CQS PET
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2026
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| _version_ | 1866901345594769408 |
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| author | Paradise, Chris |
| author_facet | Paradise, Chris |
| contents | <p>This deposit provides the <strong>Conscious Quantum Solver (CQS) v0.1 — Formal Specification</strong> 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</p> <p><span><span><span>F(x)=12x⊤Lαx+λ∑cϕc(x),F(x)=\tfrac12 x^\top L^\alpha x+\lambda\sum_c \phi_c(x),</span><span><span><span>F</span><span>(</span><span>x</span><span>)</span><span>=</span></span><span><span><span><span><span><span><span><span>2</span></span><span><span>1</span></span></span><span></span></span></span></span></span><span><span>x</span><span><span><span><span><span><span>⊤</span></span></span></span></span></span></span><span><span>L</span><span><span><span><span><span><span>α</span></span></span></span></span></span></span><span>x</span><span>+</span></span><span><span>λ</span><span><span><span><span><span><span>c</span></span><span>∑</span></span><span></span></span></span></span><span><span>ϕ</span><span><span><span><span><span><span>c</span></span></span><span></span></span></span></span></span><span>(</span><span>x</span><span>)</span><span>,</span></span></span></span></span></p> <p>where <span><span>LαL^\alpha</span><span><span><span><span>L</span><span><span><span><span><span><span>α</span></span></span></span></span></span></span></span></span></span> is a fractional graph Laplacian operator and <span><span>ϕc\phi_c</span><span><span><span><span>ϕ</span><span><span><span><span><span><span>c</span></span></span><span></span></span></span></span></span></span></span></span> are smooth clause/constraint penalties. Under explicit assumptions (spectral expansion, penalty regularity, and a local smoothness basin), the energy satisfies a local <strong>Polyak–Łojasiewicz inequality</strong>, yielding exponential energy decay under gradient flow within a certified basin.</p> <p>The document further defines a phase-only physical state manifold <span><span>θ∈Tm\theta\in\mathbb{T}^m</span><span><span><span>θ</span><span>∈</span></span><span><span><span>T</span><span><span><span><span><span><span>m</span></span></span></span></span></span></span></span></span></span> and an observation/decoder map <span><span>x=D(θ)x=D(\theta)</span><span><span><span>x</span><span>=</span></span><span><span>D</span><span>(</span><span>θ</span><span>)</span></span></span></span>. Under explicit non-degeneracy (non-saturation) conditions, the pulled-back energy <span><span>E(θ)=F(D(θ))E(\theta)=F(D(\theta))</span><span><span><span>E</span><span>(</span><span>θ</span><span>)</span><span>=</span></span><span><span>F</span><span>(</span><span>D</span><span>(</span><span>θ</span><span>))</span></span></span></span> inherits a PL inequality, establishing a measurable bridge from relaxed optimization geometry to phase-space physical realizations. A scheduled-control condition is specified for <strong>adiabatic tracking</strong> under time-varying penalty weight <span><span>λ(t)\lambda(t)</span><span><span><span>λ</span><span>(</span><span>t</span><span>)</span></span></span></span>.</p> <p>Operational computation is defined not by internal relaxation alone but by <strong>certified readout</strong> before an <strong>inference horizon</strong> <span><span>TIT_I</span><span><span><span><span>T</span><span><span><span><span><span><span>I</span></span></span><span></span></span></span></span></span></span></span></span>, determined by margin growth relative to noise floor. The certification rule is an auditable inequality using decoded margin <span><span>m(t)m(t)</span><span><span><span>m</span><span>(</span><span>t</span><span>)</span></span></span></span>, residual proxies <span><span>r(t)r(t)</span><span><span><span>r</span><span>(</span><span>t</span><span>)</span></span></span></span>, and noise <span><span>σ(t)\sigma(t)</span><span><span><span>σ</span><span>(</span><span>t</span><span>)</span></span></span></span>. <strong>Phase–Echo Theory (PET)</strong> 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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18161894 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | CQS PET Paradise, Chris Conscious Quantum Solver, CQS, spectral optimization, fractional Laplacian, Polyak–Łojasiewicz inequality, local convergence, analog computation, physical optimization, phase-space dynamics, constrained optimization, spectral graph theory, nonconvex optimization, gradient flow, adiabatic tracking, computation certification, inference horizon, observer access, auditability, information-theoretic limits, Phase–Echo Theory Conscious Quantum Solver, spectral optimization, fractional