{"version":"mathematics-knowledge/1.0","releasedOn":"2026-08-18","schemaPurpose":"A connective mathematical grammar for calculations, models, formalized traditions, and empirical tests. It does not transfer scientific validity between domains.","concepts":[{"id":"concept-jacobian-elliptic","slug":"jacobian-elliptic-functions","name":"Jacobian elliptic functions","category":"Numerical methods","proofStatus":"definition","description":"Doubly periodic functions built on the complete elliptic integral, and the third kind of analytic behaviour in this corpus.","definition":"DLMF 22.2.4, 22.2.5 and 22.2.6 define sn, cn and dn as ratios of theta functions. Two quantities carry the modulus into those definitions and both are built from the complete elliptic integral of the first kind: the nome at 22.2.1 is the exponential of minus pi times the complementary complete integral over the complete integral, and the argument scaling at 22.2.3 divides z by twice the complete integral. The chapter records the analytic character directly. Each of the twelve functions is doubly periodic in z at fixed modulus, meromorphic in z, and carries simple poles and simple zeros, and for modulus between zero and one all of them are real when z is real.","notation":["sn, cn and dn, with the twelve functions named by pairs of letters","k for the modulus and q for the nome","K of k for the complete elliptic integral and K prime for its complement"],"assumptions":["The nome and the argument scaling are defined through the complete elliptic integral, so these functions inherit whatever conditions that integral carries.","Double periodicity is stated at fixed modulus; the modulus is a parameter rather than a second variable.","Reality for real argument is stated only for modulus in the closed interval from zero to one."],"invariants":["sn, cn and dn as theta ratios, DLMF 22.2.4 to 22.2.6","The nome expressed through the complete elliptic integral, DLMF 22.2.1","Doubly periodic and meromorphic with simple poles and simple zeros, DLMF 22.2 stated in prose rather than as a numbered equation"],"procedure":["Compute the complete elliptic integral first, since both the nome at 22.2.1 and the scaling at 22.2.3 depend on it.","Check the modulus against the closed unit interval before relying on real values for real argument.","Reduce an argument by the periods before evaluating, since double periodicity means a large argument carries no extra information."],"errorBounds":["The functions have simple poles, so accuracy degrades near them and no reduction of the argument avoids a pole that the argument sits on.","The nome at 22.2.1 involves a ratio of complete integrals inside an exponential, so an error in the modulus is amplified before the functions are evaluated at all.","The chapter defines the functions and their periodicity and makes no accuracy claim about any evaluation scheme."],"sourceIds":["nist-dlmf-jacobian-elliptic"],"relatedSlugs":["elliptic-integrals","periodic-functions-and-phase","airy-functions"],"doesNotEstablish":"The chapter gives the definitions, the periods and the analytic character. It does not establish that any evaluation scheme is accurate near a pole, and periodicity is not a statement about numerical conditioning."},{"id":"concept-exponential-integrals","slug":"exponential-and-logarithmic-integrals","name":"Exponential and logarithmic integrals","category":"Numerical methods","proofStatus":"definition","description":"Integrals of a decaying exponential over its argument, and the entire companion that carries the singularity away.","definition":"DLMF 6.2.1 defines the exponential integral E1 as the integral of e to the minus t over t from z to infinity, for non-zero z, with the path avoiding the negative real axis and a branch cut along the non-positive reals. 6.2.3 defines a companion, Ein, as the integral of one minus e to the minus t over t from zero to z, and records it as entire. The two are joined at 6.2.4: E1 equals Ein minus the logarithm of z minus Euler constant, which places the whole singular part in the logarithm rather than in the integral. 6.2.5 defines Ei as a principal value, 6.2.6 relates it to E1 on the negative axis, and 6.2.8 defines the logarithmic integral as a principal value equal to Ei of the logarithm. The sine and cosine integrals follow at 6.2.9 to 6.2.12, where Si and Cin are entire while Ci needs a principal value.","notation":["E1 of z and Ei of x for the exponential integrals","Ein for the entire companion and li for the logarithmic integral","Si, si, Ci and Cin for the trigonometric integrals"],"assumptions":["E1 at 6.2.1 is stated for non-zero argument with the path avoiding the negative real axis, so it is branch-dependent rather than single valued.","Ei at 6.2.5 and li at 6.2.8 are principal values, which are different objects from ordinary integrals.","The relation at 6.2.4 holds with Euler constant as a stated term, so the constant is part of the definition rather than an approximation."],"invariants":["E1 defined with a branch cut along the non-positive reals, DLMF 6.2.1","Ein is entire while E1 is not, DLMF 6.2.3","E1 equals Ein minus log z minus Euler constant, DLMF 6.2.4"],"procedure":["Choose E1 or Ei by which side of the axis the argument lies on, using the relation at 6.2.6 rather than continuing one across the cut.","Where the argument is small, work through the entire companion at 6.2.3 and add the logarithmic term from 6.2.4, so the singularity is handled in closed form instead of by the quadrature.","Treat li and Ci as principal values, and do not hand their integrands to a scheme that assumes an ordinary integral."],"errorBounds":["The singular behaviour near the origin sits in the logarithm by 6.2.4, so a scheme that integrates E1 directly near zero is fighting a singularity the identity already removes.","A principal value is not an ordinary integral. A quadrature that ignores the interior pole in 6.2.5 or 6.2.8 returns a number without meaning rather than an inaccurate one, as with the third elliptic kind.","The chapter gives definitions rather than algorithms and makes no accuracy claim about any evaluation."],"sourceIds":["nist-dlmf-exponential-integral"],"relatedSlugs":["incomplete-gamma-functions","number-theoretic-functions","elliptic-integrals"],"doesNotEstablish":"The chapter defines these integrals and their singular structure. It states no relation here between the logarithmic integral and the prime counting function, so none is claimed on this page."},{"id":"concept-confluent-hypergeometric","slug":"confluent-hypergeometric-functions","name":"Confluent hypergeometric functions","category":"Numerical methods","proofStatus":"definition","description":"One equation that Bessel and Airy are cases of, and the exceptional parameters an alternative form exists to remove.","definition":"DLMF 13.2.1 gives Kummer equation, with a regular singularity at the origin carrying indices zero and one minus b, and an irregular singularity of rank one at infinity. That is the same structural description the Bessel chapter gives at 10.2.1, which is what it means to call these confluent. The first solution at 13.2.2 is a series converging for all complex argument, entire in the argument and in a, and meromorphic in b, but the chapter states plainly that it does not exist when b is a non-positive integer. Olver form at 13.2.3 divides by a gamma function instead and is entire in all three parameters, with 13.2.4 relating the two by a factor of the gamma function at b. The second solution at 13.2.6 has a branch point at the origin and behaves like z to the minus a at infinity in a stated sector, and 13.2.7 and 13.2.8 give the parameter values at which the solution becomes a polynomial.","notation":["M of a, b, z for the first solution and U of a, b, z for the second","bold M for Olver entire form","a and b for the parameters and z for the argument"],"assumptions":["The first solution does not exist when b is a non-positive integer, which is the exception 13.2.3 exists to remove.","The relation at 13.2.4 between the two forms holds except at those same non-positive integers.","The asymptotic statement for the second solution at 13.2.6 holds in a stated sector rather than everywhere."],"invariants":["Kummer equation with a regular singularity at the origin and an irregular one of rank one at infinity, DLMF 13.2.1","The first solution converges for all complex argument but fails to exist at non-positive integer b, DLMF 13.2.2","Olver form is entire in all three parameters, DLMF 13.2.3"],"procedure":["Check b against zero and the negative integers before using the first solution, and take the entire form at 13.2.3 where it fails.","Read the second solution as branch-dependent, since 13.2.6 declares a principal branch through the principal value of z to the minus a.","Where a parameter makes the solution a polynomial by 13.2.7 or 13.2.8, use that form rather than summing a series that has already terminated."],"errorBounds":["Convergence for all argument is not accuracy at all argument. The series at 13.2.2 converges everywhere and still loses significance to cancellation for large argument, which is why the second solution carries an asymptotic description instead.","Near a non-positive integer b the first solution is ill conditioned even where it is defined, which is the practical reason for the entire form at 13.2.3, exactly as the regularized form exists in the Gauss case at 15.2.2.","The chapter states solutions and their structure and makes no accuracy claim about any implementation."],"sourceIds":["nist-dlmf-confluent"],"relatedSlugs":["hypergeometric-function","bessel-functions","airy-functions","gamma-function"],"doesNotEstablish":"The chapter defines the solutions, their exceptional parameters and their branch structure. It does not establish that any implementation is accurate, and everywhere-convergence is not everywhere-usable."},{"id":"concept-legendre-functions","slug":"legendre-functions","name":"Legendre functions","category":"Numerical methods","proofStatus":"definition","description":"The equation behind spherical harmonics, its three singular points, and a solution pair chosen per interval.","definition":"DLMF 14.2.1 gives Legendre equation, and 14.2.2 the associated form carrying a second parameter. The chapter records the singular structure directly: regular singularities at x equal to one, minus one and infinity, with exponent pairs minus mu over two and mu over two at the finite pair and nu plus one and minus nu at infinity. That structure is why the interval matters. On minus one to one, and when the real part of mu is non-negative, the Ferrers pair at plus and minus x is independent and recessive at the endpoints. On one to infinity, and when the real part of mu is non-negative and the real part of nu is at least minus one half, a different pair is recommended, recessive at one and at infinity respectively. Wronskian relations at 14.2.3 to 14.2.11 tie the pairs together.","notation":["nu for the degree and mu for the order","Ferrers functions on the cut interval and Legendre functions outside it","P and Q for the first and second kinds"],"assumptions":["The recommended pairs carry parameter conditions: a non-negative real part of mu throughout, and additionally a real part of nu at least minus one half outside the cut interval.","The two intervals are treated separately because the singular points at plus and minus one sit at their boundary.","Recessive behaviour at an endpoint is the reason a pair is recommended, and it is a statement about the endpoint rather than the whole interval."],"invariants":["Legendre equation and its associated form, DLMF 14.2.1 and 14.2.2","Regular singularities at plus one, minus one and infinity with stated exponent pairs, DLMF 14.2.2","Numerically satisfactory pairs differ between the cut interval and the outside, DLMF 14.2 stated in prose rather than as a numbered equation"],"procedure":["Decide which interval the argument lies in before choosing a pair, since the recommendation changes at the singular points.","Check the parameter conditions on mu, and on nu outside the cut interval, rather than assuming the recommended pair applies.","Use a Wronskian from 14.2.3 to 14.2.11 to check an implementation, since it relates the pair rather than testing one solution alone."],"errorBounds":["A recessive solution is small near its endpoint, so computing it by a method tuned to the dominant one loses it to cancellation. That is the reason the chapter names pairs by interval.","The singular points at plus and minus one are where the equation degenerates, and accuracy there is governed by the exponent pairs rather than by the solver.","The chapter recommends pairs and states conditions; it makes no accuracy claim about any implementation."],"sourceIds":["nist-dlmf-legendre"],"relatedSlugs":["spherical-coordinates","orthogonal-polynomials","bessel-functions"],"doesNotEstablish":"The chapter gives the equations, their singular structure and the recommended pairs. It does not establish that any implementation is accurate, and a recommendation carries conditions rather than holding universally."},{"id":"concept-number-theoretic-functions","slug":"number-theoretic-functions","name":"Number-theoretic functions","category":"Numerical methods","proofStatus":"definition","description":"Unique factorisation, the functions built on it, and one statement about primes that is asymptotic rather than exact.","definition":"DLMF 27.2.1 states the fundamental theorem of arithmetic: every integer greater than one factors uniquely into prime powers. Three functions are defined on that footing. The totient at 27.2.7 counts the integers up to n that are coprime to n. The divisor function at 27.2.9 counts the divisors of n. The Mobius function at 27.2.12 is one at n equal to one, minus one to the number of prime factors when they are all distinct, and zero when any prime is repeated. Separately, 27.2.3 states the prime counting function as asymptotic to x over the natural logarithm of x, which is a statement about a ratio in a limit rather than a count at any particular x.","notation":["phi of n for the totient","d of n for the divisor count","mu of n for the Mobius function and pi of x for the prime count"],"assumptions":["Unique factorisation at 27.2.1 is stated for integers greater than one, so the functions defined on it inherit that domain.","The Mobius function is zero exactly when a prime is repeated, which is a condition on the factorisation rather than on the size of n.","The prime counting statement at 27.2.3 is asymptotic, so it constrains a ratio as x grows and says nothing at a fixed x."],"invariants":["Unique factorisation into prime powers, DLMF 27.2.1","The prime counting function is asymptotic to x over log x, DLMF 27.2.3","The Mobius function vanishes exactly when a prime is repeated, DLMF 27.2.12"],"procedure":["Take a factorisation as the starting point, since the totient, divisor and Mobius functions are all read off it.","Treat 27.2.3 as a limiting ratio and never as an estimate with a stated error at a particular x.","Check for a repeated prime before using the Mobius function, because that case is zero rather than small."],"errorBounds":["The asymptotic law at 27.2.3 uses the relation defined at 2.1.1: the ratio tends to one. It supplies no bound on the difference at any finite x, and the difference is not small in the way the ratio suggests.","Computing the totient or divisor function from a factorisation is only as reliable as the factorisation, which for large n is the hard part rather than the counting.","The chapter gives definitions, not algorithms, and makes no claim about the cost or stability of obtaining a factorisation."],"sourceIds":["nist-dlmf-number-theory"],"relatedSlugs":["riemann-zeta-function","asymptotic-approximations","modular-arithmetic"],"doesNotEstablish":"The chapter defines the functions and states one asymptotic law. It settles no open question about how primes are distributed, and an asymptotic law is not a formula for the prime count."