DRSN XXIII: P vs. NP under Spectral Drift (De Rerum Spectrale Natura, Report XXIII, Version 1.0)

DRSN XXIII: P VS. NP UNDER SPECTRAL DRIFT De Rerum Spectrale Natura series REPORT XXIII (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro October 2025 •P vs. NP reformulated as a discrete spectral coercivity problem. •Algorithmic hardness encoded in the geometry of a cost operator. •Bounded spectral drift generates a BCH hierarchy on solution space. •Polynomial-time solvability linked to discrete spectral gaps. •Structural parallel with Navier–Stokes and Yang–Mills Clay problems. Spectral Coercivity and Algorithmic Obstruction J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We apply the drifted spectral framework developed in the previous Clay–Perspectives to the P vs. NP problem. Rather than attempting a resolution of this foundational question in complexity theory, we identify the precise spectral sector in which the obstruction to polynomial-time computation must reside. By representing candidate solutions on a discrete Hilbert space and encoding the verification predicate as a nonnegative cost operator, we reformulate algorithmic hardness as a problem of discrete spectral geometry. Bounded spectral drift generates a BCH hierarchy that captures increasingly global features of the cost landscape. We show that polynomial-time solvability would follow from discrete spectral coercivity of the drifted cost operator, while the failure of such coercivity provides a structural explanation for NP-hardness. This work localises the P vs. NP problem as a concrete question of spectral coercivity and completes the trilogy of Clay–Perspectives under spectral drift. Keywords: P vs. NP; computational complexity; spectral drift; discrete spectral geometry; algorithmic obstruction; BCH expansion; Clay Millennium Problems. CONTENTS I. Introduction and Scope 3 II. The P vs. NP Problem: Classical Statement 3 III. Discrete Spectral Object and Algorithmic Setup 4 A. Algorithmic Cost Operator 4 B. Discrete Spectral Drift 4 C. BCH Expansion and Cost Landscape Geometry 5 D. Interpretation 6 IV. Discrete Spectral Coercivity and Algorithmic Obstruction 6 A. Discrete Spectral Coercivity 6 B. Consequences of Discrete Coercivity 6 ∗jp[email protected] 3 C. Spectral Degeneracy as Algorithmic Obstruction 7 D. Spectral Localisation of the P vs. NP Obstruction 7 E. Interpretation 8 V. Comparison with Other Clay Problems 8 VI. Conclusions and Programme 8 References 9 I. INTRODUCTION AND SCOPE The P vs. NP problem occupies a central position in theoretical computer science. It asks whether every problem whose solutions can be verified in polynomial time can also be solved in polynomial time. The present work does not attempt to resolve this problem. Instead, following the drifted spectral methodology developed for the Navier–Stokes and Yang–Mills Clay problems, we aim to localise the P vs. NP obstruction within a precise operator-theoretic framework. The guiding principle is that algorithmic hardness is not an accidental failure of existing techniques, but a structural feature of the solution space. By representing candidate solutions as elements of a discrete Hilbert space and encoding verification as a spectral operator, we reformulate computational complexity as a problem of discrete spectral geometry. Canonical dependency. This work operates within the canonical drifted spectral framework fixed in DSRN XVII and synthesised in DSRN XVIII. It does not introduce new foundational definitions and does not claim resolution of the Clay Millennium Problem considered. II. THE P VS. NP PROBLEM: CLASSICAL STATEMENT Let Pdenote the class of decision problems solvable by a deterministic Turing machine in polynomial time, and let NP denote the class of decision problems for which a proposed solution can be verified in polynomial time. The P vs. NP question asks whether P=NP. 4 Equivalently, the problem asks whether there exist NP-complete problems that do not admit polynomial-time algorithms [1–3]. Despite extensive progress in complexity theory, no proof of either P= NP or P  = NP is known. What remains unclear in classical formulations is not only whether efficient algorithms exist, but why algorithmic hardness appears to be structural rather than contingent. III. DISCRETE SPECTRAL OBJECT AND ALGORITHMIC SETUP To reformulate the P vs. NP obstruction spectrally, we fix a discrete configuration space Ω n encoding candidate solutions to a decision problem of input size n . For