information–loss procedures. This report contains no physical interpretation and serves solely as the methodological foundation for all subsequent developments. Keywords: drift geometry; bounded similarity; Dirac operators; spectral rigidity; operator theory; heat kernel; zeta functions. CONTENTS I. Scope and Non-Claims 3 II. Functional-Analytic Setting 3 III. Drifted Families and Similarity Transformations 4 IV. Isospectrality Under Similarity 4 V. Spectral Invariants 5 VI. Heat Kernel Invariants (Bridge to DSRN XII) 5 VII. Spectral Rigidity versus Effective Dynamics 5 A. Motivation 5 B. Canonical Separation Principle 6 C. Effective Quantities 6 ∗jp[email protected] 3 D. Non-Contradiction with Isospectrality 6 E. Formal Consequence 6 VIII. Reusable Results and Canonical Dependencies 7 A. Canonical Results 7 B. Dependency Map (summary) 7 C. Closure 7 A. Minimal Analytic Hypotheses for Trace-Based Invariants 7 1. Heat trace 8 2. Zeta regularisation 8 Acknowledgement of Sources 8 References 8 I. SCOPE AND NON-CLAIMS This report is foundational and methodological. Its purpose is to fix a precise mathematical language and a strict separation between: (i) spectral invariants—quantities depending only on the spectrum of an operator—and (ii) effective constructs that may vary along a family but necessarily arise from explicit information-loss operations such as truncation, projection, regularisation, or coarse-graining. Non-claims. DSRN XI makes no claim of physical uniqueness, no claim of quantisation, and no claim that a drift parameter should be interpreted as physical time. Any physical interpretation that appears in later reports must cite the methodological constraints established here. II. FUNCTIONAL-ANALYTIC SETTING Let H be a separable Hilbert space. Operators are linear and may be unbounded. We use standard conventions: •An operator H: Dom(H)⊂ H → H is densely defined if Dom(H)is dense. •His closed if its graph is closed in H × H. 4 •When relevant, self-adjointness is understood in the usual Hilbert-space sense. •For λ∈C, the resolvent is (H−λ)−1when it exists as a bounded operator. •Spec(H)denotes the spectrum of H. Whenever traces are used (heat trace, zeta trace), we will state the analytic hypotheses under which such traces are well-defined. Standard sources include heat-kernel and spectral theory references such as [1,2] and semigroup foundations such as [3]. III. DRIFTED FAMILIES AND SIMILARITY TRANSFORMATIONS Let H be a separable Hilbert space and let H : Dom ( H ) ⊂ H → H be a closed, densely defined operator. Definition 1 (Drifted Operator Family).Let {Us}s∈I be a family of invertible operators on H , with Usand U−1 sbounded for each s. The associated drifted family is defined by Hs:= U−1 sHUs,Dom(Hs):=U−1 sDom(H). Remark 2. The drift is purely functional-analytic. No assumption of unitarity is imposed on Us . The parameter sis not interpreted as a time variable unless explicitly stated in later reports. IV. ISOSPECTRALITY UNDER SIMILARITY Lemma 3 (Isospectrality under similarity).Let H be a closed operator on H and let U be a bounded invertible operator with bounded inverse. Define HU:= U−1HU with domain U−1Dom(H). Then Spec(HU) = Spec(H). Proof. Let λ∈C. Then H−λis invertible if and only if HU−λ=U−1(H−λ)U is invertible, with (HU−λ)−1=U−1(H−λ)−1U. Thus λ belongs to the resolvent set of H if and only if it belongs to the resolvent set of HU , hence Spec(HU) = Spec(H). Remark 4. Lemma 3applies equally to point, continuous, and essential spectrum. 5 V. SPECTRAL INVARIANTS Definition 5 (Spectral invariant).A functional I ( H )is called a spectral invariant if I ( H )depends only on Spec ( H )(counted with multiplicities where appropriate) and not on eigenvectors or representation. Examples include spectral measures, eigenvalue counting functions (when defined), and trace functionals built from functional calculus under appropriate hypotheses. Lemma 6 (Rigidity of spectral invariants).Let Hs = U−1 sHUs be a drifted family with Us bounded and invertible. Then any spectral invariant Isatisfies I(Hs)=I(H) for all sfor which I(H)is well-defined. Proof. By Lemma 3, Spec ( Hs ) = Spec ( H ). Since I depends only on the spectrum, I ( Hs ) = I(H). Remark 7 (Trace-based invariants).When I is defined via a trace, additional analytic conditions are