},{"id":"concept-airy-functions","slug":"airy-functions","name":"Airy functions","category":"Numerical methods","proofStatus":"definition","description":"The simplest equation whose behaviour changes character across a point, and the pair chosen to describe it.","definition":"DLMF 9.2.1 gives Airy equation as the second derivative of w equal to z times w, and records that all its solutions are entire functions. That is a sharper statement than it looks: the equation changes character at the origin, oscillating on one side and growing or decaying on the other, yet its solutions have no singularity anywhere. The standard solutions are named at 9.2.2, and their values at the origin are given at 9.2.3 through 9.2.6 in terms of the gamma function at one third and two thirds, which is why a chapter on a differential equation depends on one about a factorial. Table 9.2.1 records that Ai and Bi are the numerically satisfactory pair on the whole real line, with other pairs preferred in other sectors.","notation":["Ai of z and Bi of z for the standard pair","Ai prime and Bi prime for their derivatives","the gamma function at one third and two thirds in the initial values"],"assumptions":["All solutions are entire, so unlike the Bessel case there is no branch point and no cut to respect.","Which pair is numerically satisfactory depends on the sector, and Table 9.2.1 gives the real line rather than the whole plane.","The values at the origin are exact expressions in the gamma function rather than decimal approximations."],"invariants":["Airy equation, DLMF 9.2.1","The values at the origin in terms of the gamma function, DLMF 9.2.3 to 9.2.6","Ai and Bi are numerically satisfactory on the real line, DLMF Table 9.2.1"],"procedure":["Use Ai and Bi on the real line, per Table 9.2.1, and check the table before carrying that choice into the complex plane.","Take the initial values from 9.2.3 to 9.2.6 in gamma form rather than from rounded decimals, so precision is set by the gamma evaluation and not by the transcription.","Expect different behaviour either side of the origin, since the equation changes character there even though the solutions do not."],"errorBounds":["Bi grows rapidly for positive argument while Ai decays, so a scheme that computes one accurately can lose the other entirely to cancellation.","Numerical satisfactoriness is a property of a pair in a region, not of a solution: the same functions can be a poor basis in a sector where the table names a different pair.","The chapter defines the solutions and tabulates the choice; it makes no accuracy claim about any evaluation method."],"sourceIds":["nist-dlmf-airy"],"relatedSlugs":["gamma-function","bessel-functions","asymptotic-approximations"],"doesNotEstablish":"The chapter gives the equation, its solutions and where each pair is usable. It does not establish that any implementation is accurate, and entirety of the solutions is not stability of a computation."},{"id":"concept-calendrical-reconciliation","slug":"calendrical-reconciliation","name":"Calendrical reconciliation","category":"Time and periodicity","proofStatus":"method","description":"Two astronomical periods that do not divide each other, and the three families of arithmetic built to live with that.","definition":"A calendar reconciles two measured periods that share no common multiple. Doggett gives the tropical year as 365.2421896698 days and the mean synodic month as 29.5305888531 days, so twelve lunar months come to 354.36707 days and fall short of the year by about 10.875 days. Three families answer that differently. A solar calendar tracks the tropical year and intercalates days, as the Gregorian does with a 400-year cycle of 146,097 days. A lunar calendar follows the phase cycle and lets the months move through the seasons, as the Islamic calendar does. A lunisolar calendar keeps lunar months but intercalates a whole month every few years, using the Metonic relation that 235 lunations occupy nineteen years, as the Hebrew and Chinese calendars do.","notation":["the tropical year in days","the mean synodic month in days","the Metonic relation of 235 lunations to 19 years"],"assumptions":["The stated periods are mean values. Doggett records that an individual synodic month departs from the mean, so a mean period fixes a scheme rather than a date.","The tropical year and synodic month share no common multiple, which is why every scheme is an approximation rather than a fix.","Which family a calendar belongs to is a design choice about what to keep synchronised, not a fact about the sky."],"invariants":["Tropical year 365.2421896698 days and mean synodic month 29.5305888531 days, per Doggett","Twelve synodic months come to 354.36707 days, per Doggett","The Metonic relation, 235 lunations in nineteen years, 6939.688 days, per Doggett"],"procedure":["Decide which period is to be tracked, because no scheme tracks both exactly.","For a solar scheme, choose an intercalation cycle and state its residual error against the tropical year.","For a lunisolar scheme, choose an intercalation rule such as the Metonic and state the residual, which is about two hours per nineteen years.","Treat a computed date as the output of the declared scheme rather than as an astronomical event, since mean periods do not locate an individual new moon."],"errorBounds":["Every scheme carries a residual because the two periods are incommensurable. The Metonic relation leaves 235 lunations about 0.087 days, roughly two hours, longer than nineteen tropical years.","The Gregorian 400-year cycle averages 365.2425 days. Against the tropical year given here that is an excess of about 0.00031 days per year, one day in roughly 3200 years; Doggett states the error as about one day in 2500 years, and the difference reflects which definition of the tropical year is used rather than an arithmetic disagreement.","Mean periods do not predict an individual month. A scheme built on them fixes a rule, and a rule is not an observation."],"sourceIds":["doggett-calendars"],"relatedSlugs":["calendar-and-timescale-mappings","modular-arithmetic","periodic-functions-and-phase"],"doesNotEstablish":"This is the arithmetic of reconciling two periods. It does not determine any religious observance, which is fixed by a community rule that may use observation rather than computation, and it settles nothing about the meaning of any calendar."},{"id":"concept-elliptic-integrals","slug":"elliptic-integrals","name":"Elliptic integrals","category":"Numerical methods","proofStatus":"definition","description":"Three Legendre integrals, the conditions that keep them defined, and the one that needs a principal value.","definition":"DLMF 19.2.4 defines the incomplete integral of the first kind as the integral from zero to phi of d theta over the square root of one minus k squared sine squared theta. 19.2.5 defines the second kind with that square root in the numerator instead. Both carry the same domain conditions: one minus sine squared phi and one minus k squared sine squared phi must each avoid the cut along the non-positive reals, and at most one may be zero. 19.2.7 defines the third kind with an extra factor of one minus alpha squared sine squared theta in the denominator, requiring that expression to be non-zero, and the chapter states that a Cauchy principal value is taken when it vanishes at an interior point. 19.2.8 defines the complete integrals as the incomplete ones evaluated at phi equal to pi over two.","notation":["F of phi and k for the first kind, E for the second, capital Pi for the third","K of k, E of k and Pi of alpha squared and k for the complete forms","k for the modulus and alpha squared for the characteristic"],"assumptions":["The first and second kinds require both one minus sine squared phi and one minus k squared sine squared phi to avoid the cut along the non-positive reals, with at most one of them zero.","The third kind requires one minus alpha squared sine squared phi to be non-zero, and takes a principal value when it vanishes inside the range.","The principal branch is declared where the phase of one minus k squared is at most pi, with cuts on the reals outside minus one to one."],"invariants":["First and second kinds with their shared domain conditions, DLMF 19.2.4 and 19.2.5","Third kind with its non-vanishing condition and principal value, DLMF 19.2.7","Complete integrals are the incomplete ones at quarter period, DLMF 19.2.8"],"procedure":["Check both domain conditions before evaluating the first or second kind, since the definition excludes the cut rather than merely warning about it.","For the third kind, test whether the characteristic factor vanishes inside the range, and take a principal value where it does.","Reach the complete integrals through 19.2.8 rather than by pushing an incomplete evaluation to the endpoint."],"errorBounds":["The integrands become singular as the modulus approaches one, so accuracy degrades near that limit however the quadrature is arranged.","A principal value is not an ordinary integral, and a scheme that ignores the interior singularity in the third kind returns a number without meaning rather than an inaccurate one.","The chapter defines the integrals and declares a branch; it makes no accuracy claim about any evaluation scheme."],"sourceIds":["nist-dlmf-elliptic"],"relatedSlugs":["numerical-integration","hypergeometric-function"],"doesNotEstablish":"The chapter defines the three kinds and their domains. It does not establish that any quadrature scheme is accurate near the singular limit, and a principal value is a different object from the integral it replaces."},{"id":"concept-incomplete-gamma","slug":"incomplete-gamma-functions","name":"Incomplete gamma functions","category":"Numerical methods","proofStatus":"definition","description":"Split the gamma integral at a point, and keep the two halves and their asymmetry straight.","definition":"DLMF 8.2.1 defines the lower incomplete gamma as the integral of t to the a minus one times e to the minus t from zero to z, for real part of a greater than zero. 8.2.2 defines the upper form as the same integrand taken from z to infinity. 8.2.3 states the identity that binds them: the lower plus the upper equals the complete gamma at a, for a not zero or a negative integer. The normalized pair at 8.2.4 divides each by the complete gamma, and 8.2.5 states that they sum to one. The two halves are not symmetric in the parameter: the chapter records that the upper form is entire in a when z is non-zero, while the lower form is meromorphic with simple poles at the non-positive integers.","notation":["lower case gamma of a, z for the lower form and capital Gamma of a, z for the upper","P of a, z and Q of a, z for the normalized pair","a for the parameter and z for the split point"],"assumptions":["The lower definition at 8.2.1 is stated for real part of a greater than zero.","The identity at 8.2.3 excludes a equal to zero and the negative integers, which are exactly the poles of the complete gamma.","The two halves differ in the parameter: the upper is entire in a for non-zero z while the lower has simple poles at the non-positive integers."],"invariants":["Lower and upper integral definitions, DLMF 8.2.1 and 8.2.2","The two halves sum to the complete gamma, DLMF 8.2.3","The normalized pair sums to one, DLMF 8.2.5"],"procedure":["Choose the half whose value is the smaller of the two, and take the other by subtraction only when precision allows.","Use the normalized pair from 8.2.4 when comparing across parameters, since the complete gamma has been divided out.","Check the parameter against zero and the negative integers before relying on 8.2.3, which excludes them."],"errorBounds":["The identity at 8.2.5 says the normalized pair sums to one exactly, which is why computing the small member by subtracting the large one from one destroys it. The same device appears at 7.2.2 for the error function, and for the same reason.","Near the poles of the lower form the value is unbounded, so an algorithm tuned for one half is not automatically usable for the other.","The chapter gives definitions rather than algorithms and makes no accuracy claim for any evaluation method."],"sourceIds":["nist-dlmf-incomplete-gamma"],"relatedSlugs":["gamma-function","error-function-and-related-integrals"],"doesNotEstablish":"The chapter defines the two halves and their exact relation. It does not establish the accuracy of any algorithm, and the exactness of the identity is a mathematical statement rather than a numerical guarantee."},{"id":"concept-riemann-zeta","slug":"riemann-zeta-function","name":"Riemann zeta function","category":"Numerical methods","proofStatus":"definition","description":"A series that converges in a half-plane, a product over primes that matches it there, and a function defined everywhere else by continuation.","definition":"DLMF 25.2.1 defines the zeta function by the Dirichlet series, the sum of one over n to the s, and states it for real part of s greater than one. 25.2.11 gives the Euler product, the product over all primes of one minus p to the minus s, inverted, under the same condition. Elsewhere the function is defined by analytic continuation, and the chapter records that the continued function is meromorphic with a single singularity in the complex plane: a simple pole at s equals one with residue one. 25.2.4 gives the Laurent expansion about that pole, one over s minus one plus a series in the Stieltjes constants, which are themselves defined as limits at 25.2.5.","notation":["zeta of s","the product index p running over primes","gamma sub n for the Stieltjes constants"],"assumptions":["Both the series at 25.2.1 and the product at 25.2.11 are stated only for real part of s greater than one; neither defines the function elsewhere.","Values outside that half-plane come from analytic continuation, which is a different operation from evaluating the series.","The Laurent expansion at 25.2.4 is local to the pole at s equals one."],"invariants":["The Dirichlet series in the half-plane, DLMF 25.2.1","The Euler product over primes in the same half-plane, DLMF 25.2.11","A single simple pole at s equals one with residue one, DLMF 25.2 stated in prose rather than as a numbered equation"],"procedure":["Check the real part of the argument before using either representation, since both stop at one.","Treat a value at real part below one as a continuation rather than a sum, because the series diverges there.","Near s equals one, use the Laurent form at 25.2.4 rather than the series, which is where the pole sits."],"errorBounds":["The Dirichlet series converges slowly near the boundary of its half-plane, so truncating it close to real part one gives poor accuracy long before it fails outright.","The Euler product is over infinitely many primes; truncating it at a finite prime is an approximation the chapter does not bound.","The Stieltjes constants at 25.2.5 are defined as limits of differences that cancel, so computing them naively loses significance."],"sourceIds":["nist-dlmf-zeta"],"relatedSlugs":["bernoulli-and-euler-numbers","convergence-precision-and-error"],"doesNotEstablish":"The chapter states representations and the pole. It settles no question about the location of the zeros, and neither representation defines the function outside its half-plane."},{"id":"concept-hypergeometric-function","slug":"hypergeometric-function","name":"Hypergeometric function","category":"Numerical methods","proofStatus":"definition","description":"One series that many named functions are special cases of, with the parameter values where it stops making sense.","definition":"DLMF 15.2.1 defines the Gauss hypergeometric function as a series in Pochhammer symbols over the unit disk, extended elsewhere by analytic continuation, with the principal branch taken in the sector where the phase of one minus z is at most pi and a cut running from one to infinity along the real axis. The chapter states plainly that the function does not in general exist when the lower parameter c is zero or a negative integer. The regularized form at 15.2.2 divides by a gamma function instead and is valid for all c, which is why it exists. 15.2.4 records that the series terminates into a polynomial when the upper parameter is a non-positive integer.","notation":["F of a, b; c; z for the Gauss function","bold F for the regularized form","the Pochhammer symbol for the rising factorial"],"assumptions":["The series at 15.2.1 converges on the open unit disk; values outside come from continuation, not summation.","The function is generally undefined when c is zero or a negative integer, which is the exception the regularized form at 15.2.2 removes.","A principal branch is declared with a cut from one to infinity, so a value is branch-dependent rather than absolute."],"invariants":["The Gauss series on the unit disk, DLMF 15.2.1","The regularized form valid for all c, DLMF 15.2.2","Termination into a polynomial at non-positive integer upper parameter, DLMF 15.2.4"],"procedure":["Check c against zero and the negative integers before using the unregularized form.","On the boundary circle, classify by the real part of c minus a minus b: above zero converges absolutely, between minus one and zero converges conditionally away from z equals one, and at or below minus one diverges.","Where a parameter makes the series terminate, use the polynomial form rather than summing an infinite series that has already stopped."],"errorBounds":["The three boundary regimes are stated by condition rather than pointwise, so conditional convergence near the circle is slow and the chapter offers no rate.","Continuation outside the unit disk is an analytic statement, not a numerical method; the chapter does not bound the accuracy of any particular continuation scheme.","Near c equal to a non-positive integer the unregularized form is ill conditioned even where it is defined, which is the practical reason for 15.2.2."],"sourceIds":["nist-dlmf-hypergeometric"],"relatedSlugs":["bessel-functions","gamma-function","convergence-precision-and-error"],"doesNotEstablish":"The chapter defines the function, its branch and its exceptional parameters. It does not establish the accuracy of any continuation scheme outside the unit disk."