Universal Bicone Deformation, Hyperbolic Saddle Stretch, and Twin-Pinch Instability Across Scales up to 𝑋 = 𝑒 20 X=e 20

We formulate and certify the Universal Bicone Deformation and Twin-Pinch Saddle Instability across scales L∈[1.00,10.00]L \in [1.00, 10.00]L∈[1.00,10.00] (X=e20≈485.17×106X = e^{20} \approx 485.17 \times 10^6X=e20≈485.17×106, encompassing 25,617,27225,617,27225,617,272 active prime powers) and dimension N=256N = 256N=256 in global Vasyunin space under a strict Zero-Free Firewall (assert ZERO_INPUT_COUNT == 0). In this Release v1.1.0, all hardcoded zero ordinates and synthetic magic constants are removed: test frequencies ω0\omega_0ω0 and off-critical shifts δ\deltaδ are treated purely as free geometric parameters. We prove and certify: (1) First-principles Archimedean scale frequency ωarch(L)≈20.0969\omega_{\mathrm{arch}}(L) \approx 20.0969ωarch(L)≈20.0969 computed with 0 zeros used; (2) Twin-Pinch Hyperbolic Saddle Stretch: off-axis displacement δ>0\delta > 0δ>0 forces a symmetric quartet {12±δ±iω0}\{\frac{1}{2} \pm \delta \pm i\omega_0\}{21±δ±iω0} inducing negative Gaussian curvature Ksaddle<0\mathcal{K}_{\mathrm{saddle}} < 0Ksaddle<0 and exponential energy stretch Asaddle(δ)∼cosh⁡(2δLeff)\mathcal{A}_{\mathrm{saddle}}(\delta) \sim \cosh(2\delta L_{\mathrm{eff}})Asaddle(δ)∼cosh(2δLeff), tearing positive definiteness at Ltear(δ)≈2.14/δL_{\mathrm{tear}}(\delta) \approx 2.14/\deltaLtear(δ)≈2.14/δ; (3) Critical-Strip Capping: enforcing δ≤0.50\delta \le 0.50δ≤0.50 contracts the permissible corridor as δmax⁡strip(L)≈2.14/L→0\delta_{\max}^{\mathrm{strip}}(L) \approx 2.14/L \to 0δmaxstrip(L)≈2.14/L→0 for L≥5.00L \ge 5.00L≥5.00; and (4) Cutoff-free Vasyunin ring scaling G~pr,pr=p−(r−1)G~p,p\widetilde{G}_{p^r, p^r} = p^{-(r-1)}\widetilde{G}_{p, p}Gpr,pr=p−(r−1)Gp,p holds to 512-bit precision at x=∞x = \inftyx=∞.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-31
DOI
https://doi.org/10.5281/zenodo.23839651
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Universal Bicone Deformation, Hyperbolic Saddle Stretch, and Twin-Pinch Instability Across Scales up to 𝑋 = 𝑒 20 X=e 20

Qamar Munawer
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Universal Bicone Deformation, Hyperbolic Saddle Stretch, and Twin-Pinch Instability Across Scales up to 𝑋 = 𝑒 20 X=e 20

Qamar Munawer
preprint en

Abstract

We formulate and certify the Universal Bicone Deformation and Twin-Pinch Saddle Instability across scales L∈[1.00,10.00]L \in [1.00, 10.00]L∈[1.00,10.00] (X=e20≈485.17×106X = e^{20} \approx 485.17 \times 10^6X=e20≈485.17×106, encompassing 25,617,27225,617,27225,617,272 active prime powers) and dimension N=256N = 256N=256 in global Vasyunin space under a strict Zero-Free Firewall (assert ZERO_INPUT_COUNT == 0). In this Release v1.1.0, all hardcoded zero ordinates and synthetic magic constants are removed: test frequencies ω0\omega_0ω0 and off-critical shifts δ\deltaδ are treated purely as free geometric parameters. We prove and certify: (1) First-principles Archimedean scale frequency ωarch(L)≈20.0969\omega_{\mathrm{arch}}(L) \approx 20.0969ωarch(L)≈20.0969 computed with 0 zeros used; (2) Twin-Pinch Hyperbolic Saddle Stretch: off-axis displacement δ>0\delta > 0δ>0 forces a symmetric quartet {12±δ±iω0}\{\frac{1}{2} \pm \delta \pm i\omega_0\}{21±δ±iω0} inducing negative Gaussian curvature Ksaddle<0\mathcal{K}_{\mathrm{saddle}} < 0Ksaddle<0 and exponential energy stretch Asaddle(δ)∼cosh⁡(2δLeff)\mathcal{A}_{\mathrm{saddle}}(\delta) \sim \cosh(2\delta L_{\mathrm{eff}})Asaddle(δ)∼cosh(2δLeff), tearing positive definiteness at Ltear(δ)≈2.14/δL_{\mathrm{tear}}(\delta) \approx 2.14/\deltaLtear(δ)≈2.14/δ; (3) Critical-Strip Capping: enforcing δ≤0.50\delta \le 0.50δ≤0.50 contracts the permissible corridor as δmax⁡strip(L)≈2.14/L→0\delta_{\max}^{\mathrm{strip}}(L) \approx 2.14/L \to 0δmaxstrip(L)≈2.14/L→0 for L≥5.00L \ge 5.00L≥5.00; and (4) Cutoff-free Vasyunin ring scaling G~pr,pr=p−(r−1)G~p,p\widetilde{G}_{p^r, p^r} = p^{-(r-1)}\widetilde{G}_{p, p}Gpr,pr=p−(r−1)Gp,p holds to 512-bit precision at x=∞x = \inftyx=∞.

Zenodo (CERN European Organization for Nuclear Research)
Brandenburg University of Technology Cottbus-Senftenberg (DE)
Analytic Number Theory Research
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