Encyclopedia/All topics/Numerics

Numerics

Articles 1–34 of 34. Alphabetical by title.

• Numerics Interval Alpha Bounds Alpha Inv Gt 137031

  A machine-checked proof places the inverse fine-structure constant above 137.031, a bound far too loose to test the theory behind it.

• Numerics Interval Alpha Bounds Alpha Inv Lt Strong

  A machine-checked proof pins the framework's inverse fine-structure constant below 137.039, a bound far looser than measurement.

• Numerics Interval Alpha Bounds Alpha Seed Gt

  A machine-checked proof pins the Recognition Science seed for the inverse fine-structure constant above 138.230048, a bound far too loose to count as agreement with measurement.

• Numerics Interval Alpha Bounds F Gap Gt Strong

  A machine-checked theorem pins down a number that appears in the Recognition Science account of the inverse fine-structure constant, and the honest reading is that the number is fa

• Numerics Interval Basic

  Interval arithmetic computes with ranges instead of single numbers, so a result comes with a guaranteed bound on its error.

• Numerics Interval Basic Hi Le Implies Contains Le

  A small theorem about intervals guarantees that any number inside a range stays below the range's upper edge, a workhorse fact for verified numerical computation.

• Numerics Interval Basic Hi Lt Implies Contains Lt

  A small lemma about intervals guarantees that if an interval's top edge sits below a rational number, then every real value inside that interval also sits below it.

• Numerics Interval Basic Lo Ge Implies Contains Ge

  A simple theorem about intervals guarantees that if a rational number sits at or below an interval's lower edge, every real number inside that interval is at least that ration

• Numerics Interval Basic Lo Gt Implies Contains Gt

  A small theorem about intervals with rational endpoints guarantees a strict lower bound for every real number inside them.

• Numerics Interval Exp

  A machine-checked method for pinning down the exponential function and the number e inside guaranteed bounds, with no floating-point guesswork.

• Numerics Interval Exp E In E Interval

  A machine-checked proof pins Euler's number e between 2.718 and 2.719, a tiny interval with a rigorous guarantee.

• Numerics Interval Exp Exp Interval Simple

  For inputs between 0 and 1, the function exp(x) is trapped between x+1 and 1/(1-x), a fact a machine-checked library proves.

• Numerics Interval Exp Exp Interval Simple Contains Exp

  A machine-checked proof that a simple interval arithmetic method always contains the true value of the exponential function.

• Numerics Interval Exp Exp Lower Simple

  The exponential function always sits above the line x + 1, a fact the framework's machine-checked library records as a building block for rigorous interval arithmetic.

• Numerics Interval Log

  A machine-checked library pins down the natural logarithm of key constants to a few decimal places, with no floating-point guesswork.

• Numerics Interval Log Log 10 In Interval

  The natural logarithm of 10 is an irrational number, but a machine-checked proof pins it between 2.30 and 2.31.

• Numerics Interval Log Log 2 In Interval

  A machine-checked proof pins the natural logarithm of 2 between 0.693 and 0.694, a rigorous interval that ordinary floating-point arithmetic cannot guarantee.

• Numerics Interval Log Log Interval Mono Contains Log

  A machine-checked theorem wraps the natural logarithm in a narrow interval, proving exactly where its value must lie.

• Numerics Interval Log Log Phi In Interval

  A machine-checked proof pins the natural logarithm of the golden ratio between 0.48 and 0.483, with no approximation left to faith.

• Numerics Interval Phi Bounds

  The golden ratio is known to about one part in a trillion, and a machine-checked library of formal theorems now proves it.

• Numerics Interval Phi Bounds Phi Inv3 In Interval Proven

  A machine-checked proof pins the cube of the golden ratio's reciprocal between 0.2359 and 0.237, a decimal bound that needs no calculator.

• Numerics Interval Phi Bounds Phi Inv5 In Interval Proven

  A machine-checked proof pins the fifth power of the golden ratio's reciprocal between 0.089 and 0.091, a bound that later physics constants lean on.

• Numerics Interval Phi Bounds Phi Pow51 In Interval Proven

  The golden ratio's 51st power is pinned between two 20-digit integers, and a machine-checked proof certifies the window.

• Numerics Interval Phi Bounds Phi Pow8 In Interval Proven

  A machine-checked proof pins the eighth power of the golden ratio between 46.97 and 46.99, a narrow window that later calculations rely on.

• Numerics Interval Pow

  Interval arithmetic computes a power like x^y not as a single number but as a guaranteed range that contains the true value.

• Numerics Interval Pow Phi Pow Neg3 In Interval

  The golden ratio raised to the power minus three is a number, and this page forces the result to a narrow interval with machine-checked certainty.

• Numerics Interval Pow Phi Pow Neg5 In Interval

  A machine-checked proof pins the golden ratio raised to the minus fifth power between two rational numbers, a small but exact step in a larger program.

• Numerics Interval Pow Rpow Interval Simple Contains Rpow

  A machine-checked theorem states a simple rule about when a number raised to a power lies inside a given interval.

• Numerics Interval Pow Two Pow Neg22 In Interval

  A machine-checked theorem pins down 2 to the power minus 22 inside a narrow interval, a routine numerical fact with a precise scope.

• Numerics Interval W8 Bounds

  A closed-form constant from the framework's eight-tick cycle is pinned down to nine decimal places by a machine-checked proof.

• Numerics Interval W8 Bounds Phi Gt 161803395

  A machine-checked proof pins the golden ratio between 1.61803395 and 1.6180340, a narrow window that later calculations rely on.

• Numerics Interval W8 Bounds Phi Lt 16180340

  A machine-checked proof pins the golden ratio below 1.6180340, a decimal bound that later numerical work depends on.

• Numerics Interval W8 Bounds Sqrt2 Gt 14142

  A machine-checked proof pins √2 between 1.4142 and 1.4143, a small but exact step in a larger computation.

• Numerics Interval W8 Bounds Sqrt2 Lt 14143

  The square root of 2 is less than 1.4143, and a machine-checked proof pins that down exactly.