},{"id":"concept-asymptotic-approximations","slug":"asymptotic-approximations","name":"Asymptotic approximations","category":"Numerical methods","proofStatus":"definition","description":"Say precisely what an approximation claims in a limit, and what it still leaves undetermined.","definition":"DLMF 2.1.1 defines asymptotic equality by the ratio: f is asymptotically equal to phi when the ratio of the two tends to one in the limit. The order symbols follow, with little-o at 2.1.2 when the ratio tends to zero and big-O at 2.1.3 when the ratio stays bounded. A Poincare asymptotic expansion is defined at 2.1.13 and 2.1.14 by a condition on every truncation: for each n the function equals the first n terms plus a remainder of order x to the minus n. The chapter states that a convergent series is automatically the asymptotic expansion of its sum, but that the converse fails, and it records that the functions zero, e to the minus z, and e to the minus z cosine z all share one null expansion in suitable sectors.","notation":["The tilde for asymptotic equality","big-O for a bounded ratio and little-o for a vanishing one","x tending to c within a declared point set"],"assumptions":["Every one of these statements is relative to a limit point and a point set, so an order symbol without a stated limit says nothing.","The expansion condition at 2.1.13 is imposed on each truncation separately rather than on the infinite series.","A convergent series is an asymptotic expansion of its sum, but an asymptotic expansion need not converge."],"invariants":["Asymptotic equality is the ratio tending to one, DLMF 2.1.1","Big-O is a bounded ratio and little-o a vanishing one, DLMF 2.1.2 and 2.1.3","A Poincare expansion is a condition on every truncation, DLMF 2.1.13 and 2.1.14"],"procedure":["State the limit point and the point set before writing an order symbol, since the symbol is meaningless without them.","Read an expansion as a family of statements about truncations rather than as a series to be summed.","Where an expansion is used numerically, truncate at the smallest term rather than at as many terms as are available."],"errorBounds":["An asymptotic expansion does not determine its function. The chapter gives zero, e to the minus z, and e to the minus z cosine z as three functions sharing one null expansion, so agreement of expansions is not agreement of functions.","The defining condition bounds the remainder for each fixed truncation as the argument goes to the limit. It says nothing about accuracy at a fixed argument as more terms are taken.","A divergent expansion has a smallest term, and accuracy past that point degrades however carefully the arithmetic is done."],"sourceIds":["nist-dlmf-asymptotics"],"relatedSlugs":["gamma-function","convergence-precision-and-error","bessel-functions"],"doesNotEstablish":"The chapter defines what asymptotic statements mean. It does not establish accuracy at any particular argument, and it states explicitly that an expansion is shared by more than one function."},{"id":"concept-orthogonal-polynomials","slug":"orthogonal-polynomials","name":"Orthogonal polynomials","category":"Numerical methods","proofStatus":"definition","description":"Orthogonality is a relation to a declared weight, not a property a family owns, and a recurrence carries it.","definition":"DLMF 18.2.1 defines orthogonality on an interval by an integral: the product of two distinct members against a weight function integrates to zero. The weight is constrained at 18.2.1_5, which requires it to be non-negative, to have positive total mass, and to have all moments finite. Discrete analogues replace the integral by a sum over an infinite set at 18.2.2 or a finite one at 18.2.3, and 18.2.4_5 states the general form against a Lebesgue-Stieltjes measure. Two three-term recurrences follow, at 18.2.8 and 18.2.10, each carrying a strict positivity condition under which Favard theorem gives the converse, so that a family satisfying such a recurrence is orthogonal for some measure.","notation":["p sub n of x for the n-th member of the family","w of x for the weight and mu for the general measure","A sub n, B sub n and C sub n for the recurrence coefficients"],"assumptions":["Orthogonality is relative to a declared weight or measure; the same polynomials are not orthogonal against a different one.","The weight conditions at 18.2.1_5 require non-negativity, positive total mass and finite moments, so a weight failing any of these defines no such family.","The converse direction depends on the strict positivity conditions at 18.2.9_5 and 18.2.11_2, not on the recurrence shape alone."],"invariants":["Continuous orthogonality against a weight, DLMF 18.2.1","The weight must be non-negative with positive mass and finite moments, DLMF 18.2.1_5","The three-term recurrences, DLMF 18.2.8 and 18.2.10"],"procedure":["Name the weight or measure before calling a family orthogonal, because the term is otherwise incomplete.","Check the weight against the conditions at 18.2.1_5 rather than assuming a positive-looking function qualifies.","Where a family is generated by recurrence, confirm the positivity conditions if orthogonality is being inferred from the recurrence rather than the other way round."],"errorBounds":["The chapter states the recurrences without claiming they are numerically stable. Forward recurrence can amplify rounding, and the reference does not settle the direction to use.","A finite discrete family at 18.2.3 is orthogonal only up to degree N; beyond that the relation does not hold and is not claimed to.","Orthogonality against a weight says nothing about conditioning of the resulting expansion coefficients for a particular function."],"sourceIds":["nist-dlmf-orthogonal-polynomials"],"relatedSlugs":["interpolation","numerical-integration"],"doesNotEstablish":"The chapter defines orthogonality and its recurrences. It does not establish the numerical stability of evaluating them, and orthogonality holds against a declared weight rather than as an intrinsic property."},{"id":"concept-bessel-functions","slug":"bessel-functions","name":"Bessel functions","category":"Numerical methods","proofStatus":"definition","description":"Solve the equation that appears whenever a problem is posed on a disk or a cylinder, and pick a solution pair that behaves.","definition":"Bessel equation is stated at DLMF 10.2.1 as z squared times the second derivative, plus z times the first derivative, plus the quantity z squared minus nu squared times the function, equal to zero. The chapter records its analytic structure directly: a regular singularity at the origin with indices plus and minus nu, and an irregular singularity of rank one at infinity. The first-kind solution at 10.2.2 is a power series whose coefficients divide by the gamma function at nu plus k plus one, which is why the gamma function turns up in a problem that began as a differential equation. The second-kind solution at 10.2.3 is built from the first at plus and minus nu, and 10.2.4 gives the limiting form when nu is an integer and that construction degenerates.","notation":["J of nu at z for the first kind","Y of nu at z for the second kind","nu for the order and z for the argument"],"assumptions":["The first-kind solution at 10.2.2 is analytic except for a branch point at the origin when nu is not an integer.","The second-kind solution has a branch point at the origin whether or not nu is an integer, with a cut along the negative real axis.","The construction at 10.2.3 divides by sine of nu pi, so it degenerates at integer order and 10.2.4 supplies the limit instead.","The equation has a regular singularity at zero and an irregular one at infinity, so behaviour at the two ends is not governed by the same expansion."],"invariants":["Bessel equation, DLMF 10.2.1","The first-kind series with its gamma-function coefficients, DLMF 10.2.2","The second-kind solution built from first-kind solutions of opposite order, DLMF 10.2.3"],"procedure":["Fix the order nu, and note whether it is an integer, because that decides whether 10.2.3 or 10.2.4 applies.","Choose a solution pair from Table 10.2.1 for the region in question rather than assuming one pair works everywhere.","Respect the branch cut along the negative real axis when continuing either solution.","Where the order is large or the argument small, check that the chosen pair is still the numerically satisfactory one for that region."],"errorBounds":["Linear independence and numerical satisfactoriness are different properties. A pair can be independent and still lose all its accuracy to cancellation, which is why Table 10.2.1 is organised by region.","The series at 10.2.2 alternates, so for large argument it subtracts nearly equal terms and loses significance long before it stops converging.","The construction at 10.2.3 divides by sine of nu pi, so near integer order it is ill conditioned even where it is defined."],"sourceIds":["nist-dlmf-bessel"],"relatedSlugs":["gamma-function","convergence-precision-and-error"],"doesNotEstablish":"The chapter gives the solutions and their analytic structure. It does not select a pair for a given computation, and it makes no claim that any particular implementation is accurate over any particular range."},{"id":"concept-bernoulli-euler-numbers","slug":"bernoulli-and-euler-numbers","name":"Bernoulli and Euler numbers","category":"Numerical methods","proofStatus":"definition","description":"Define the coefficients that keep appearing in expansions, and note where their generating series stop converging.","definition":"The Bernoulli numbers are defined at DLMF 24.2.1 by the generating function t over the quantity e to the t minus one, expanded as a sum of B sub n times t to the n over n factorial, converging for the modulus of t less than two pi. The polynomials follow at 24.2.3 from the same function multiplied by e to the x t, in the same disk, and 24.2.4 records that B sub n is the polynomial evaluated at zero. The Euler numbers are defined at 24.2.6 by two e to the t over the quantity e to the two t plus one, converging only for the modulus of t less than pi over two. The vanishing rules at 24.2.2 and 24.2.7 say that odd-indexed Bernoulli numbers vanish apart from the first, and that odd-indexed Euler numbers vanish.","notation":["B sub n for the Bernoulli numbers and B sub n of x for the polynomials","E sub n for the Euler numbers and E sub n of x for the polynomials","t for the generating variable"],"assumptions":["The Bernoulli generating function at 24.2.1 converges only for the modulus of t less than two pi, which is the distance to the nearest pole of the generating expression.","The Euler generating function at 24.2.6 converges only for the modulus of t less than pi over two, a strictly smaller disk than the Bernoulli one.","The vanishing rule at 24.2.2 exempts the first Bernoulli number, so the odd indices are not uniformly zero.","The polynomials at 24.2.3 reduce to the numbers only at argument zero, by 24.2.4."],"invariants":["Bernoulli generating function with its radius, DLMF 24.2.1","Odd-indexed Bernoulli numbers vanish apart from the first, and the even ones alternate in sign, DLMF 24.2.2","Euler generating function with its smaller radius, DLMF 24.2.6"],"procedure":["Take the numbers from the generating function at 24.2.1 rather than from a recurrence, so the convergence radius stays visible.","Check the index parity before assuming a coefficient is present: by 24.2.2 the odd Bernoulli terms are absent above the first.","Where an expansion is indexed by even numbers only, look for a Bernoulli coefficient behind it rather than treating the gap as a convention."],"errorBounds":["The generating series say nothing outside their disks. Beyond two pi for Bernoulli, or pi over two for Euler, the definition still holds but the series is not a usable expansion.","The even-indexed Bernoulli numbers grow rapidly and alternate in sign, so a series carrying them is a candidate for catastrophic cancellation rather than a safe summation.","The chapter defines the coefficients and makes no claim about the stability of any recurrence used to generate them numerically."],"sourceIds":["nist-dlmf-bernoulli"],"relatedSlugs":["gamma-function","convergence-precision-and-error"],"doesNotEstablish":"These are definitions with stated convergence radii. They do not establish that any recurrence for computing the coefficients is numerically stable, and they carry no claim about series that use them."},{"id":"concept-gamma-function","slug":"gamma-function","name":"Gamma function","category":"Numerical methods","proofStatus":"definition","description":"Extend the factorial to complex arguments, and carry the identities that make it computable.","definition":"The gamma function extends the factorial off the integers. DLMF 5.5.1 states the recurrence that fixes its character: Gamma(z+1) equals z times Gamma(z), which reproduces the factorial on the positive integers and defines the function elsewhere by continuation. Two further relations do the work in practice. The reflection formula at 5.5.3, Gamma(z) times Gamma(1-z) equals pi over sin(pi z), converts an argument in one half-plane into one in the other and holds for z not a non-positive integer. The duplication formula at 5.5.5 relates Gamma(2z) to Gamma(z) and Gamma(z+1/2). Large arguments are handled by the Stirling expansion at 5.11.1, valid as z tends to infinity in the sector where the phase of z is at most pi minus delta.","notation":["Gamma(z)","psi(z) for the digamma function","B_2k for the Bernoulli numbers appearing in the Stirling series"],"assumptions":["The reflection formula at 5.5.3 requires z to be neither zero nor a negative integer.","The duplication formula at 5.5.5 requires 2z to be neither zero nor a negative integer, which is a stricter condition than the one on 5.5.3.","The Stirling expansion at 5.11.1 is stated for z tending to infinity within a sector bounded away from the negative real axis, so it says nothing about behaviour near the poles.","The recurrence at 5.5.1 is what carries the function off the positive integers; treating the factorial as the definition leaves the rest of the plane undefined."],"invariants":["Gamma(z+1) = z Gamma(z), DLMF 5.5.1","Gamma(z) Gamma(1-z) = pi / sin(pi z), DLMF 5.5.3","Gamma(2z) expressed through Gamma(z) and Gamma(z+1/2), DLMF 5.5.5"],"procedure":["Reduce an awkward argument using the recurrence at 5.5.1.","Move an argument across the half-plane with the reflection formula at 5.5.3, observing its excluded points.","For large argument inside the stated sector, use the Stirling expansion at 5.11.1 and truncate before the terms begin to grow."],"errorBounds":["The Stirling series at 5.11.1 is a Poincare asymptotic expansion, not a convergent series. Adding terms indefinitely makes the approximation worse, so useful accuracy is bounded by the smallest term rather than by the number of terms taken.","The digamma expansion at 5.11.2 carries the same sector restriction as 5.11.1, so neither says anything about the excluded region.","The points excluded by 5.5.3 are poles. The function is unbounded near zero and the negative integers, so an expansion valid for large argument gives no information there.","Reducing a large argument by repeated application of the recurrence at 5.5.1 accumulates one rounding error per step, so the reduction is not free."],"sourceIds":["nist-dlmf-gamma"],"relatedSlugs":["convergence-precision-and-error","numerical-integration"],"doesNotEstablish":"The chapter states identities and their conditions. It does not establish that a given library computes them to any stated accuracy, and an asymptotic expansion carries no error bound from the number of terms alone."