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Also rerun original RK4 at the same RK4 resolution as a control. This is a finite-grid selection, not global optimization.","mr":"आधी मोजलेल्या १२ गणनांपैकी अचूक अंतिम मर्यादा 1/200 किंवा कमी असणाऱ्या गणना निवडल्या. प्रत्येक पद्धतीसाठी गणना आणि प्रमाणपत्र यांच्या मध्यम वेळांची बेरीज सर्वात कमी असलेली गणना घेतली. मूळ RK4ही त्याच RK4 पायरीवर नियंत्रण म्हणून पुन्हा चालवला. ही मोजलेल्या संचापुरती निवड आहे."},"measurement":{"en":"Six new cold full-pipeline trials per method in balanced cyclic order, all in one Python process: method setup, endpoint generation, compressed input write, full residual certificate read/reconstruction/calculation/report writes. Interpreter/import startup excluded.","mr":"प्रत्येक धावेची नवी पूर्वतयारी, endpoint गणना, compressed input लिहिणे आणि पूर्ण प्रमाणपत्रासाठी वाचन, पुनर्रचना, गणना व अहवाल लिहिणे यांचा वेळ धरला आहे. Python सुरू होण्याचा व importचा वेळ वगळला आहे. तीन पद्धती प्रत्येकी सहा वेळा, समान वारंवारतेने क्रम बदलून चालवल्या."},"cases":[{"id":"w2_dt256","method":"w2","stepDenominator":256,"label":"Paper formula 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implementation timings on one short-time W test. Optimized RK4 uses real-odd coefficients and interior pairs during evolution. Full residual certification is unchanged. This does not establish general superiority, true-error ordering or the paper global theorem.","mr":"वेळा या संगणकावरील prototypeच्या आहेत. तिन्ही गणना एकच प्रमाणित लक्ष्य पूर्ण करतात. यावरून प्रत्यक्ष त्रुटीचा क्रम, नवीन शोध, पूर्ण theoremची वैधता किंवा औद्योगिक श्रेष्ठता ठरत नाही."}},"rk4Label":{"en":"Optimized RK4","mr":"सुधारित RK4"},"history":{"en":"The earlier 29.5% figure compared w₂ with the original Python RK4 implementation. This page now uses a faster interior-pair, real-odd RK4 baseline and freshly measured timings. The original result remains available as a dated experimental snapshot.","mr":"पूर्वीचा २९.५% आकडा मूळ Python RK4 implementationशी केलेल्या तुलनेचा होता. आता आवश्यक mode-जोड्या आणि real-odd गणना वापरणारा सुधारित RK4 व नव्याने मोजलेले वेळ वापरले आहेत. आधीचा निकाल स्वतंत्र नोंद म्हणून उपलब्ध आहे.","url":"comparison-original-kit.zip"},"conclusion":{"en":"At the common certified tolerance 0.005, the paper’s w₂ reference (Δt=1/256) completed setup, generation and certification in 0.808 s median, versus 0.989 s for optimized RK4 (Δt=1/128): about 18.4% less measured time. Six full trials were measured for each of w₂, optimized RK4 and the original RK4 in the same batch. The observed timing ranges overlap. At identical step size, RK4 gives the tighter bound. This is a local timing observation for one short-time test, not a general speed guarantee.","mr":"एकच प्रमाणित त्रुटी-लक्ष्य ०.००५ घेतल्यावर, पेपरमधील w₂ (Δt=1/256) साठी तयारी, गणना व प्रमाणपत्र मिळून मध्यम वेळ 0.808 सेकंद आला; सुधारित RK4 (Δt=1/128) साठी 0.989 सेकंद. या चाचणीत सुमारे 18.4% कमी वेळ लागला. तिन्ही पद्धतींच्या सहा पूर्ण चाचण्या एकाच सत्रात केल्या. वेळांच्या किमान–कमाल पट्ट्यांत काही आच्छादन आहे. समान वेळेच्या पावलावर RK4ची सीमा अधिक घट्ट होती. हा या मर्यादित प्रयोगातील मध्यम वेळेचा निकाल आहे; सर्वत्र वेगाचा फायदा मिळेल अशी हमी नाही."},"timingNote":{"en":"Measured on the same shared Python server, not your browser. Nine generation trials, three certificate trials, seven setup trials; six directly measured cold end-to-end trials per method at target 0.005. All individual samples are downloadable. End-to-end excludes Python import/startup time.","mr":"वेळा त्याच सामायिक Python serverवर मोजल्या आहेत; तुमच्या browserवर नाहीत. गणनेच्या ९, प्रमाणपत्राच्या ३, तयारीच्या ७ आणि समान ०.००५ लक्ष्यावरील एकूण वेळेच्या प्रत्येक पद्धतीसाठी ६ चाचण्या केल्या. सर्व कच्चे आकडे डाउनलोडमध्ये आहेत. Python सुरू होण्याचा व importचा वेळ धरलेला नाही."},"scope":{"en":"All bounds belong to exact rational projected piecewise-linear reconstructions, checked against the full PDE residual. They are normalized spatial L² bounds, not pointwise errors, actual measured errors, percentages, or the preprint’s global X-norm convergence claim. Independent here means a separately implemented numerical consistency check; it is not external peer review.","mr":"सर्व सीमा अचूक परिमेय, divergence-free, वेळेनुसार तुकड्यांत रेषीय संदर्भ गणनेसाठी आहेत; पूर्ण PDE residual तपासला आहे. त्या normalized spatial L² सीमा आहेत—प्रत्येक बिंदूची त्रुटी, प्रत्यक्ष खरी त्रुटी, टक्केवारी किंवा पेपरचा global X-norm निष्कर्ष नाहीत. स्वतंत्र तपासणी म्हणजे वेगळ्या codeने केलेली numerical पडताळणी; बाह्य peer review 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