DRSN XX: Drifted Spectral Applications II. Amplitudes, Combinatorial Superstrings and Computational Complexity (De Rerum Spectrale Natura, Report XX, Version 1.0)

DSRN XX: DRIFTED SPECTRAL APPLICATIONS II Amplitudes, Combinatorial Superstrings and Computational Complexity De Rerum Spectrale Natura series REPORT XX (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro October 2025 •Drifted spectral formulation of string scattering amplitudes. •Veneziano bootstrap and Regge behaviour from operator geometry. •Combinatorial superstrings and masked BWT structures. •Spectral reformulation of NP and NP-complete problems. •Unified operator-theoretic picture of amplitudes, strings and complexity. Drifted Spectral Geometry: String Amplitudes, Combinatorial Superstrings and Computational Complexity J. Pinho-da-Cruz 1 1 Department of Mechanical Engineering, University of Aveiro We develop the second set of applications of the Drifted Spectral Universe, focusing on string scattering amplitudes, combinatorial superstrings and computational complexity. Classical results such as the Veneziano amplitude, Regge behaviour and bootstrap consistency are reformulated in terms of drifted spectral operators acting on suitable Hilbert spaces. We further introduce a spectral framework for combinatorial superstring problems and show how NP and NP-complete structures admit a unified operator-theoretic description under spectral drift. These results complement the physical applications presented in DSRN XIX and complete the Applications block of the De Rerum Spectrale Natura collection. Keywords: Spectral Action; Drift Geometry; String Scattering Amplitudes; Veneziano Bootstrap; Combinatorial Superstrings; Burrows–Wheeler Transform; Computational Complexity; NPComplete Problems; Operator Theory. CONTENTS I. Introduction 3 II. Drifted Spectral Amplitudes and the Veneziano Bootstrap 4 A. Classical Veneziano amplitude and duality 4 B. Spectral formulation of amplitudes 5 C. Drift and Regge behaviour 5 D. Bootstrap as spectral consistency 6 E. Summary 6 III. Drifted Geometry of String Amplitudes 6 A. Factorisation as operator decomposition 6 B. Worldsheet–target reinterpretation 7 C. Crossing symmetry from spectral analyticity 7 D. Drifted geometry versus classical geometry 7 E. Summary 7 3 IV. Combinatorial Superstrings and Discrete Spectral Operators 8 A. From continuous strings to combinatorial superstrings 8 B. Discrete Hilbert spaces and overlap operators 8 C. Drifted discrete operators 8 D. Masked BWT and spectral ordering 9 E. Summary 9 V. Spectral Reformulation of NP and NP-Complete Problems 9 A. Complexity classes and decision problems 9 B. Decision problems as spectral constraints 10 C. Drift, optimisation and hardness 10 D. NP-completeness as spectral obstruction 10 E. Summary 11 VI. Unified Operator-Theoretic Picture 11 VII. Conclusions 12 References 12 I. INTRODUCTION The Drifted Spectral Universe developed in DSRN I–III establishes a unified operator-theoretic framework in which geometry, amplitudes and discrete structures arise from a controlled similarity deformation of Dirac-type operators, Ds=esϕ D e−sϕ, preserving the principal symbol and ellipticity while modifying lower-order terms. The associated spectral action generates universal analytic structures and drift-induced operator flows that have been shown to govern geometry, cosmology and field dynamics [1–3]. In DSRN XIX, the drifted framework was applied to physical cosmology, cosmic (super)strings, gravitational-wave backgrounds and string-inspired phenomenology. The present report, DSRN XX, constitutes Applications II and shifts the focus to a different but structurally related domain: string scattering amplitudes, combinatorial superstrings and computational complexity. While these topics 4 are traditionally treated as separate subjects, we show that they admit a common formulation in terms of drifted spectral operators. The first part of this report reformulates classical string scattering amplitudes in spectral terms. The Veneziano amplitude, Regge behaviour and bootstrap consistency are interpreted as consequences of operator geometry and spectral factorisation, rather than as purely worldsheet constructions [ 6 , 7 ]. In this perspective, duality and pole structure emerge naturally from spectral properties of drifted operators acting on appropriate Hilbert spaces. The second part introduces combinatorial superstrings, including shortest superstring problems and masked Burrows–Wheeler transform (BWT) structures. These discrete objects, central in information theory and bioinformatics, are embedded into a spectral framework where overlaps, concatenations and masks are represented by bounded operators on discrete Hilbert spaces. This operator viewpoint reveals deep analogies between string amplitudes and combinatorial