{
  "id": "oai:papers.qnfo.org:when-does-spectral-data-constructively-determine-arithmetic-sequences",
  "slug": "when-does-spectral-data-constructively-determine-arithmetic-sequences",
  "title": "When Does Spectral Data Constructively Determine Arithmetic Sequences?",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Cryptographic key generation depends on primality testing, and every widely deployed test is probabilistic, leaving a small but nonzero error probability in each certified RSA prime. A deterministic alternative driven by global spectral data—the non-trivial zeros of the Riemann zeta function—has been conjectured but never ruled out or established. The specific gap is that the explicit formula shows the zeros encode the prime-counting function π(x) distributionally, yet no quantitative bound exists on how much spectral data a pointwise, integer-exact reconstruction requires. We formalize constructive emergence as algorithmic recovery of sequence terms from truncated spectral data, and derive the height T of zeros needed to evaluate π(x) with error at most 1/2 from the truncated explicit formula under a termwise error model: T≥ 2 x¹/2log²x. For x=10⁶ this gives T_min≈ 3.82× 10⁵, i.e. N(T_min)≈ 6.69× 10⁵ zeros—exponential in log x, hence super-polynomial in the input size log x. A worked Bernoulli power-sum identity and the spectral-gap constants λ₁^prime≥ 0.17157288 versus λ₁^comp≤ 0.16666667 (gap δ≈ 0.00490621) illustrate the framework. By contrast, Pisot-type recurrences such as the Fibonacci sequence admit exact reconstruction from finitely many spectral parameters. The result rules out, at current error-model strength, a polynomial-size spectral replacement for probabilistic primality testing.",
  "abstract_markdown": "Cryptographic key generation depends on primality testing, and every widely deployed test is probabilistic, leaving a small but nonzero error probability in each certified RSA prime. A deterministic alternative driven by global spectral data—the non-trivial zeros of the Riemann zeta function—has been conjectured but never ruled out or established. The specific gap is that the explicit formula shows the zeros encode the prime-counting function $\\pi(x)$ distributionally, yet no quantitative bound exists on how much spectral data a pointwise, integer-exact reconstruction requires. We formalize constructive emergence as algorithmic recovery of sequence terms from truncated spectral data, and derive the height $T$ of zeros needed to evaluate $\\pi(x)$ with error at most $\\tfrac12$ from the truncated explicit formula under a termwise error model: $T\\ge 2\\,x^{1/2}\\log^{2}x$. For $x=10^{6}$ this gives $T_{\\min}\\approx 3.82\\times 10^{5}$, i.e. $N(T_{\\min})\\approx 6.69\\times 10^{5}$ zeros—exponential in $\\log x$, hence super-polynomial in the input size $\\log x$. A worked Bernoulli power-sum identity and the spectral-gap constants $\\lambda_{1}^{\\text{prime}}\\ge 0.17157288$ versus $\\lambda_{1}^{\\text{comp}}\\le 0.16666667$ (gap $\\delta\\approx 0.00490621$) illustrate the framework. By contrast, Pisot-type recurrences such as the Fibonacci sequence admit exact reconstruction from finitely many spectral parameters. The result rules out, at current error-model strength, a polynomial-size spectral replacement for probabilistic primality testing.",
  "date": "2026-11-02",
  "first_published": "2026-11-02",
  "last_revised": "2026-11-02T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/when-does-spectral-data-constructively-determine-arithmetic-sequences",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "When Does Spectral Data Constructively Determine Arithmetic Sequences?"
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        "description": "Cryptographic key generation depends on primality testing, and every widely deployed test is probabilistic, leaving a small but nonzero error probability in each certified RSA prime. A deterministic alternative driven by global spectral data—the non-trivial zeros of the Riemann zeta function—has been conjectured but never ruled out or established. The specific gap is that the explicit formula shows the zeros encode the prime-counting function π(x) distributionally, yet no quantitative bound exists on how much spectral data a pointwise, integer-exact reconstruction requires. We formalize constructive emergence as algorithmic recovery of sequence terms from truncated spectral data, and derive the height T of zeros needed to evaluate π(x) with error at most 1/2 from the truncated explicit formula under a termwise error model: T≥ 2 x¹/2log²x. For x=10⁶ this gives T_min≈ 3.82× 10⁵, i.e. N(T_min)≈ 6.69× 10⁵ zeros—exponential in log x, hence super-polynomial in the input size log x. A worked Bernoulli power-sum identity and the spectral-gap constants λ₁^prime≥ 0.17157288 versus λ₁^comp≤ 0.16666667 (gap δ≈ 0.00490621) illustrate the framework. By contrast, Pisot-type recurrences such as the Fibonacci sequence admit exact reconstruction from finitely many spectral parameters. The result rules out, at current error-model strength, a polynomial-size spectral replacement for probabilistic primality testing.",
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