},{"id":"concept-error-function","slug":"error-function-and-related-integrals","name":"Error function and related integrals","category":"Numerical methods","proofStatus":"definition","description":"Name the integrals of the Gaussian and Fresnel kernels, and keep their definitions exact.","definition":"The error function is defined at DLMF 7.2.1 as two over the square root of pi, times the integral of exp of minus t squared from zero to z. Its complement is defined at 7.2.2 as the same normalised integral taken from z to infinity, and that equation also states the identity that erfc z equals one minus erf z. Two relatives share the chapter. The Dawson integral, at 7.2.5, is exp of minus z squared times the integral of exp of t squared from zero to z. The Fresnel integrals at 7.2.7 and 7.2.8 integrate cosine and sine of pi t squared over two. All are recorded as entire functions of their complex argument.","notation":["erf z","erfc z","F(z) for the Dawson integral","C(z) and S(z) for the Fresnel integrals"],"assumptions":["The definitions are stated for complex argument, and the chapter records all of these as entire functions.","Because the integrands are entire, the integral from zero to z does not depend on the path taken between them.","The factor of two over the square root of pi in 7.2.1 is part of the definition; an unnormalised Gaussian integral is a different function.","The identity in 7.2.2 relating erfc to erf holds as stated and is not an approximation."],"invariants":["erf z = (2/sqrt(pi)) times the integral of exp(-t^2) from 0 to z, DLMF 7.2.1","erfc z = 1 - erf z, DLMF 7.2.2","F(z) = exp(-z^2) times the integral of exp(t^2) from 0 to z, DLMF 7.2.5"],"procedure":["Take the definition from 7.2.1 or 7.2.2 rather than from a normal-distribution table, so the normalisation is explicit.","Convert between erf and erfc with the identity in 7.2.2 rather than by subtracting rounded values.","For a tail probability, evaluate erfc directly rather than forming one minus erf, so the small quantity is never the difference of two near-equal numbers.","Before reading a Fresnel value from another reference, check whether it uses the pi t squared over two convention of 7.2.7 and 7.2.8."],"errorBounds":["Computing erfc as one minus erf loses significance when erf is close to one. The complementary form at 7.2.2 exists so that the small quantity can be computed directly.","The Dawson integral at 7.2.5 is written as a product of a decaying exponential and a growing one. The product stays bounded while the factors do not, so evaluating them separately overflows for moderate argument even though the function is well behaved.","The Fresnel integrals at 7.2.7 and 7.2.8 use the argument convention pi t squared over two. Other references omit that factor, and reading a value under the wrong convention rescales the argument.","These are definitions rather than algorithms, so they carry no accuracy statement for any particular evaluation method."],"sourceIds":["nist-dlmf-error-function"],"relatedSlugs":["uncertainty-propagation","numerical-integration"],"doesNotEstablish":"These are definitions of integrals. They do not establish that an observed distribution is Gaussian, and no statistical interpretation follows from the definition alone."},{"id":"mathematics-angle-normalization","slug":"angle-normalization","name":"Angle normalization","category":"Geometry and coordinates","proofStatus":"method","description":"Represent cyclic angles in one declared interval without losing the original frame or direction of travel.","definition":"Angle normalization maps equivalent directions that differ by integer turns into a chosen half-open interval, commonly [0°, 360°). The modulo operation makes comparison deterministic, while metadata must preserve units, reference axis, orientation, and any unwrapped value needed to distinguish repeated crossings.","notation":["θ̄ = ((θ mod 360°) + 360°) mod 360°","θ ∈ ℝ; θ̄ ∈ [0°, 360°)"],"assumptions":["The angular unit is declared.","The zero direction and positive orientation are fixed.","A half-open output interval is chosen."],"invariants":["θ and θ + 360°k represent the same direction.","Normalization is idempotent.","The output remains inside the declared interval."],"procedure":["Convert the input to the canonical angular unit.","Apply floor-modulo rather than language remainder for negative inputs.","Store unwrapped angle or crossing index when chronology matters."],"errorBounds":["Modulo does not reduce upstream coordinate uncertainty.","Values near 0° require boundary-aware tolerances.","Rounded display values must not replace full-precision inputs."],"sourceIds":["iau-sofa","nist-dlmf-numerical"],"relatedSlugs":["modular-arithmetic","spherical-coordinates","root-finding"],"doesNotEstablish":"A normalized longitude is geometry. A sign, mansion, aspect, or interpretive label derived from it remains convention-dependent and does not gain predictive validity from the normalization."},{"id":"mathematics-spherical-coordinates","slug":"spherical-coordinates","name":"Spherical coordinates","category":"Geometry and coordinates","proofStatus":"definition","description":"Represent directions on a sphere with radius, longitude-like angle, and latitude-like angle under an explicit convention.","definition":"Spherical coordinates represent a point by radial distance and two angular coordinates. Multiple conventions swap angle names or origins, so a reproducible record must declare axis orientation, longitude range, latitude or colatitude, frame, origin, epoch, and whether the position is geometric, astrometric, or apparent.","notation":["x = r cos φ cos λ","y = r cos φ sin λ","z = r sin φ"],"assumptions":["The coordinate origin and frame are fixed.","Angular conventions are explicit.","Singularities at the poles are handled."],"invariants":["Rotation preserves radial distance.","Cartesian conversion preserves the represented point within numerical error.","Longitude is undefined at a coordinate pole."],"procedure":["Declare the spherical convention and frame.","Convert to Cartesian form for robust transformations.","Convert back with quadrant-aware inverse trigonometric functions."],"errorBounds":["Longitude becomes ill-conditioned near the poles.","Parallax matters when the observer origin changes.","Rounding near classification boundaries can change labels."],"sourceIds":["iau-sofa","jpl-horizons"],"relatedSlugs":["angle-normalization","reference-frame-transformations"],"doesNotEstablish":"Spherical coordinates locate a direction. They do not supply physical explanation or astrological interpretation."},{"id":"mathematics-reference-frame-transformations","slug":"reference-frame-transformations","name":"Reference-frame transformations","category":"Geometry and coordinates","proofStatus":"method","description":"Transform the same physical direction between declared origins, axes, epochs, and correction states.","definition":"A reference-frame transformation maps coordinates between explicitly defined systems while preserving the represented physical direction to the accuracy of the transformation model. In astronomy this may include rotations, precession-nutation, Earth rotation, aberration, parallax, and origin changes in a specified sequence.","notation":["x_target = Rₙ ··· R₂R₁ x_source","RᵀR = I for an ideal rotation"],"assumptions":["Source and target frames are identified.","Epoch and timescale inputs are available.","Required physical corrections are declared."],"invariants":["An exact rotation preserves vector norm.","Round-trip transforms agree within tolerance.","Transformation order is part of the method."],"procedure":["Construct the source state and metadata.","Apply authoritative transformations in prescribed order.","Record target frame and verify a round trip."],"errorBounds":["Earth-orientation uncertainty can dominate topocentric work.","Approximate precession models have validity intervals.","Omitted corrections create systematic, not random, error."],"sourceIds":["iau-sofa","jpl-horizons"],"relatedSlugs":["angle-normalization","spherical-coordinates","uncertainty-propagation"],"doesNotEstablish":"Frame conversion can explain why two systems assign different labels to one direction; it cannot decide which symbolic tradition is true."},{"id":"mathematics-modular-arithmetic","slug":"modular-arithmetic","name":"Modular arithmetic","category":"Time and periodicity","proofStatus":"framework","description":"Reason about equivalence classes and repeating cycles without mistaking a wrapped label for elapsed distance.","definition":"Modular arithmetic identifies numbers that differ by integer multiples of a modulus. It is the natural grammar for clock time, angular cycles, weekday indices, lunar-limb boundaries, and ring buffers, but calculations that need elapsed turns must retain an unwrapped counter alongside the residue.","notation":["a ≡ b (mod n)","[a]ₙ = {a + kn : k ∈ ℤ}"],"assumptions":["The modulus is positive and fixed.","Residue convention is declared.","Wrapped and unwrapped quantities are distinguished."],"invariants":["Congruence is preserved by addition and multiplication.","Every integer has one canonical residue in a chosen complete system.","A residue alone does not reveal cycle count."],"procedure":["Choose the modulus from the actual cycle.","Compute a canonical residue.","Retain epoch or cycle index for ordering across wraps."],"errorBounds":["Boundary rounding can select the adjacent residue class.","A variable physical period cannot be modeled as fixed modulo without residual error.","Modulo comparisons need circular rather than linear distance."],"sourceIds":["nist-dads","iau-sofa"],"relatedSlugs":["angle-normalization","periodic-functions-and-phase","calendar-and-timescale-mappings"],"doesNotEstablish":"A repeating mathematical index does not establish that historical outcomes repeat or that a symbolic cycle causes events."},{"id":"mathematics-periodic-functions-and-phase","slug":"periodic-functions-and-phase","name":"Periodic functions and phase","category":"Time and periodicity","proofStatus":"framework","description":"Describe repeating variation through period, frequency, amplitude, and phase while testing whether periodicity is actually stable.","definition":"A periodic function repeats after a period T, while phase identifies position within that cycle. Fourier representations decompose suitable signals into sinusoidal components, but finite, noisy, drifting, or irregularly sampled observations require uncertainty estimates and tests against non-periodic alternatives.","notation":["f(t + T) = f(t)","φ(t) = 2πt/T + φ₀"],"assumptions":["The proposed period is stable over the analyzed interval.","Sampling can resolve the frequency.","Trend and seasonality are not conflated."],"invariants":["Phase is equivalent modulo 2π.","Frequency is the reciprocal of period.","A time shift produces a phase shift."],"procedure":["Inspect sampling and detrend only under a declared model.","Estimate candidate frequency and phase.","Validate out of sample and test aliases."],"errorBounds":["Aliasing can create false periods.","Short windows yield broad frequency uncertainty.","Phase drift invalidates fixed-period extrapolation."],"sourceIds":["nist-statistical-handbook","nist-dlmf-numerical"],"relatedSlugs":["modular-arithmetic","time-series-models","change-point-detection"],"doesNotEstablish":"Detecting periodic structure does not identify a celestial cause, and a visually aligned cycle is not evidence of forecast skill."},{"id":"mathematics-calendar-and-timescale-mappings","slug":"calendar-and-timescale-mappings","name":"Calendar and timescale mappings","category":"Time and periodicity","proofStatus":"method","description":"Map civil labels, atomic scales, rotational time, and ephemeris arguments without treating them as interchangeable.","definition":"A calendar date and clock reading are labels that map to an instant only with a calendar, timezone history, offset policy, and fold or gap resolution. Astronomical work then converts among UTC, TAI, TT, UT1, and other scales using leap-second and Earth-orientation data.","notation":["instant = resolve(calendar, local time, zone, fold)","TT = TAI + 32.184 s"],"assumptions":["Calendar system and timezone identifier are known.","Leap-second and Earth-orientation tables are versioned.","Ambiguous or nonexistent local times are resolved explicitly."],"invariants":["A resolved instant can be represented in multiple scales.","Timezone display changes do not change the instant.","Scale offsets follow their defining standards."],"procedure":["Resolve the civil timestamp to an instant.","Convert with authoritative scale tables.","Store input label, resolution decision, instant, scale, and data version."],"errorBounds":["Historical timezone records can be uncertain.","UT1 prediction degrades beyond measured Earth orientation.","Date-only records imply an interval, not an exact instant."],"sourceIds":["iau-sofa","jpl-horizons"],"relatedSlugs":["modular-arithmetic","uncertainty-propagation","cryptographic-commitments"],"doesNotEstablish":"A precise event time improves reproducibility but does not make a natal, corporate, or electional interpretation empirically valid."},{"id":"mathematics-interpolation","slug":"interpolation","name":"Interpolation","category":"Numerical methods","proofStatus":"method","description":"Estimate values between computed or measured samples under a declared local model and bounded domain.","definition":"Interpolation constructs an approximating function that agrees with known samples and estimates values between them. Polynomial, spline, rational, and trigonometric methods make different smoothness and stability assumptions; interpolation should not be silently extended into extrapolation.","notation":["p(xᵢ) = yᵢ","f(x) = p(x) + R(x)"],"assumptions":["The target lies inside the supported sample range.","Sampling resolves relevant variation.","The selected interpolant matches local smoothness."],"invariants":["The interpolant reproduces declared nodes within tolerance.","Units are preserved.","Method and node set determine the result."],"procedure":["Select bracketing samples.","Choose a stable interpolation family.","Estimate or test residual error against denser references."],"errorBounds":["High-degree equispaced polynomials can oscillate.","Sparse nodes miss sharp changes.","Interpolation error is separate from source-data error."],"sourceIds":["nist-dlmf-numerical","jpl-horizons"],"relatedSlugs":["root-finding","convergence-precision-and-error","time-series-models"],"doesNotEstablish":"Interpolation fills a numerical gap under a model; it does not create observations or justify extrapolated predictions."},{"id":"mathematics-root-finding","slug":"root-finding","name":"Root finding and event location","category":"Numerical methods","proofStatus":"method","description":"Locate times or states where a continuous residual crosses a target, with bracketing and convergence evidence.","definition":"Root finding solves f(x)=0 numerically. Bracketing methods preserve an interval containing a sign-changing root under continuity; open methods can converge faster but require stronger local conditions and may converge to an unintended root.","notation":["find x*: f(x*) = 0","aₖ ≤ x* ≤ bₖ"],"assumptions":["The residual is defined in the search region.","Continuity or differentiability matches the algorithm.","Repeated and tangent roots are considered."],"invariants":["A valid sign-change bracket retains at least one root for a continuous function.","Stopping criteria are declared in input and residual units.","Multiple crossings require separate brackets."],"procedure":["Define a continuous residual and scan for candidate brackets.","Refine each candidate with a safeguarded solver.","Verify residual, bracket width, direction, and duplicate handling."],"errorBounds":["Sampling can miss tangent or rapid roots.","Time error depends on local slope.","Ephemeris and timescale errors remain in the located event."],"sourceIds":["nist-dlmf-numerical","jpl-horizons"],"relatedSlugs":["interpolation","uncertainty-propagation","angle-normalization"],"doesNotEstablish":"A precisely located ingress, station, phase, or boundary is an event calculation; it does not prove an interpretation attached to that event."},{"id":"mathematics-numerical-integration","slug":"numerical-integration","name":"Numerical integration","category":"Numerical methods","proofStatus":"method","description":"Approximate accumulated quantity over an interval with a stated quadrature rule and convergence check.","definition":"Numerical integration approximates a definite integral from sampled function values. Accuracy depends on smoothness, interval partition, singularities, oscillation, and the chosen quadrature rule; adaptive methods allocate evaluations according to estimated local error.","notation":["I = ∫ₐᵇ f(x) dx","I ≈ Σ wᵢf(xᵢ)"],"assumptions":["The integral exists under the chosen definition.","The integrand can be evaluated accurately.","Discontinuities and singularities are isolated."],"invariants":["Additivity holds across exact subinterval partitions.","Units equal integrand units times integration-variable units.","Convergence should stabilize under refinement."],"procedure":["Partition at known discontinuities.","Apply an appropriate fixed or adaptive rule.","Repeat with tighter tolerance or an independent rule."],"errorBounds":["Quadrature estimates can fail on unresolved spikes.","Cancellation can hide large local errors.","Model and measurement uncertainty remain separate."],"sourceIds":["nist-dlmf-numerical","nist-gum"],"relatedSlugs":["convergence-precision-and-error","uncertainty-propagation","dynamical-systems"],"doesNotEstablish":"An accurately accumulated exposure or signal does not by itself identify causation or predictive usefulness."