{
  "id": "oai:papers.qnfo.org:constant-gap-sculpted-hamiltonians-for-factoring-and-np-hard-optimization-analyt",
  "slug": "constant-gap-sculpted-hamiltonians-for-factoring-and-np-hard-optimization-analyt",
  "title": "Constant-Gap Sculpted Hamiltonians for Factoring and NP-Hard Optimization: Analytic Gap Scaling and the Location of the Exponential Cost",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Analog Hamiltonian devices are widely dismissed because encoding unstructured computational problems is believed to demand control precision that grows exponentially with problem size. A proposed escape is to sculpt the Hamiltonian so that every invalid configuration is lifted by a constant energy gap Δ E, making correctness a matter of energetic protection (k_BT ≪ Δ E) rather than of analog precision. Whether such a constant-gap construction genuinely avoids exponential resources has not been settled by a single explicit derivation. We analyze a sculpted factoring Hamiltonian H_fac that combines a logarithmic product constraint with integer-valued penalty terms enforcing integrality and primality. Under the discrete-penalty convention, in which the product penalty |pq-N|² carries unit coefficient, we derive a ground-state energy E₀=0, a first excited state at E₁=1, and a constant gap Δ E=1 independent of the integer N; the corresponding measurement time at the experimentally reported spectroscopic transition rate of 70 s^-1 is T_meas≈1.43×10^-2 s, negligible against the 4,500 h aggregate budget reported for a long-horizon analog design benchmark. Under the alternative smooth-encoding convention, in which only the logarithmic constraint is engineered, we show that the minimal non-zero constraint value scales as 1/N², so that maintaining a thermal-protection gap at 10 mK for a 128-bit integer requires a coefficient α≈1.6×10⁵2 J. The exponential cost of analog factoring therefore does not vanish; it migrates from parameter precision to the physical implementation of arithmetic penalties. This reframes analog computation as a compilation-style engineering problem whose binding constraint is identifiable and testable.",
  "abstract_markdown": "Analog Hamiltonian devices are widely dismissed because encoding unstructured computational problems is believed to demand control precision that grows exponentially with problem size. A proposed escape is to sculpt the Hamiltonian so that every invalid configuration is lifted by a constant energy gap $\\Delta E$, making correctness a matter of energetic protection ($k_{B}T \\ll \\Delta E$) rather than of analog precision. Whether such a constant-gap construction genuinely avoids exponential resources has not been settled by a single explicit derivation. We analyze a sculpted factoring Hamiltonian $H_{\\text{fac}}$ that combines a logarithmic product constraint with integer-valued penalty terms enforcing integrality and primality. Under the discrete-penalty convention, in which the product penalty $|pq-N|^{2}$ carries unit coefficient, we derive a ground-state energy $E_{0}=0$, a first excited state at $E_{1}=1$, and a constant gap $\\Delta E=1$ independent of the integer $N$; the corresponding measurement time at the experimentally reported spectroscopic transition rate of $70\\;\\text{s}^{-1}$ is $T_{\\text{meas}}\\approx1.43\\times10^{-2}\\;\\text{s}$, negligible against the $4{,}500\\;\\text{h}$ aggregate budget reported for a long-horizon analog design benchmark. Under the alternative smooth-encoding convention, in which only the logarithmic constraint is engineered, we show that the minimal non-zero constraint value scales as $1/N^{2}$, so that maintaining a thermal-protection gap at $10\\;\\text{mK}$ for a 128-bit integer requires a coefficient $\\alpha\\approx1.6\\times10^{52}\\;\\text{J}$. The exponential cost of analog factoring therefore does not vanish; it migrates from parameter precision to the physical implementation of arithmetic penalties. This reframes analog computation as a compilation-style engineering problem whose binding constraint is identifiable and testable.",
  "date": "2026-11-03",
  "first_published": "2026-11-03",
  "last_revised": "2026-11-03T00:00:00Z",
  "modified": "2026-11-03T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/constant-gap-sculpted-hamiltonians-for-factoring-and-np-hard-optimization-analyt",
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  "version_url": "https://papers.qnfo.org/papers/constant-gap-sculpted-hamiltonians-for-factoring-and-np-hard-optimization-analyt/v/1.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Constant-Gap Sculpted Hamiltonians for Factoring and NP-Hard Optimization: Analytic Gap Scaling and the Location of the Exponential Cost"
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        "description": "Analog Hamiltonian devices are widely dismissed because encoding unstructured computational problems is believed to demand control precision that grows exponentially with problem size. A proposed escape is to sculpt the Hamiltonian so that every invalid configuration is lifted by a constant energy gap Δ E, making correctness a matter of energetic protection (k_BT ≪ Δ E) rather than of analog precision. Whether such a constant-gap construction genuinely avoids exponential resources has not been settled by a single explicit derivation. We analyze a sculpted factoring Hamiltonian H_fac that combines a logarithmic product constraint with integer-valued penalty terms enforcing integrality and primality. Under the discrete-penalty convention, in which the product penalty |pq-N|² carries unit coefficient, we derive a ground-state energy E₀=0, a first excited state at E₁=1, and a constant gap Δ E=1 independent of the integer N; the corresponding measurement time at the experimentally reported spectroscopic transition rate of 70 s^-1 is T_meas≈1.43×10^-2 s, negligible against the 4,500 h aggregate budget reported for a long-horizon analog design benchmark. Under the alternative smooth-encoding convention, in which only the logarithmic constraint is engineered, we show that the minimal non-zero constraint value scales as 1/N², so that maintaining a thermal-protection gap at 10 mK for a 128-bit integer requires a coefficient α≈1.6×10⁵2 J. The exponential cost of analog factoring therefore does not vanish; it migrates from parameter precision to the physical implementation of arithmetic penalties. This reframes analog computation as a compilation-style engineering problem whose binding constraint is identifiable and testable.",
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