required (e.g. trace-class, θ -summability, or suitable regularisation). Lemma 6asserts invariance whenever the quantity is defined. VI. HEAT KERNEL INVARIANTS (BRIDGE TO DSRN XII) Assume H≥ 0is self-adjoint and that e−tH is trace class for t > 0. Then the heat trace Tr ( e−tH ) is a spectral invariant [1,2]. Hence, for any drifted family satisfying the hypotheses of Lemma 3, Tr(e−tHs) = Tr(e−tH ) (t>0). This rigidity underlies the invariance of Seeley–DeWitt coefficients in the heat-kernel asymptotic expansion (developed systematically in DSRN XII). VII. SPECTRAL RIGIDITY VERSUS EFFECTIVE DYNAMICS A. Motivation Sections III–VI show that drifted similarity preserves the full spectrum and all spectral invariants. Yet many constructions of interest produce s -dependent quantities. This is resolved by distinguishing spectral data from effective descriptions. 6 B. Canonical Separation Principle Axiom 8 (DSRN Separation Principle).Let Hs = U−1 sHUs be a drifted family with Us bounded and invertible. Then: 1. Any spectral invariant of Hsis independent of s. 2. Any s -dependent quantity associated with Hs must arise from truncation, projection, regularisation, coarse-graining, or other explicit loss of spectral information. C. Effective Quantities We call E(Hs)an effective quantity if it satisfies at least one of: •it depends on a finite subset of the spectrum; •it depends on a cutoff or scale parameter; •it is defined after projection to a subspace of H; •it is obtained from an asymptotic series truncated at finite order. Such quantities are not spectral invariants and may vary with s without contradicting isospectrality. D. Non-Contradiction with Isospectrality Key point. Isospectrality does not imply dynamical triviality. It implies rigidity only at the level of spectral data. Effective dynamics necessarily live in reduced or approximate descriptions and therefore are compatible with spectral invariance. E. Formal Consequence Let Ibe a spectral invariant and Ean effective quantity. Then d dsI(Hs)=0,d dsE(Hs)= 0 in general. Any claim of s -dependent spectral change must identify explicitly which spectral information has been discarded. 7 VIII. REUSABLE RESULTS AND CANONICAL DEPENDENCIES A. Canonical Results The following are fixed and will not be reproved in subsequent DSRN reports: •Drifted similarity framework (Section III); •Isospectrality under bounded similarity (Lemma 3); •Rigidity of spectral invariants (Lemma 6); •Heat trace invariance under drift (Section VI); •DSRN Separation Principle (Axiom 8). B. Dependency Map (summary) •DSRN XII: uses Lemmas 3–6. •DSRN XIII: uses Sections V–VII. •DSRN XIV–XVI: must respect Axiom 8in all interpretations. •DSRN XVII–XVIII: uses Axiom 8to localise all emergent dynamics. •DSRN XXI–XXIV: use the full methodological layer of DSRN XI. C. Closure With the results above fixed, all subsequent DSRN reports can unambiguously separate what is spectrally invariant from what is effective or emergent. Appendix A: Minimal Analytic Hypotheses for Trace-Based Invariants This appendix records minimal hypotheses under which common trace-based spectral invariants are well-defined and hence covered by Lemma 6. 8 1. Heat trace If H≥ 0is self-adjoint and e−tH is trace class for all t > 0, then Tr ( e−tH )is well-defined and depends only on Spec ( H )[ 1 , 2 ]. In geometric settings (e.g. elliptic operators on compact manifolds), such trace-class properties are standard. 2. Zeta regularisation If H is positive self-adjoint with discrete spectrum accumulating only at infinity, one defines ζH ( s ) = Tr ( H−s )for ℜ ( s )sufficiently large and extends meromorphically under standard conditions [ 1 , 2 ]. Any quantity defined purely from ζH (e.g. regularised determinants) is spectral and hence drift-invariant whenever defined. ACKNOWLEDGEMENT OF SOURCES The general analytic background for heat kernels and spectral invariants follows standard references such as [ 1 , 2 ]. Semigroup foundations are standard [ 3 ]. Noncommutative geometry context is classical [ 4 ], though DSRN XI itself only uses functional-analytic facts and does not rely on NCG-specific axioms. [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] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. [4] A. Connes, Noncommutative Geometry, Academic Press, 1994.