Laplacian, Polyak–Łojasiewicz inequality, analog computation, physical optimization, computation certification, inference horizon, auditability, Phase–Echo Theory <p>This deposit provides the <strong>Conscious Quantum Solver (CQS) v0.1 — Formal Specification</strong> 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</p> <p><span><span><span>F(x)=12x⊤Lαx+λ∑cϕc(x),F(x)=\tfrac12 x^\top L^\alpha x+\lambda\sum_c \phi_c(x),</span><span><span><span>F</span><span>(</span><span>x</span><span>)</span><span>=</span></span><span><span><span><span><span><span><span><span>2</span></span><span><span>1</span></span></span><span></span></span></span></span></span><span><span>x</span><span><span><span><span><span><span>⊤</span></span></span></span></span></span></span><span><span>L</span><span><span><span><span><span><span>α</span></span></span></span></span></span></span><span>x</span><span>+</span></span><span><span>λ</span><span><span><span><span><span><span>c</span></span><span>∑</span></span><span></span></span></span></span><span><span>ϕ</span><span><span><span><span><span><span>c</span></span></span><span></span></span></span></span></span><span>(</span><span>x</span><span>)</span><span>,</span></span></span></span></span></p> <p>where <span><span>LαL^\alpha</span><span><span><span><span>L</span><span><span><span><span><span><span>α</span></span></span></span></span></span></span></span></span></span> is a fractional graph Laplacian operator and <span><span>ϕc\phi_c</span><span><span><span><span>ϕ</span><span><span><span><span><span><span>c</span></span></span><span></span></span></span></span></span></span></span></span> are smooth clause/constraint penalties. Under explicit assumptions (spectral expansion, penalty regularity, and a local smoothness basin), the energy satisfies a local <strong>Polyak–Łojasiewicz inequality</strong>, yielding exponential energy decay under gradient flow within a certified basin.</p> <p>The document further defines a phase-only physical state manifold <span><span>θ∈Tm\theta\in\mathbb{T}^m</span><span><span><span>θ</span><span>∈</span></span><span><span><span>T</span><span><span><span><span><span><span>m</span></span></span></span></span></span></span></span></span></span> and an observation/decoder map <span><span>x=D(θ)x=D(\theta)</span><span><span><span>x</span><span>=</span></span><span><span>D</span><span>(</span><span>θ</span><span>)</span></span></span></span>. Under explicit non-degeneracy (non-saturation) conditions, the pulled-back energy <span><span>E(θ)=F(D(θ))E(\theta)=F(D(\theta))</span><span><span><span>E</span><span>(</span><span>θ</span><span>)</span><span>=</span></span><span><span>F</span><span>(</span><span>D</span><span>(</span><span>θ</span><span>))</span></span></span></span> inherits a PL inequality, establishing a measurable bridge from relaxed optimization geometry to phase-space physical realizations. A scheduled-control condition is specified for <strong>adiabatic tracking</strong> under time-varying penalty weight <span><span>λ(t)\lambda(t)</span><span><span><span>λ</span><span>(</span><span>t</span><span>)</span></span></span></span>.</p> <p>Operational computation is defined not by internal relaxation alone but by <strong>certified readout</strong> before an <strong>inference horizon</strong> <span><span>TIT_I</span><span><span><span><span>T</span><span><span><span><span><span><span>I</span></span></span><span></span></span></span></span></span></span></span></span>, determined by margin growth relative to noise floor. The certification rule is an auditable inequality using decoded margin <span><span>m(t)m(t)</span><span><span><span>m</span><span>(</span><span>t</span><span>)</span></span></span></span>, residual proxies <span><span>r(t)r(t)</span><span><span><span>r</span><span>(</span><span>t</span><span>)</span></span></span></span>, and noise <span><span>σ(t)\sigma(t)</span><span><span><span>σ</span><span>(</span><span>t</span><span>)</span></span></span></span>. <strong>Phase–Echo Theory (PET)</strong> 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.</p> |
| title | CQS PET |
| topic | Conscious Quantum Solver, CQS, spectral optimization, fractional Laplacian, Polyak–Łojasiewicz inequality, local convergence, analog computation, physical optimization, phase-space dynamics, constrained optimization, spectral graph theory, nonconvex optimization, gradient flow, adiabatic tracking, computation certification, inference horizon, observer access, auditability, information-theoretic limits, Phase–Echo Theory Conscious Quantum Solver, spectral optimization, fractional Laplacian, Polyak–Łojasiewicz inequality, analog computation, physical optimization, computation certification, inference horizon, auditability, Phase–Echo Theory |
| url | https://doi.org/10.5281/zenodo.18161894 |