},{"id":"mathematics-convergence-precision-and-error","slug":"convergence-precision-and-error","name":"Convergence, precision, and error","category":"Numerical methods","proofStatus":"framework","description":"Separate approximation error, floating-point behavior, input uncertainty, and model discrepancy.","definition":"Convergence describes whether an approximation approaches a limiting value as resolution or iteration changes. Precision describes numerical representation or repeatability, while error is deviation from a reference quantity. Reproducible software must not collapse truncation, roundoff, measurement uncertainty, and model discrepancy into one number.","notation":["eₙ = xₙ − x*","|eₙ₊₁| ≤ C|eₙ|ᵖ"],"assumptions":["A target quantity and reference meaning are defined.","Stopping criteria are scale-aware.","Arithmetic and library versions are recorded."],"invariants":["More printed digits do not imply lower error.","Convergence to a value does not imply convergence to the correct model.","Tolerance is not identical to uncertainty."],"procedure":["Identify error sources before computation.","Run refinement and independent-reference checks.","Report precision, tolerance, residual, and uncertainty separately."],"errorBounds":["Catastrophic cancellation can dominate.","Ill-conditioned problems amplify tiny perturbations.","Unknown model error cannot be inferred from solver residual alone."],"sourceIds":["nist-dlmf-numerical","nist-gum"],"relatedSlugs":["interpolation","root-finding","uncertainty-propagation"],"doesNotEstablish":"Numerical agreement establishes implementation consistency only within tested conditions; it cannot validate a symbolic or causal claim."},{"id":"mathematics-uncertainty-propagation","slug":"uncertainty-propagation","name":"Uncertainty propagation","category":"Probability and statistics","proofStatus":"method","description":"Carry declared input uncertainty and covariance through a measurement or calculation model.","definition":"Uncertainty propagation estimates the distribution or standard uncertainty of an output produced by uncertain inputs. Linearized covariance propagation, interval methods, and Monte Carlo simulation answer different questions and require declared distributions, dependencies, and model equations.","notation":["Σᵧ ≈ JΣₓJᵀ","y = g(x)"],"assumptions":["Input uncertainty models are defensible.","Dependencies and covariance are represented.","The forward model covers material effects."],"invariants":["Units and covariance dimensions remain consistent.","Perfectly shared error must not be counted as independent.","Output uncertainty is conditional on the model."],"procedure":["Inventory uncertain inputs and correlations.","Select linearized, interval, or simulation propagation.","Check sensitivity and convergence, then report included components."],"errorBounds":["Linearization fails for strong nonlinearity or boundaries.","Unknown systematics remain outside the budget.","Monte Carlo sampling error must be quantified."],"sourceIds":["nist-gum","nist-statistical-handbook"],"relatedSlugs":["convergence-precision-and-error","reference-frame-transformations","calibration-and-reliability"],"doesNotEstablish":"A quantified uncertainty budget does not certify that omitted variables, interpretations, or causal assumptions are correct."},{"id":"mathematics-bayesian-updating","slug":"bayesian-updating","name":"Bayesian updating","category":"Probability and statistics","proofStatus":"framework","description":"Update a declared prior distribution with a likelihood generated by observed data.","definition":"Bayesian inference combines a prior distribution and likelihood to produce a posterior distribution. The result is conditional on the model, prior, data-generating assumptions, and observation process; posterior concentration does not protect against misspecification or leakage.","notation":["p(θ|D) ∝ p(D|θ)p(θ)","posterior ∝ likelihood × prior"],"assumptions":["Prior and likelihood are declared before evaluation.","The observation model reflects sampling and censoring.","Model comparison accounts for complexity."],"invariants":["The posterior normalizes to one.","Sequential updating is coherent for conditionally independent batches.","Changing the prior can change sparse-data conclusions."],"procedure":["Pre-register parameterization and priors.","Compute and diagnose the posterior.","Run prior sensitivity and posterior predictive checks."],"errorBounds":["Approximate inference adds computational error.","Misspecified likelihoods produce misleading certainty.","Selection bias is not removed by Bayes rule."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["calibration-and-reliability","proper-scoring-rules","causal-inference"],"doesNotEstablish":"Bayesian updating can measure how evidence changes belief under a model; it cannot turn repeated post-hoc astrological correlations into prospective evidence."},{"id":"mathematics-calibration-and-reliability","slug":"calibration-and-reliability","name":"Calibration and reliability","category":"Probability and statistics","proofStatus":"method","description":"Test whether stated probabilities agree with observed frequencies and remain stable across relevant groups and time.","definition":"A probabilistic forecaster is calibrated when events assigned probability p occur at approximately frequency p over an appropriate reference class. Reliability diagrams and calibration error summarize agreement, but calibration must be evaluated with discrimination, sample size, dependence, and subgroup stability.","notation":["P(Y=1 | p̂=p) ≈ p","calibration error = observed frequency − forecast probability"],"assumptions":["Forecasts are locked before outcomes.","Outcome definitions and horizons are stable.","Reference classes are large enough to estimate frequencies."],"invariants":["A constant base-rate forecast can be calibrated but uninformative.","Calibration depends on the evaluated population.","Retrospective relabeling invalidates the test."],"procedure":["Bin or smooth locked forecasts without viewing outcomes during design.","Compare predicted and observed frequencies with uncertainty.","Assess discrimination, sharpness, and subgroup drift."],"errorBounds":["Small bins create noisy estimates.","Adaptive binning can bias summaries.","Non-stationarity can make historical calibration stale."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["proper-scoring-rules","bayesian-updating","change-point-detection"],"doesNotEstablish":"Calibration alone does not show useful skill over a baseline, causation, or transportability to a new decision context."},{"id":"mathematics-proper-scoring-rules","slug":"proper-scoring-rules","name":"Proper scoring rules and multiplicity","category":"Probability and statistics","proofStatus":"method","description":"Score probabilistic forecasts honestly while controlling the many-comparisons problem created by large rule libraries.","definition":"A proper scoring rule gives an expected optimum when the forecaster reports its true probability distribution. Forecast evaluation must also account for multiplicity: searching many planets, windows, outcomes, and subgroups inflates false discoveries unless the analysis plan, correction, or held-out evaluation is fixed in advance.","notation":["Brier = (p − y)²","log score = −log p(y)"],"assumptions":["Forecast probabilities and outcomes are valid.","The score and baseline are selected before outcomes.","The family of tested hypotheses is declared."],"invariants":["Proper scores reward honest probabilities in expectation.","Lower Brier and log loss are better under their standard definitions.","Adding undisclosed tests changes the error budget."],"procedure":["Lock forecast, outcome, horizon, score, and baseline.","Compute paired score differences.","Report uncertainty, multiplicity controls, and all registered analyses."],"errorBounds":["Rare outcomes require large samples.","Log loss is sensitive to overconfident errors.","Repeated observations may violate independence."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["calibration-and-reliability","bayesian-updating","cryptographic-commitments"],"doesNotEstablish":"A score measures performance on a specified task; it does not prove a universal mechanism or justify claims beyond the registered population and horizon."},{"id":"mathematics-graph-theory","slug":"graph-theory","name":"Graph theory","category":"Systems and networks","proofStatus":"framework","description":"Represent entities and typed relationships as nodes and edges while keeping topology distinct from causality.","definition":"A graph consists of vertices connected by edges, optionally directed, weighted, temporal, or typed. Graph algorithms expose reachability, dependency, centrality, paths, and communities, but those structural properties inherit the meaning and quality of the encoded edges.","notation":["G = (V, E)","Aᵢⱼ = edge weight from i to j"],"assumptions":["Node and edge semantics are explicit.","Direction and time are represented where material.","Missing edges are not automatically negative evidence."],"invariants":["Isomorphic graphs preserve topology.","A directed path encodes reachability under edge semantics.","Graph metrics depend on graph construction."],"procedure":["Define a typed node and edge schema.","Build the graph from provenance-bearing records.","Run algorithms appropriate to edge meaning and validate sensitivity."],"errorBounds":["Incomplete graphs bias centrality.","Projection can erase edge types.","Correlation edges must not be rendered as causal links."],"sourceIds":["nist-dads"],"relatedSlugs":["causal-inference","constraint-satisfaction","information-theory"],"doesNotEstablish":"A knowledge graph connects claims and methods; visual proximity or centrality does not establish truth, causation, or predictive power."},{"id":"mathematics-dynamical-systems","slug":"dynamical-systems","name":"Dynamical systems","category":"Systems and networks","proofStatus":"framework","description":"Model how state evolves under explicit equations, parameters, inputs, and boundary conditions.","definition":"A dynamical system specifies a state space and a rule for evolution through continuous or discrete time. Stability, attractors, sensitivity, and bifurcations are properties of that model; applying them to a real system requires measured state variables and validated equations.","notation":["dx/dt = F(x,t;θ)","xₜ₊₁ = F(xₜ,uₜ)"],"assumptions":["State variables are sufficient for the intended scale.","Evolution law and boundary conditions are declared.","Parameter stability is tested."],"invariants":["State plus inputs determines modeled evolution.","Conserved quantities follow only from the equations.","Sensitivity depends on metric and initial uncertainty."],"procedure":["Define state, inputs, parameters, and observation model.","Estimate or derive the evolution rule.","Test trajectories and stability out of sample."],"errorBounds":["Chaotic sensitivity limits long-range point forecasts.","Unobserved state creates apparent noise.","Structural model error can dominate numerical error."],"sourceIds":["nist-dlmf-numerical","nist-statistical-handbook"],"relatedSlugs":["numerical-integration","time-series-models","change-point-detection"],"doesNotEstablish":"Calling a business, sky, or symbolic system dynamic does not supply equations or evidence that celestial variables drive its evolution."},{"id":"mathematics-time-series-models","slug":"time-series-models","name":"Time-series models","category":"Systems and networks","proofStatus":"method","description":"Model ordered observations while preserving trend, seasonality, dependence, interventions, and forecast origin.","definition":"A time series is an ordered sequence whose observations may depend on prior values, time-varying inputs, seasonality, and structural changes. Valid forecasting separates training from future evaluation and compares against simple persistence, seasonal, and base-rate baselines.","notation":["yₜ = f(yₜ₋₁,…,xₜ)+εₜ","forecast made at origin t₀"],"assumptions":["Timestamps and observation intervals are trustworthy.","Missingness and revisions are modeled.","Evaluation uses information available at forecast time."],"invariants":["Temporal order cannot be shuffled without changing the problem.","Lagged features must precede the target.","Backtests must reproduce historical information sets."],"procedure":["Define target, cadence, horizon, and forecast origin.","Split data chronologically and fit candidate models.","Evaluate against naive baselines across rolling origins."],"errorBounds":["Autocorrelation reduces effective sample size.","Revisions can leak future data.","Regime changes can invalidate fitted parameters."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["periodic-functions-and-phase","change-point-detection","proper-scoring-rules"],"doesNotEstablish":"A historical fit or attractive overlay is not a reliable forecast; prospective, leakage-free performance is required."},{"id":"mathematics-change-point-detection","slug":"change-point-detection","name":"Change-point detection","category":"Systems and networks","proofStatus":"method","description":"Detect candidate shifts in level, variance, trend, or model parameters with controlled false alarms.","definition":"Change-point methods search an ordered series for times where a statistical property changes. Offline methods segment a completed record; online methods monitor sequentially. Penalties, priors, minimum segment lengths, and detection delays determine what counts as a change.","notation":["τ = argmin segmented loss + penalty","H₀: θ₁ = θ₂; H₁: θ₁ ≠ θ₂"],"assumptions":["The changing property is defined.","Noise and dependence are modeled.","Detection thresholds are selected without outcome cherry-picking."],"invariants":["A detected point is conditional on model and threshold.","Online detection occurs after evidence accumulates.","Multiple candidate points consume an error budget."],"procedure":["Specify change type and minimum duration.","Fit null and segmented alternatives.","Validate with sensitivity, false-alarm simulation, and external evidence."],"errorBounds":["Gradual drift can resemble multiple abrupt changes.","Outliers can trigger false points.","Small samples make location uncertainty wide."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["time-series-models","calibration-and-reliability","periodic-functions-and-phase"],"doesNotEstablish":"A change near a transit or ingress does not establish correspondence beyond chance; event windows and controls must be pre-registered."},{"id":"mathematics-optimization","slug":"optimization","name":"Optimization","category":"Decisions and computation","proofStatus":"framework","description":"Choose feasible inputs that maximize or minimize a declared objective, with sensitivity to assumptions and constraints.","definition":"Optimization selects a decision variable to minimize or maximize an objective subject to constraints. A solution is only as meaningful as the objective, feasible set, data, and uncertainty model; mathematical optimality is not the same as safety, fairness, causal effectiveness, or business value.","notation":["minimize f(x) subject to g(x) ≤ 0","x* ∈ argminₓ f(x)"],"assumptions":["Objective and constraints represent the decision.","Feasibility can be evaluated.","Uncertainty and trade-offs are declared."],"invariants":["An optimum is relative to a feasible set and objective.","Adding constraints cannot improve a minimization optimum value.","Equivalent scaling should preserve the decision when modeled consistently."],"procedure":["Define variables, objective, constraints, and uncertainty.","Solve with an algorithm matched to structure.","Test feasibility, sensitivity, and alternative objectives."],"errorBounds":["Local methods may miss global optima.","Data uncertainty can reorder candidates.","Proxy objectives can induce harmful solutions."],"sourceIds":["nist-dads","nist-statistical-handbook"],"relatedSlugs":["constraint-satisfaction","proper-scoring-rules","causal-inference"],"doesNotEstablish":"An optimized electional time is optimal only under the encoded tradition rules and weights; it is not thereby proven to improve real outcomes."