optimisation. The final part connects these constructions to computational complexity. Classical NP and NPcomplete problems, including shortest superstring and related decision problems, are reformulated in spectral terms. Drifted operators provide a unified language in which hardness, approximation and structural constraints can be analysed without reliance on problem-specific combinatorics, following the standard complexity-theoretic classification [8]. The unifying theme throughout is that amplitudes, combinatorial strings and complexity classes are not disparate topics, but different manifestations of a single operator-theoretic structure under spectral drift. Technical details and auxiliary constructions are deferred to later sections and appendices. a. Canonical dependency. This report develops applications of the Drifted Spectral Universe based on the canonical framework fixed in DSRN XVII and the synthesis presented in DSRN XVIII. No new foundational operator-theoretic definitions are introduced. This is Report XX of the De Rerum Spectrale Natura collection. II. DRIFTED SPECTRAL AMPLITUDES AND THE VENEZIANO BOOTSTRAP A. Classical Veneziano amplitude and duality The Veneziano amplitude, A(s, t) = Γ(−α′s)Γ(−α′t) Γ(−α′(s+t)) , 5 is the prototypical tree-level open-string scattering amplitude, exhibiting crossing symmetry, Regge behaviour and channel duality [ 5 – 7 ]. Its pole structure encodes an infinite tower of resonances, while factorisation ensures consistency with string unitarity. Traditionally, these properties are derived from worldsheet conformal field theory. Here we reformulate them in purely operator-theoretic terms. B. Spectral formulation of amplitudes Let H be a Hilbert space carrying a representation of the kinematic data of the scattering process, and let D be a self-adjoint operator whose spectrum organises the resonance structure. We introduce the drifted operator Ds=esϕ D e−sϕ, where ϕ is a bounded generator. The scattering amplitude is encoded in a spectral trace or resolvent functional of Ds, schematically, A(s, t)∼Tr(f(Ds;s, t)) , for a suitable analytic function f. In this formulation: •poles of the amplitude correspond to discrete spectral values of Ds, •residues arise from spectral projections, •factorisation follows from the spectral decomposition of Ds. C. Drift and Regge behaviour Regge behaviour emerges from controlled drift in the complex spectral plane. For large s at fixed t, the amplitude exhibits A(s, t)∼sα(t), where the Regge trajectory α ( t )is determined by the asymptotic spectral distribution of Ds . Drift modifies the effective spectral density without altering the principal symbol, yielding Regge growth from operator geometry rather than explicit worldsheet arguments. 6 a. Remark. The drift flow discussed throughout this report is an exact similarity flow of operators (Kato type-(A)), not a coupling-space renormalisation group flow. D. Bootstrap as spectral consistency The bootstrap conditions—crossing symmetry, factorisation and analyticity—are reinterpreted as spectral consistency requirements: (i) self-adjointness and analyticity of Dsensure crossing symmetry, (ii) spectral projectors guarantee factorisation in all channels, (iii) bounded drift preserves analyticity and avoids spurious singularities. Thus the Veneziano bootstrap is recast as a statement about the admissible class of drifted spectral operators. E. Summary In the drifted spectral framework, classical string amplitudes are no longer primitive objects but manifestations of operator geometry. Duality, Regge behaviour and bootstrap consistency arise from spectral properties of Ds , providing a unifying viewpoint that prepares the ground for discrete and computational generalisations developed in subsequent sections. III. DRIFTED GEOMETRY OF STRING AMPLITUDES A. Factorisation as operator decomposition A central property of string scattering amplitudes is factorisation: residues at poles decompose into products of lower-point amplitudes. In the spectral framework this property is reinterpreted as operator decomposition. Let Ds be the drifted spectral operator encoding the amplitude data. Near a simple pole λn of the spectrum, the resolvent admits the expansion (Ds−λ)−1∼Pn λn−λ+regular, where Pn is the spectral projector onto the eigenspace associated with λn . Factorisation of amplitudes corresponds precisely to the insertion of such projectors between external states. 