Conscious Quantum Solver (CQS) v0.1 — Formal Specification

Conscious Quantum Solver (CQS) v0.1 — Formal Specification is a technical whitepaper that defines an auditable framework for local optimization in structured instance families using fractional spectral regularization and Polyak–Łojasiewicz (PL) geometry. The specification includes explicit pullback conditions from relaxed variable space to a phase-only physical manifold, a scheduled-control (adiabatic tracking) condition, and an operational definition of computation based on certified readout before an inference horizon. Phase–Echo Theory (PET) is included only to formalize observer-access limits and to distinguish internal relaxation from observable computation. All guarantees are local, instance-restricted, and falsifiable; no claims are made regarding worst-case complexity, global optimization, or violations of known physical laws. This deposit provides the Conscious Quantum Solver (CQS) v0.1 — Formal Specification and associated diagrams. CQS is defined as a local optimization framework for restricted, structured instance families, grounded in spectral–topological geometry. The specification models a relaxed assignment space with energy F(x)=12x⊤Lαx+λ∑cϕc(x),F(x)=\tfrac12 x^\top L^\alpha x+\lambda\sum_c \phi_c(x),F(x)=21x⊤Lαx+λc∑ϕc(x), where LαL^\alphaLα is a fractional graph Laplacian operator and ϕc\phi_cϕc are smooth clause/constraint penalties. Under explicit assumptions (spectral expansion, penalty regularity, and a local smoothness basin), the energy satisfies a local Polyak–Łojasiewicz inequality, yielding exponential energy decay under gradient flow within a certified basin. The document further defines a phase-only physical state manifold θ∈Tm\theta\in\mathbb{T}^mθ∈Tm and an observation/decoder map x=D(θ)x=D(\theta)x=D(θ). Under explicit non-degeneracy (non-saturation) conditions, the pulled-back energy E(θ)=F(D(θ))E(\theta)=F(D(\theta))E(θ)=F(D(θ)) inherits a PL inequality, establishing a measurable bridge from relaxed optimization geometry to phase-space physical realizations. A scheduled-control condition is specified for adiabatic tracking under time-varying penalty weight λ(t)\lambda(t)λ(t). Operational computation is defined not by internal relaxation alone but by certified readout before an inference horizon TIT_ITI, determined by margin growth relative to noise floor. The certification rule is an auditable inequality using decoded margin m(t)m(t)m(t), residual proxies r(t)r(t)r(t), and noise σ(t)\sigma(t)σ(t). Phase–Echo Theory (PET) is included only to formalize observer-access limits (recoverability bounds and echo/variance tradeoffs) and to distinguish internal convergence from observable computation. The specification is intentionally conservative: all guarantees are local, instance-restricted, and falsifiable; no worst-case complexity claims, global optimization claims, or nonstandard-physics claims are made.