},{"id":"mathematics-constraint-satisfaction","slug":"constraint-satisfaction","name":"Constraint satisfaction","category":"Decisions and computation","proofStatus":"framework","description":"Find assignments that satisfy explicit hard rules while distinguishing them from preferences and evidence weights.","definition":"A constraint-satisfaction problem defines variables, domains, and constraints, then seeks assignments satisfying all hard constraints. Soft constraints and weighted preferences require a separate optimization or ranking layer so that exceptions and disagreements remain visible.","notation":["find x such that Cᵢ(x)=true for all hard constraints","xᵢ ∈ Dᵢ"],"assumptions":["Variables and domains are finite or searchable.","Hard and soft rules are distinguished.","Conflicts have a declared resolution policy."],"invariants":["Every returned assignment satisfies all active hard constraints.","An unsatisfiable core identifies jointly conflicting constraints.","Removing constraints cannot reduce the feasible set."],"procedure":["Compile rules into typed predicates.","Check consistency and extract conflicts.","Enumerate or optimize feasible assignments with a complete audit trail."],"errorBounds":["Natural-language rules may be mistranscribed.","Incomplete constraints create false feasibility.","Search cutoffs can hide valid assignments."],"sourceIds":["nist-dads"],"relatedSlugs":["optimization","graph-theory","formal-logic-and-rule-compilation"],"doesNotEstablish":"Faithfully satisfying traditional rules establishes internal consistency, not that those rules predict or cause the desired outcome."},{"id":"mathematics-information-theory","slug":"information-theory","name":"Information theory","category":"Decisions and computation","proofStatus":"framework","description":"Quantify uncertainty, coding cost, and predictive information without confusing compression with understanding.","definition":"Information theory measures uncertainty and dependence through quantities such as entropy, cross-entropy, and mutual information. Estimates depend on distributions, sample size, discretization, and conditioning; apparent information can arise from leakage or shared trends.","notation":["H(X) = −Σ p(x) log p(x)","I(X;Y) = H(Y) − H(Y|X)"],"assumptions":["Probability distributions are defined.","Sampling supports the estimator.","Conditioning variables prevent obvious confounding where possible."],"invariants":["Entropy is nonnegative for discrete variables.","Mutual information is symmetric and nonnegative.","Deterministic invertible recoding preserves information."],"procedure":["Define variables and estimation method.","Estimate against shuffled and simple baselines.","Use held-out data and report estimator bias."],"errorBounds":["High-dimensional estimates are sample hungry.","Binning changes estimates.","Mutual information does not identify causal direction."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["proper-scoring-rules","graph-theory","causal-inference"],"doesNotEstablish":"Statistical dependence between planetary features and outcomes does not establish a causal celestial mechanism or robust future utility."},{"id":"mathematics-cryptographic-commitments","slug":"cryptographic-commitments","name":"Cryptographic commitments","category":"Decisions and computation","proofStatus":"method","description":"Commit to canonical inputs, rules, and forecasts before outcomes while making later changes detectable.","definition":"A cryptographic commitment workflow canonicalizes a record, hashes the exact bytes, and binds the digest to an external timestamp or signed registry entry. Later disclosure can prove that the opened record matches the commitment, provided the canonicalization and identity protocol are preserved.","notation":["digest = SHA-256(canonical(record))","verify(opening) = committed digest"],"assumptions":["Canonical serialization is deterministic.","Hash and signature algorithms are appropriate.","Timestamp and identity controls are independently auditable."],"invariants":["Identical canonical bytes produce identical digests.","Any byte change should change the digest with overwhelming probability.","Verification does not require trusting the opening party."],"procedure":["Validate and canonicalize the full record.","Hash, sign, and timestamp the digest.","Publish the commitment before outcomes and preserve the opening bundle."],"errorBounds":["A commitment cannot reveal omitted fields.","Weak identity controls permit attribution disputes.","Hashing false data preserves false data."],"sourceIds":["fips-180-4","rfc-8785"],"relatedSlugs":["proper-scoring-rules","calendar-and-timescale-mappings","formal-logic-and-rule-compilation"],"doesNotEstablish":"Immutability prevents hindsight editing; it does not establish that the committed prediction is accurate or the underlying theory is valid."},{"id":"mathematics-formal-logic-and-rule-compilation","slug":"formal-logic-and-rule-compilation","name":"Formal logic and rule compilation","category":"Decisions and computation","proofStatus":"method","description":"Translate bounded source statements into typed conditions, conclusions, exceptions, and conflicts that can be audited.","definition":"Rule compilation maps a source-bounded statement into explicit predicates and outputs while retaining scope, exceptions, provenance, and disagreements. A compiler can test applicability and contradictions, but fidelity requires human source review and does not imply truth of the proposition.","notation":["conditions ∧ scope ∧ ¬exception → interpretation","rule = (predicate, output, provenance)"],"assumptions":["Source scope and translation are known.","Predicates preserve the source meaning.","Conflict policy is published."],"invariants":["A rule cannot fire outside its declared scope.","Every output retains its source and version.","Withheld rules remain visible to audit."],"procedure":["Extract bounded source claim and context.","Encode typed conditions, exceptions, and output.","Review fidelity, test fixtures, and conflict behavior."],"errorBounds":["Ambiguous language may resist deterministic encoding.","OCR and translation errors propagate.","Rule coverage can create false completeness."],"sourceIds":["nist-dads"],"relatedSlugs":["constraint-satisfaction","graph-theory","cryptographic-commitments"],"doesNotEstablish":"Faithful formalization establishes provenance and reproducibility only; the empirical status of an astrological proposition remains unvalidated until tested."},{"id":"mathematics-causal-inference","slug":"causal-inference","name":"Causal inference and counterfactuals","category":"Decisions and computation","proofStatus":"framework","description":"Distinguish prediction and association from claims about what would happen under an intervention.","definition":"Causal inference asks how an outcome would differ under alternative interventions, using a declared causal graph, identification assumptions, and study design. Randomization can identify effects under compliance and measurement conditions; observational analyses require stronger, testable and untestable assumptions.","notation":["ATE = E[Y(1) − Y(0)]","Y ⟂ T | X under conditional exchangeability"],"assumptions":["Treatment, outcome, and intervention are well defined.","Confounders required for identification are addressed.","Interference and selection are considered."],"invariants":["Association alone does not identify intervention effect.","Adjustment follows the causal graph rather than predictive importance.","A counterfactual contrast requires a target population."],"procedure":["Draw the assumed causal structure.","Choose a design and identification strategy.","Estimate effects with falsification and sensitivity analyses."],"errorBounds":["Unmeasured confounding may dominate.","Positivity failures prevent comparison.","Measurement and selection bias can reverse estimates."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["graph-theory","time-series-models","optimization"],"doesNotEstablish":"Unless celestial timing is manipulated or otherwise identified under a defensible design, predictive association must not be described as celestial causation."}],"bridges":[{"id":"bridge-panchanga-timing-lunisolar-intercalation","conceptId":"concept-calendrical-reconciliation","domain":"panchanga-timing","title":"Lunisolar intercalation","application":"Keep a month sequence built on the lunar phase cycle from drifting out of the tropical year, by inserting a whole month on a declared rule.","inputs":["mean synodic month","tropical year","a declared intercalation rule such as the Metonic"],"transformation":"Compare accumulated lunar months against elapsed tropical years and insert a month when the declared rule fires.","outputs":["an intercalation schedule","the residual error of the scheme","a computed month index"],"evidenceRole":"calculation","targetPath":"/knowledge/panchanga","limitations":"The arithmetic fixes a schedule, not an observance. A tradition may set its months by sighting rather than by computation, and where it does, the computed date is a prediction about a rule and not about the practice."},{"id":"bridge-astrology-traditions-lunar-seasonal-drift","conceptId":"concept-calendrical-reconciliation","domain":"astrology-traditions","title":"Seasonal drift of a lunar year","application":"Quantify how far a twelve-month lunar year falls short of the tropical year, and how long a purely lunar calendar takes to return to the same season.","inputs":["mean synodic month of 29.5305888531 days","tropical year of 365.2421896698 days","a twelve-month lunar year as the scheme under test"],"transformation":"Multiply the synodic month by twelve, subtract from the tropical year, and divide the year by the shortfall.","outputs":["a lunar year of 354.36707 days","a shortfall of about 10.875 days per year","a return period of about 33.6 years"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology","limitations":"Mean periods give a mean drift. An individual month departs from the mean, so the figure describes the scheme rather than any particular year, and it carries no claim about what a tradition does with the drift."},{"id":"bridge-astronomy-metonic-residual","conceptId":"concept-calendrical-reconciliation","domain":"astronomy","title":"Residual of the Metonic relation","application":"Show why nineteen years of lunar months nearly close, and by how much the approximation misses.","inputs":["235 mean synodic months","19 tropical years","the mean periods stated by the source"],"transformation":"Evaluate both products in days and take the difference.","outputs":["6939.688 days for 235 lunations","6939.602 days for nineteen tropical years","a residual near 0.087 days, about two hours, per nineteen years"],"evidenceRole":"calculation","targetPath":"/knowledge/astronomy/coordinates-reference-frames-and-sky-position","limitations":"The two periods are incommensurable, so the relation is an approximation that accumulates. The residual is arithmetic and says nothing about which calendars adopted the cycle or why."},{"id":"bridge-celestial-facts-calendar-family-choice","conceptId":"concept-calendrical-reconciliation","domain":"celestial-facts","title":"What a calendar chooses to track","application":"Separate the three calendar families by which period each keeps synchronised, since no scheme keeps both.","inputs":["the tropical year","the lunar phase cycle","a declared intercalation policy"],"transformation":"Classify by which period the scheme preserves and where it absorbs the mismatch.","outputs":["solar, lunar or lunisolar classification","which period the scheme preserves","the residual error each family carries"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/calculations/civil-time-to-utc","limitations":"A classification of arithmetic. Which family a community uses is a historical and religious fact about that community, and this bridge neither explains nor evaluates that choice."},{"id":"bridge-semiconductor-overlay-registration","conceptId":"mathematics-reference-frame-transformations","domain":"semiconductor","title":"Lithography overlay registration","application":"Represent translation, rotation, magnification, and distortion corrections between wafer and reticle coordinate systems.","inputs":["alignment marks","measured residual vectors","tool coordinate basis"],"transformation":"Fit a declared linear or higher-order registration model and preserve residuals.","outputs":["correction matrix","overlay residual map","condition diagnostics"],"evidenceRole":"measurement","targetPath":"/knowledge/processes/photolithography","limitations":"Higher-order wafer distortion and tool drift can violate a simple linear model."},{"id":"bridge-semiconductor-yield-spc","conceptId":"mathematics-calibration-and-reliability","domain":"semiconductor","title":"Yield learning and process control","application":"Compare measured defect and yield behavior with stable process limits and calibrated metrology.","inputs":["lot measurements","control limits","tool and recipe identifiers"],"transformation":"Estimate reliability and monitor departures from the qualified process distribution.","outputs":["control signals","calibration status","subgroup diagnostics"],"evidenceRole":"measurement","targetPath":"/knowledge/concepts/yield-learning-and-statistical-process-control","limitations":"Control limits detect distributional change; they do not identify the physical root cause."},{"id":"bridge-semiconductor-process-dependency-graph","conceptId":"mathematics-graph-theory","domain":"semiconductor","title":"Manufacturing dependency graph","application":"Connect process steps, equipment, materials, defects, metrology, suppliers, and downstream failure modes as typed edges.","inputs":["process nodes","typed dependencies","source records"],"transformation":"Build a directed provenance graph and compute paths only under declared edge semantics.","outputs":["dependency paths","critical interfaces","evidence gaps"],"evidenceRole":"formalization-only","targetPath":"/knowledge/maps/semiconductor-manufacturing-process-map","limitations":"Graph centrality is not proof of physical causation or commercial importance."},{"id":"bridge-semiconductor-thermal-integration","conceptId":"mathematics-numerical-integration","domain":"semiconductor","title":"Accumulated thermal exposure","application":"Integrate a time-varying temperature or power profile to compare qualified process exposure.","inputs":["temperature time series","time intervals","response model"],"transformation":"Apply declared quadrature over valid segments and propagate sensor uncertainty.","outputs":["integrated exposure","numerical error estimate","coverage gaps"],"evidenceRole":"physical-model","targetPath":"/knowledge/processes/thermal-oxidation-diffusion-and-furnace-processing","limitations":"Equal integrated exposure need not imply equal material response when kinetics are nonlinear."},{"id":"bridge-semiconductor-fab-change-points","conceptId":"mathematics-change-point-detection","domain":"semiconductor","title":"Tool and lot regime shifts","application":"Detect candidate shifts in defect rate, critical dimension, or equipment telemetry.","inputs":["ordered fab measurements","lot boundaries","maintenance events"],"transformation":"Compare stable and segmented process models under a preselected false-alarm policy.","outputs":["candidate shift times","location uncertainty","external-evidence checklist"],"evidenceRole":"measurement","targetPath":"/knowledge/concepts/semiconductor-metrology-and-defect-inspection","limitations":"A detected shift localizes a change but does not identify the responsible chamber, material, or recipe."},{"id":"bridge-semiconductor-package-optimization","conceptId":"mathematics-optimization","domain":"semiconductor","title":"Packaging trade-space optimization","application":"Explore thermal, electrical, mechanical, yield, cost, and supply constraints without collapsing them into one hidden score.","inputs":["design variables","constraint models","declared objectives"],"transformation":"Construct a constrained multi-objective optimization and report Pareto alternatives.","outputs":["feasible designs","trade-off frontier","sensitivity results"],"evidenceRole":"decision-method","targetPath":"/knowledge/processes/advanced-packaging-and-heterogeneous-integration","limitations":"A computed optimum depends on proxy models and cannot replace qualification evidence."},{"id":"bridge-semiconductor-metrology-uncertainty","conceptId":"mathematics-uncertainty-propagation","domain":"semiconductor","title":"Metrology uncertainty budget","application":"Propagate instrument, calibration, sampling, and model components into a reported process measurement.","inputs":["instrument readings","calibration covariance","sampling model"],"transformation":"Combine correlated uncertainty components through the measurement equation.","outputs":["measurand estimate","combined uncertainty","dominant sensitivities"],"evidenceRole":"measurement","targetPath":"/knowledge/concepts/semiconductor-metrology-and-defect-inspection","limitations":"Unmodeled systematic effects remain outside the reported budget."