7 Thus, factorisation is not an additional assumption but a direct consequence of spectral theory. B. Worldsheet–target reinterpretation In conventional string theory, amplitudes are computed via integrals over worldsheet moduli spaces. In the drifted spectral picture, this rôle is replaced by spectral integration over operator data. Worldsheet moduli correspond to continuous spectral parameters, while target-space kinematics is encoded in the choice of representation of Ds on the Hilbert space H . Drift acts by modifying lower-order operator terms without changing the principal symbol, ensuring consistency with ultraviolet behaviour. This provides a clean operator-theoretic reinterpretation of the worldsheet–target correspondence. C. Crossing symmetry from spectral analyticity Crossing symmetry of amplitudes follows from analyticity properties of the spectral trace functionals associated with Ds . Because drift preserves analyticity and self-adjointness, continuation between channels corresponds to analytic continuation in the spectral plane. Hence crossing symmetry is encoded in the analytic structure of operator-valued functions rather than imposed as a separate condition. D. Drifted geometry versus classical geometry Classical geometry enters string amplitudes through target-space metrics and background fields. In the drifted spectral framework, geometry is encoded operatorially: curvature, fluxes and background data appear as lower-order terms in Ds. Drift modifies these terms while preserving the principal symbol, providing a controlled deformation of the geometric background that leaves the ultraviolet structure intact. This parallels the rôle of background deformations in string perturbation theory, now expressed in spectral language. E. Summary The geometry underlying string amplitudes admits a natural reformulation in terms of drifted spectral operators. Factorisation, worldsheet–target correspondence and crossing symmetry arise 8 from operator decomposition and analyticity. This geometric reinterpretation prepares the transition to discrete and combinatorial settings developed in the next section. IV. COMBINATORIAL SUPERSTRINGS AND DISCRETE SPECTRAL OPERATORS A. From continuous strings to combinatorial superstrings Combinatorial superstrings arise in problems where one seeks a shortest string containing a given family of substrings as factors. Classical instances include the Shortest Superstring Problem, masked superstrings and variants related to the Burrows–Wheeler transform (BWT). These problems play a central role in information theory, data compression and computational biology. In contrast with continuous string amplitudes, combinatorial superstrings are discrete objects. Nevertheless, they admit a natural operator-theoretic representation once one replaces continuous Hilbert spaces by finite or countable discrete Hilbert spaces. B. Discrete Hilbert spaces and overlap operators Let Hdisc be the Hilbert space spanned by basis vectors {|wi⟩} associated with a finite set of words {wi}. Define the overlap operator Oby O |wi⟩=X j ωij |wj⟩, where ωij measures the overlap between the suffix of wiand the prefix of wj. Concatenation of words corresponds to operator composition, while maximal overlap corresponds to selecting dominant spectral components of O . Thus, combinatorial superstring construction is reformulated as a spectral optimisation problem. C. Drifted discrete operators We introduce a drifted discrete operator Os=esϕ Oe−sϕ, where ϕ is a bounded diagonal operator encoding word weights or masks. Drift modifies overlap weights without changing the combinatorial support of the operator. In this setting: 9 •overlaps are encoded spectrally, •masks and constraints are implemented via ϕ, •optimisation corresponds to spectral selection under drift. This mirrors the rôle of drift in continuous spectral geometry, now applied to purely discrete structures. D. Masked BWT and spectral ordering Masked Burrows–Wheeler transforms can be interpreted as reorderings of words according to cyclic shifts subject to constraints. Spectrally, such reorderings correspond to diagonalisation or partial diagonalisation of overlap operators with drift-induced weights. Thus BWT-based constructions admit a unified description as spectral flows on Hdisc , controlled by drift parameters encoding masks and priorities. E. Summary Combinatorial superstrings, masked BWT structures and related optimisation problems admit a clean reformulation in terms of discrete spectral operators. Drift provides a unifying mechanism to encode overlaps, constraints and ordering, establishing a direct bridge between continuous string amplitudes and discrete combinatorial strings. This prepares the transition to computational complexity, where hardness and optimality are analysed in spectral terms. V. SPECTRAL REFORMULATION OF NP AND NP-COMPLETE PROBLEMS A. Complexity classes and decision problems Computational complexity theory classifies decision problems according to the resources required to solve them. The class NP consists of problems whose solutions can be verified in polynomial time, while NP-complete problems are those that are both in NP and at least as hard as any other problem in NP [8].