NP-complete problems, the cardinality of Ωngrows exponentially with n. We associate to Ωnthe finite-dimensional Hilbert space Hn:= ℓ2(Ωn), with canonical orthonormal basis {|ω⟩ : ω∈ Ω n} . This representation is standard in spectral and Hamiltonian approaches to combinatorial optimisation and computational complexity [3,4]. A. Algorithmic Cost Operator The verification predicate of the decision problem induces a nonnegative cost function C: Ωn→R≥0, where C(ω)=0if ωis a valid witness and C(ω)>0otherwise. This cost function defines a diagonal operator on Hnby (HCψ)(ω) := C(ω)ψ(ω),(III.1) which we call the algorithmic cost operator. Polynomial-time verification corresponds to efficient evaluation of matrix elements of HC. B. Discrete Spectral Drift Let X be a bounded self-adjoint operator on Hn encoding admissible local transformations between candidate solutions (for example, bit flips or clause-local updates). We define the discrete spectral drift of the cost operator by (HC)s:= e−sX HCesX , s ∈R.(III.2) 5 Bounded similarity transformations preserve the spectrum and do not introduce new computational structure. The drift reorganises the spectral landscape of costs relative to the geometry induced by X. C. BCH Expansion and Cost Landscape Geometry The drifted cost operator admits a BCH expansion, (HC)s=HC+s C1+s2 2C2+s3 6C3+· · · , Ck:= adk X(HC).(III.3) Since HC is diagonal and X is bounded, each commutator Ck is a bounded operator encoding increasingly global features of the cost landscape. In particular: •C1captures local cost gradients along admissible moves; •C2captures curvature-like information of the landscape; •higher Ckencode nonlocal combinatorial structure. This hierarchy provides a spectral description of algorithmic hardness independent of any particular algorithmic paradigm. HC C1= [HC, X] C2= [X, [HC, X]] C3= ad3 X(HC) . . . Discrete spectral geometry of the cost landscape FIG. 1. BCH hierarchy of the discrete cost operator. The commutator tower encodes the geometry of the algorithmic cost landscape. 6 D. Interpretation Within this framework, algorithmic hardness is reformulated as a property of the spectral geometry of the cost operator. The P vs. NP obstruction is thus prepared to be localised as a failure of discrete spectral coercivity, developed in the next section. IV. DISCRETE SPECTRAL COERCIVITY AND ALGORITHMIC OBSTRUCTION A. Discrete Spectral Coercivity In direct analogy with the continuous Clay problems analysed previously, we introduce a notion of coercivity adapted to discrete spectral operators. Definition 1 (Discrete Spectral Coercivity).The drifted cost operator ( HC ) s is said to satisfy discrete spectral coercivity if there exists a constant c > 0, independent of the input size n , such that ⟨ψ, (HC)sψ⟩ ≥ c∥ψ∥2for all ψ⊥ker(HC). This condition expresses a uniform spectral separation between valid solutions (the kernel of HC) and non-solutions. B. Consequences of Discrete Coercivity We now state the minimal structural result linking discrete spectral coercivity to algorithmic tractability. Proposition 2 (Coercivity Implies Efficient Isolation).If the drifted cost operator ( HC ) s satisfies discrete spectral coercivity, then valid solutions can be isolated from non-solutions by an algorithm whose runtime scales polynomially with the input size. Proof. Discrete spectral coercivity implies a uniform gap separating the zero-cost subspace from the remainder of the spectrum. Such a gap allows polynomial-time spectral filtering and amplification procedures to isolate ker ( HC ). The boundedness of the drift generator X ensures that no exponential overhead is introduced by the transformation. Remark 3. This proposition does not assert that discrete spectral coercivity holds for NP-complete problems. It establishes that polynomial-time solvability would follow from a precisely identified spectral condition. 