},{"id":"bridge-celestial-facts-longitude-normalization","conceptId":"mathematics-angle-normalization","domain":"celestial-facts","title":"Canonical celestial longitude","application":"Normalize continuous ecliptic longitude while retaining full precision, frame, and unwrapped motion.","inputs":["raw longitude","angular unit","reference frame"],"transformation":"Apply floor-modulo to the declared interval and retain provenance.","outputs":["canonical longitude","boundary distance","unwrapped companion value"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/longitude-normalization","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-celestial-facts-coordinate-frame","conceptId":"mathematics-reference-frame-transformations","domain":"celestial-facts","title":"Ecliptic and equatorial frame conversion","application":"Transform one physical direction between frame, epoch, origin, and correction conventions.","inputs":["source coordinates","time and epoch","frame conventions"],"transformation":"Apply versioned SOFA-compatible transformation sequence.","outputs":["target coordinates","round-trip residual","convention record"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/ecliptic-vs-equatorial","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-celestial-facts-timescale-map","conceptId":"mathematics-calendar-and-timescale-mappings","domain":"celestial-facts","title":"Civil time to ephemeris time","application":"Resolve a local civil label and map the instant to the scale required by an ephemeris.","inputs":["local date and time","IANA timezone","fold or gap policy"],"transformation":"Resolve the instant, then apply versioned leap-second and scale offsets.","outputs":["UTC instant","ephemeris-scale argument","resolution audit"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/civil-time-to-utc","limitations":"Historical zone uncertainty can be wider than numerical precision."},{"id":"bridge-celestial-facts-ephemeris-interpolation","conceptId":"mathematics-interpolation","domain":"celestial-facts","title":"Ephemeris state interpolation","application":"Estimate a body state between tabulated or integrated ephemeris samples.","inputs":["bracketing state vectors","target epoch","interpolation method"],"transformation":"Interpolate within the supported interval and compare against denser reference output.","outputs":["state estimate","method identifier","residual bound"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/ephemeris-versioning","limitations":"Interpolation cannot repair an incorrect ephemeris, timescale, or observer origin."},{"id":"bridge-celestial-facts-event-root","conceptId":"mathematics-root-finding","domain":"celestial-facts","title":"Ingress and station event location","application":"Locate when a continuous angular residual reaches a boundary or apparent speed reaches zero.","inputs":["ephemeris function","search interval","target residual"],"transformation":"Bracket and refine each crossing with direction and duplicate controls.","outputs":["event instant","crossing direction","time tolerance"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/timing/jupiter-station-reference","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-celestial-facts-calculation-uncertainty","conceptId":"mathematics-convergence-precision-and-error","domain":"celestial-facts","title":"Calculation conformance and precision","application":"Separate solver tolerance, floating-point precision, source uncertainty, and convention disagreement.","inputs":["implementation output","independent reference","declared tolerances"],"transformation":"Compare continuous values before classifications and attribute discrepancies.","outputs":["residuals","pass or explain verdict","disagreement category"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/precision-rounding-and-uncertainty","limitations":"Conformance within tolerance validates implementation behavior, not interpretive claims."},{"id":"bridge-celestial-facts-fact-digest","conceptId":"mathematics-cryptographic-commitments","domain":"celestial-facts","title":"Reproducibility digest","application":"Bind exact inputs, versions, conventions, and outputs to a deterministic record digest.","inputs":["canonical fact bundle","software version","calculation conventions"],"transformation":"Canonicalize the record and compute a standard cryptographic digest.","outputs":["digest","canonical payload","verification metadata"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/calculations/reproducibility-digests","limitations":"The digest proves byte-level integrity, not truth of the submitted time or location."},{"id":"bridge-astronomy-sky-separation","conceptId":"mathematics-spherical-coordinates","domain":"astronomy","title":"Angular separation on the celestial sphere","application":"Compute observer-relative separation between measured or modeled directions.","inputs":["two direction vectors","reference frame","epoch"],"transformation":"Convert to a common Cartesian frame and evaluate the stable great-circle separation.","outputs":["angular separation","frame record","uncertainty estimate"],"evidenceRole":"measurement","targetPath":"/knowledge/astronomy/coordinates-reference-frames-and-sky-position","limitations":"Angular proximity does not imply physical proximity."},{"id":"bridge-astronomy-astrometric-covariance","conceptId":"mathematics-uncertainty-propagation","domain":"astronomy","title":"Astrometric covariance propagation","application":"Carry correlated position and motion uncertainty to another epoch or frame.","inputs":["state estimate","covariance matrix","transformation Jacobian"],"transformation":"Propagate covariance through the declared astrometric model.","outputs":["transformed state","output covariance","sensitivity diagnostics"],"evidenceRole":"physical-model","targetPath":"/knowledge/astronomy/astrometry-parallax-and-proper-motion","limitations":"Linear covariance propagation may fail across nonlinear or multimodal uncertainty."},{"id":"bridge-astronomy-orbit-dynamics","conceptId":"mathematics-dynamical-systems","domain":"astronomy","title":"Orbital state evolution","application":"Evolve a physical state under a declared gravitational model and initial conditions.","inputs":["initial state","force model","integration interval"],"transformation":"Numerically integrate equations of motion with convergence checks.","outputs":["trajectory","state uncertainty","model version"],"evidenceRole":"physical-model","targetPath":"/knowledge/astronomy/orbits-gravity-and-ephemerides","limitations":"Long-horizon uncertainty includes both initial-state and force-model error."},{"id":"bridge-astronomy-light-curve-series","conceptId":"mathematics-time-series-models","domain":"astronomy","title":"Time-domain light-curve analysis","application":"Model ordered brightness observations with cadence, missingness, noise, and transient structure intact.","inputs":["timestamped flux","measurement uncertainty","survey cadence"],"transformation":"Fit temporal models using chronology-preserving validation.","outputs":["period or transient candidates","forecast residuals","selection function"],"evidenceRole":"measurement","targetPath":"/knowledge/astronomy/time-domain-and-multimessenger-astronomy","limitations":"Survey cadence and selection effects can create or hide apparent variability."},{"id":"bridge-astronomy-period-search","conceptId":"mathematics-periodic-functions-and-phase","domain":"astronomy","title":"Periodic-signal search","application":"Estimate candidate periods in irregular observations while testing aliases and phase stability.","inputs":["observation times","signal values","noise model"],"transformation":"Evaluate a declared period estimator against shuffled and simulated null data.","outputs":["candidate period","phase model","false-alarm assessment"],"evidenceRole":"measurement","targetPath":"/knowledge/astronomy/time-domain-and-multimessenger-astronomy","limitations":"A periodogram peak is not a physical mechanism and may be an alias."},{"id":"bridge-astronomy-model-graph","conceptId":"mathematics-graph-theory","domain":"astronomy","title":"Observation-to-model provenance graph","application":"Connect detector products, calibrations, measured quantities, model assumptions, and inferred claims.","inputs":["data products","calibration records","claims and citations"],"transformation":"Build typed provenance paths that keep observation and inference edges distinct.","outputs":["claim lineage","missing dependencies","model boundaries"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astronomy/evidence-uncertainty-and-model-comparison","limitations":"Complete lineage improves auditability but does not guarantee a model is correct."},{"id":"bridge-astronomy-model-comparison","conceptId":"mathematics-bayesian-updating","domain":"astronomy","title":"Model-conditioned parameter inference","application":"Update parameter distributions from calibrated observations under explicit physical likelihoods and priors.","inputs":["calibrated data","physical likelihood","parameter priors"],"transformation":"Compute posterior distributions and predictive diagnostics.","outputs":["posterior estimates","model checks","prior sensitivity"],"evidenceRole":"physical-model","targetPath":"/knowledge/astronomy/evidence-uncertainty-and-model-comparison","limitations":"Posterior precision is conditional on the selected model family."},{"id":"bridge-astrology-traditions-rule-predicates","conceptId":"mathematics-formal-logic-and-rule-compilation","domain":"astrology-traditions","title":"Source-bound interpretation predicates","application":"Compile a bounded passage into explicit chart conditions, scope, exceptions, and attributed output.","inputs":["source passage","tradition identifier","chart facts"],"transformation":"Encode reviewed predicates without adding unstated conditions.","outputs":["applicable rules","withheld rules","source lineage"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-astrology-traditions-rule-conflicts","conceptId":"mathematics-constraint-satisfaction","domain":"astrology-traditions","title":"Rule conflict and satisfiability audit","application":"Expose incompatible conditions, exceptions, and precedence policies across named traditions.","inputs":["typed rules","active chart facts","conflict policy"],"transformation":"Evaluate hard predicates and extract unsatisfied or conflicting cores.","outputs":["applicable set","conflict set","resolution trace"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/registry","limitations":"Logical consistency is not empirical confirmation."},{"id":"bridge-astrology-traditions-tradition-graph","conceptId":"mathematics-graph-theory","domain":"astrology-traditions","title":"Tradition and source lineage graph","application":"Connect rules to passages, editions, reviewers, variants, techniques, and report modules.","inputs":["source catalog","rule records","review records"],"transformation":"Create typed provenance edges without merging disagreements.","outputs":["lineage paths","variant clusters","coverage gaps"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology","limitations":"Historical influence and graph proximity do not prove predictive efficacy."},{"id":"bridge-astrology-traditions-zodiac-frame","conceptId":"mathematics-reference-frame-transformations","domain":"astrology-traditions","title":"Tropical and sidereal label comparison","application":"Show how one celestial direction receives different longitude labels under declared zero points.","inputs":["continuous direction","tropical origin","sidereal ayanāṁśa"],"transformation":"Apply each named frame conversion before any interpretation rules.","outputs":["parallel labels","offset record","boundary flags"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/tropical-vs-sidereal","limitations":"Coordinate conversion explains disagreement but cannot adjudicate symbolic correctness."},{"id":"bridge-astrology-traditions-house-boundaries","conceptId":"mathematics-angle-normalization","domain":"astrology-traditions","title":"House and aspect boundary classification","application":"Classify continuous angular geometry under a named house or aspect convention.","inputs":["continuous longitudes","house system","orb policy"],"transformation":"Normalize and classify only after preserving boundary distance.","outputs":["derived labels","boundary uncertainty","convention ID"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/calculations/house-cusp-boundaries","limitations":"Classification is convention-dependent and is not a measured physical property."},{"id":"bridge-astrology-traditions-rule-information","conceptId":"mathematics-information-theory","domain":"astrology-traditions","title":"Incremental rule information audit","application":"Measure whether a rule pack adds out-of-sample information beyond base rates and ordinary covariates.","inputs":["locked rule features","outcomes","baseline predictions"],"transformation":"Estimate held-out score or information gain with leakage controls.","outputs":["incremental information","uncertainty","null comparison"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"Dependence does not identify cause and must replicate prospectively."},{"id":"bridge-astrology-traditions-corporate-rules","conceptId":"mathematics-optimization","domain":"astrology-traditions","title":"Corporate-report decision boundary","application":"Separate reflective interpretation from operational actions, losses, and prohibited high-stakes uses.","inputs":["report claims","decision context","risk policy"],"transformation":"Apply explicit utility and safety constraints outside the interpretation compiler.","outputs":["reflective statements","withheld directives","escalation conditions"],"evidenceRole":"decision-method","targetPath":"/knowledge/astrology/corporate-mundane","limitations":"Astrological interpretation must not be the sole basis for investment, legal, employment, or safety decisions."},{"id":"bridge-panchanga-timing-limb-modulo","conceptId":"mathematics-modular-arithmetic","domain":"panchanga-timing","title":"Pañcāṅga limb indexing","application":"Convert continuous Sun–Moon geometry and weekday cycles into tithi, yoga, karaṇa, nakṣatra, and vāra indices.","inputs":["sidereal longitudes","local day boundary","index convention"],"transformation":"Apply declared modular divisions while preserving continuous parent values.","outputs":["limb indices","fractional progress","boundary distance"],"evidenceRole":"calculation","targetPath":"/knowledge/panchanga","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-panchanga-timing-limb-boundary","conceptId":"mathematics-root-finding","domain":"panchanga-timing","title":"Limb boundary time search","application":"Locate when a tithi, nakṣatra, yoga, or karaṇa boundary occurs.","inputs":["continuous limb phase","search window","ephemeris convention"],"transformation":"Bracket each modular crossing and refine its instant.","outputs":["boundary instant","direction","time tolerance"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/tithi","limitations":"A computed boundary does not establish auspiciousness."},{"id":"bridge-panchanga-timing-dasha-intervals","conceptId":"mathematics-uncertainty-propagation","domain":"panchanga-timing","title":"Daśā timing under uncertain birth time","application":"Propagate a birth-time interval through lunar mansion position and period-boundary calculations.","inputs":["birth-time interval","Moon longitude function","daśā convention"],"transformation":"Evaluate boundary ranges across the admissible input interval.","outputs":["period date ranges","boundary sensitivity","stable and unstable labels"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/timing/vimshottari-birth-balance-reference","limitations":"Timing ranges quantify input sensitivity, not predictive validity."},{"id":"bridge-panchanga-timing-local-day","conceptId":"mathematics-calendar-and-timescale-mappings","domain":"panchanga-timing","title":"Local sunrise and civil-day mapping","application":"Map an instant to a locality-specific calendrical day under explicit sunrise and timezone rules.","inputs":["observer location","timezone history","sunrise convention"],"transformation":"Resolve civil time and calculate the relevant local boundary.","outputs":["local day label","boundary instant","uncertainty flag"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/sunrise-day-boundary","limitations":"Polar and near-boundary cases require an explicit fallback convention."},{"id":"bridge-panchanga-timing-election-constraints","conceptId":"mathematics-constraint-satisfaction","domain":"panchanga-timing","title":"Muhūrta eligibility compiler","application":"Find candidate windows satisfying a named tradition’s hard conditions while exposing withheld and conflicting rules.","inputs":["candidate intervals","tradition rule pack","activity type"],"transformation":"Evaluate typed predicates and compute feasible interval intersections.","outputs":["eligible windows","failed conditions","conflicts"],"evidenceRole":"formalization-only","targetPath":"/knowledge/muhurta","limitations":"Eligibility means conformity to encoded tradition, not demonstrated outcome improvement."