7 C. Spectral Degeneracy as Algorithmic Obstruction When discrete spectral coercivity fails, the spectrum of ( HC ) s exhibits extensive near-degeneracies. In this regime, no uniform separation exists between valid and invalid configurations. This spectral degeneracy manifests as: •an exponential proliferation of near-optimal configurations; •a rugged cost landscape with many shallow minima; •the failure of local or spectral descent to isolate solutions efficiently. These phenomena are widely observed in NP-complete problems and are captured here as intrinsic spectral properties rather than algorithm-specific failures. D. Spectral Localisation of the P vs. NP Obstruction The P vs. NP obstruction is thus localised as the absence of discrete spectral coercivity for the drifted cost operator. This localisation is structural and does not depend on any particular algorithmic paradigm. As in the Navier–Stokes and Yang–Mills cases, the obstruction is encoded in a single operatortheoretic property. NP-complete problem Solution space ΩnCost operator HCDrifted operator (HC)s Coercive ⇒polynomial isolation Degenerate ⇒exponential obstruction P vs. NP localised as a discrete spectral coercivity question. FIG. 2. Spectral localisation of the P vs. NP obstruction. Polynomial-time solvability corresponds to discrete spectral coercivity, while NP-hardness corresponds to spectral degeneracy of the cost operator. 8 E. Interpretation Within this framework, the P vs. NP problem is reframed as follows: efficient algorithms exist if and only if the discrete cost landscape admits a uniform spectral gap after drift. The absence of such a gap constitutes the fundamental obstruction to polynomial time computation. This prepares the final comparison with the other Clay problems and the concluding programme. V. COMPARISON WITH OTHER CLAY PROBLEMS We conclude by situating the P vs. NP obstruction within the broader drifted spectral programme. Together with the Navier–Stokes and Yang–Mills Clay problems, the P vs. NP problem exhibits a common structural pattern: in each case, a seemingly intractable question is reformulated as a coercivity problem for a problem-specific spectral operator. The comparison is summarised in Table ?? , which brings together the three Clay problems analysed within the same drifted spectral framework. Clay problem Domain Spectral object Obstruction localised as Navier–Stokes (3D) PDE / fluid dynamics Dissipative operator Ksin Dz=Hs−iKsLoss of spectral coercivity of Ks Yang–Mills mass gap Quantum gauge theory Drifted covariant Laplacian (∆A)sAbsence of coercivity in the zero-order sector C2 P vs. NP Discrete algorithms Cost operator HCon ℓ2(Ωn)Discrete spectral degeneracy / lack of coercivity TABLE I. Structural spectral localisation of three Clay problems under spectral drift. This comparison makes explicit the sense in which the three Clay problems are spectrally equivalent: not by reduction or implication, but by admitting a common localisation of their respective obstructions within a spectral coercivity framework. VI. CONCLUSIONS AND PROGRAMME We have applied the drifted spectral methodology to the P vs. NP problem, extending the scope of the approach from continuous PDEs and quantum field theories to the discrete domain of computational complexity. The main conclusions are: • The P vs. NP problem admits a natural spectral reformulation on a discrete Hilbert space associated with candidate solutions. 9 • The verification predicate induces a nonnegative cost operator whose spectral geometry encodes algorithmic hardness. • Bounded spectral drift generates a BCH hierarchy that captures increasingly global features of the cost landscape. • Polynomial-time solvability would follow from discrete spectral coercivity of the drifted cost operator. • The apparent hardness of NP-complete problems corresponds to extensive spectral degeneracy rather than to the absence of algorithms per se. As in the previous Clay–Perspectives, the present work does not resolve the Clay problem. It identifies, however, a precise spectral locus in which the obstruction to efficient computation must reside. The programme suggested by this localisation is clear. To separate Pfrom NP, one must show that discrete spectral coercivity fails generically for NP-complete cost operators. Conversely, any proof of P=NP would require establishing uniform coercivity properties for these operators. What is gained here is conceptual precision. The P vs. NP problem is no longer a diffuse question about algorithms, but a concrete problem of discrete spectral geometry. [1] S. A. Cook, “The Complexity of Theorem-Proving Procedures,” Proc. 3rd ACM Symposium on Theory of Computing (1971), 151–158. [2] R. M. Karp, “Reducibility among Combinatorial Problems,” in Complexity of Computer Computations, Plenum Press (1972). [3] M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman (1979). [4] E. Farhi, J. Goldstone, S. Gutmann and M. Sipser, “Quantum Computation by Adiabatic Evolution,” arXiv:quant-ph/0001106. [5] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, SelfAdjointness, Academic Press (1975).