},{"id":"bridge-panchanga-timing-election-optimization","conceptId":"mathematics-optimization","domain":"panchanga-timing","title":"Transparent candidate-window ranking","application":"Rank feasible windows under published soft preferences without hiding rule trade-offs in one mystical score.","inputs":["feasible windows","declared preferences","uncertainty penalties"],"transformation":"Compute a multi-criteria ranking and retain component contributions.","outputs":["ranked windows","component scores","sensitivity"],"evidenceRole":"decision-method","targetPath":"/knowledge/muhurta","limitations":"Ranking optimizes encoded preferences only and must not be marketed as proven auspiciousness."},{"id":"bridge-panchanga-timing-transit-phase","conceptId":"mathematics-periodic-functions-and-phase","domain":"panchanga-timing","title":"Transit cycle and repeated crossing model","application":"Represent direct, retrograde, and return crossings without treating mean orbital periods as exact event schedules.","inputs":["unwrapped longitude series","target boundary","motion direction"],"transformation":"Track phase continuously and segment each distinct crossing.","outputs":["crossing sequence","cycle phase","retrograde loop markers"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/timing/jupiter-ingress-reference","limitations":"Cycle phase is descriptive geometry and does not imply repeated life outcomes."},{"id":"bridge-empirical-validation-forecast-lock","conceptId":"mathematics-cryptographic-commitments","domain":"empirical-validation","title":"Pre-outcome forecast commitment","application":"Lock hypotheses, features, rules, horizons, outcomes, and analysis plans before results are observed.","inputs":["forecast record","protocol version","identity and timestamp evidence"],"transformation":"Canonicalize, hash, sign, and publish the commitment.","outputs":["public digest","sealed payload","later verification bundle"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"Pre-registration prevents some hindsight bias but cannot guarantee adherence or data quality."},{"id":"bridge-empirical-validation-forecast-score","conceptId":"mathematics-proper-scoring-rules","domain":"empirical-validation","title":"Prospective forecast scoring","application":"Compare locked probabilistic forecasts with outcomes using a preselected proper score and paired baselines.","inputs":["locked probabilities","resolved outcomes","baseline forecasts"],"transformation":"Compute paired score differences with multiplicity and dependence controls.","outputs":["Brier or log score","baseline delta","confidence or credible interval"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"Results apply only to the registered task, population, and horizon."},{"id":"bridge-empirical-validation-forecast-calibration","conceptId":"mathematics-calibration-and-reliability","domain":"empirical-validation","title":"Probability calibration audit","application":"Test whether events forecast at a stated probability occur at that frequency prospectively.","inputs":["locked forecasts","binary outcomes","forecast strata"],"transformation":"Estimate reliability curves and uncertainty alongside discrimination.","outputs":["calibration curve","calibration error","subgroup stability"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"A base-rate forecaster may be calibrated without adding useful discrimination."},{"id":"bridge-empirical-validation-rule-pruning","conceptId":"mathematics-bayesian-updating","domain":"empirical-validation","title":"Bayesian rule-pruning protocol","application":"Update bounded rule-effect estimates while shrinking noisy factors and preserving null results.","inputs":["pre-registered rule features","objective outcomes","hierarchical prior"],"transformation":"Fit a versioned hierarchical model and run prior and posterior predictive checks.","outputs":["posterior effect distributions","shrinkage diagnostics","retained and pruned candidates"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"Pruning on one dataset can overfit; retained rules require prospective replication."},{"id":"bridge-empirical-validation-ordinary-periods","conceptId":"mathematics-time-series-models","domain":"empirical-validation","title":"Milestones plus ordinary non-event periods","application":"Evaluate celestial features across complete chronological exposure rather than selected memorable events.","inputs":["timestamped outcomes","ordinary comparison periods","features available at each origin"],"transformation":"Build rolling, leakage-free forecasts across the entire observation calendar.","outputs":["prospective predictions","baseline comparisons","coverage diagnostics"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"A single organization supplies limited, dependent observations and weak external validity."},{"id":"bridge-empirical-validation-regime-analysis","conceptId":"mathematics-change-point-detection","domain":"empirical-validation","title":"Outcome regime-change audit","application":"Detect business or measurement shifts independently before testing whether registered celestial windows add information.","inputs":["objective outcome series","operational covariates","detection protocol"],"transformation":"Identify candidate changes under a controlled false-alarm process.","outputs":["change windows","location uncertainty","non-celestial explanations"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"Temporal coincidence with a planetary event is not sufficient evidence of correspondence."},{"id":"bridge-empirical-validation-causal-boundary","conceptId":"mathematics-causal-inference","domain":"empirical-validation","title":"Prediction-versus-causation boundary","application":"Specify when a benchmark can support predictive skill and why it usually cannot identify celestial causation.","inputs":["study design","assignment mechanism","outcomes and covariates"],"transformation":"Map identification assumptions and test observable implications.","outputs":["supported claim type","unidentified paths","sensitivity analysis"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"Most observational celestial-timing studies can test incremental prediction, not physical causation."}],"sources":[{"id":"nist-dlmf-jacobian-elliptic","title":"DLMF Chapter 22: Jacobian Elliptic Functions","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/22","establishes":"Definitions of the Jacobian elliptic functions through theta functions, the nome and the argument scaling both expressed through the complete elliptic integral, and their analytic character as doubly periodic meromorphic functions with simple poles and simple zeros.","boundary":"The chapter defines the functions and their periodicity. It makes no accuracy claim about any evaluation, and the reality of the functions for real argument is stated only for modulus between zero and one."},{"id":"nist-dlmf-exponential-integral","title":"DLMF Chapter 6: Exponential, Logarithmic, Sine, and Cosine Integrals","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/6","establishes":"Integral definitions for the exponential integrals with their branch cuts and principal values, the entire complementary form and the relation carrying Euler constant, the logarithmic integral, and the sine and cosine integrals with which of them are entire.","boundary":"The chapter defines the functions and states where each is singular or requires a principal value. It relates the logarithmic integral to the exponential integral and makes no claim here about approximating any counting function."},{"id":"nist-dlmf-confluent","title":"DLMF Chapter 13: Confluent Hypergeometric Functions","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/13","establishes":"Kummer equation with its singular structure, the first solution as an everywhere-convergent series with the parameter values at which it fails to exist, the entire alternative form that removes them, the second solution with its branch point and asymptotic behaviour, and the parameter values at which the series terminates.","boundary":"The chapter defines the solutions and their analytic character. Convergence for all argument is not accuracy at all argument, and the chapter makes no claim about any evaluation scheme."},{"id":"nist-dlmf-legendre","title":"DLMF Chapter 14: Legendre and Related Functions","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/14","establishes":"Legendre equation and its associated form with the location and exponent pairs of the regular singularities, and which solution pairs are numerically satisfactory on which interval.","boundary":"A reference giving the equations, their singular structure and the recommended pairs by interval. It makes no accuracy claim about any evaluation scheme, and the recommendations carry parameter conditions that a caller must check."},{"id":"nist-dlmf-number-theory","title":"DLMF Chapter 27: Functions of Number Theory","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/27","establishes":"Unique factorisation into prime powers, the asymptotic law for the prime counting function, and definitions for the totient, divisor and Mobius functions.","boundary":"The chapter states the definitions and the asymptotic law. An asymptotic statement about the prime count bounds a ratio in a limit and predicts no individual value, and the chapter settles no open question about the distribution of primes."},{"id":"nist-dlmf-airy","title":"DLMF Chapter 9: Airy and Related Functions","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/9","establishes":"Airy equation, its standard solutions and their values at the origin in terms of the gamma function, and which solution pairs are numerically satisfactory in which region.","boundary":"A reference giving the solutions and their analytic character. Numerical satisfactoriness is tabulated by region rather than settled once, and the chapter makes no accuracy claim about any evaluation scheme."},{"id":"doggett-calendars","title":"Calendars and their History","publisher":"L. E. Doggett, University Science Books, hosted by NASA Goddard Space Flight Center","url":"https://eclipse.gsfc.nasa.gov/SEhelp/calendars.html","establishes":"The three calendar families and how each handles the mismatch between the lunar phase cycle and the tropical year, with mean values for the tropical year and synodic month, the Metonic relation of 235 lunations to nineteen years, and the Gregorian 400-year cycle.","boundary":"A historical and computational account of calendar arithmetic. It fixes mean periods and reconciliation schemes; it does not determine any observance, and the mean values it gives do not predict an individual month, which varies by up to several hours from the mean."},{"id":"nist-dlmf-elliptic","title":"DLMF Chapter 19: Elliptic Integrals","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/19","establishes":"Integral definitions for the Legendre elliptic integrals of the first, second and third kinds, their complete forms at quarter period, the domain conditions each requires, and the principal branch with its cuts.","boundary":"The chapter defines the integrals and states where they are singular. It does not establish the accuracy of any evaluation scheme, and the third kind requires a principal value at parameter values the definition otherwise excludes."},{"id":"nist-dlmf-incomplete-gamma","title":"DLMF Chapter 8: Incomplete Gamma and Related Functions","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/8","establishes":"Integral definitions for the lower and upper incomplete gamma functions, the identity that splits the complete gamma between them, the normalized pair and their sum, and the differing analytic structure of the two halves in the parameter.","boundary":"The chapter defines the functions and their relation. It does not establish the accuracy of any algorithm, and the identity that the normalized pair sums to one is exact rather than a numerical guarantee."},{"id":"nist-dlmf-zeta","title":"DLMF Chapter 25: Zeta and Related Functions","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/25","establishes":"The Dirichlet series and Euler product for the Riemann zeta function with their shared half-plane of validity, the single pole of the continued function, and the Laurent expansion about it.","boundary":"The chapter states the representations and the analytic structure. The series and the product hold only where the real part exceeds one, and nothing here settles any open question about the zeros."},{"id":"nist-dlmf-hypergeometric","title":"DLMF Chapter 15: Hypergeometric Function","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/15","establishes":"The Gauss series with its unit disk of convergence, the parameter values for which it is undefined, the regularized form that removes those exceptions, the termination condition, and the three convergence regimes on the boundary circle.","boundary":"A definition with a declared principal branch and cut. It does not establish that a given numerical continuation outside the disk is accurate, and the boundary behaviour is stated by regime rather than pointwise."},{"id":"nist-dlmf-asymptotics","title":"DLMF Chapter 2: Asymptotic Approximations","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/2","establishes":"Definitions for asymptotic equality and the order symbols, the definition of a Poincare asymptotic expansion, and the statement that such an expansion does not determine the function it describes.","boundary":"The chapter defines what an asymptotic statement means. It carries no claim that any particular expansion is accurate at a particular argument, and it records explicitly that distinct functions can share one expansion."},{"id":"nist-dlmf-orthogonal-polynomials","title":"DLMF Chapter 18: Orthogonal Polynomials","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/18","establishes":"Orthogonality conditions in the continuous, discrete and measure-theoretic settings with the positivity and moment conditions attached, and the two standard three-term recurrence forms together with the positivity that makes the converse hold.","boundary":"The chapter states the conditions and the recurrences. It does not establish that evaluating a recurrence in finite precision is stable, and orthogonality is defined against a declared weight rather than being a property a polynomial family has on its own."},{"id":"nist-dlmf-bessel","title":"DLMF Chapter 10: Bessel Functions","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/10","establishes":"Bessel equation with its singularity structure, the series definition of the first-kind solution, the construction of the second-kind solution, and which solution pairs are numerically satisfactory in which region.","boundary":"A reference states the solutions and their analytic structure. It does not choose a pair for a particular computation: linear independence and numerical satisfactoriness are different properties, and the reference tabulates the second by region rather than asserting one pair is always right."},{"id":"nist-dlmf-bernoulli","title":"DLMF Chapter 24: Bernoulli and Euler Polynomials","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/24","establishes":"Generating-function definitions for the Bernoulli and Euler numbers and polynomials, each with the radius in which the generating series converges, and the vanishing and sign rules for the odd and even indices.","boundary":"The generating functions converge only inside the stated radii, so they define the coefficients without providing a usable expansion outside those disks, and the chapter makes no claim about the numerical stability of any recurrence used to generate them."},{"id":"nist-dlmf-gamma","title":"DLMF Chapter 5: Gamma Function","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/5","establishes":"Numbered functional relations for the gamma and digamma functions with the conditions each requires, and the Stirling asymptotic expansions with their sector of validity.","boundary":"A reference states identities at the scope its conditions declare. The reflection formula excludes the non-positive integers, and the Stirling series is a Poincare asymptotic expansion rather than a convergent one, so taking more terms eventually makes the approximation worse."},{"id":"nist-dlmf-error-function","title":"DLMF Chapter 7: Error Function, Dawson Integral, and Fresnel Integrals","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/7","establishes":"Integral definitions for the error function, the complementary error function, the Dawson integral and the Fresnel integrals, and the identity relating erf to erfc.","boundary":"The chapter defines the functions and their relations. It does not establish that any numerical implementation attains a stated accuracy, and it carries no claim about statistical interpretation in an applied setting."},{"id":"nist-dlmf-numerical","title":"DLMF Chapter 3: Numerical Methods","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/3","establishes":"Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.","boundary":"A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation."},{"id":"nist-statistical-handbook","title":"NIST/SEMATECH e-Handbook of Statistical Methods","publisher":"National Institute of Standards and Technology","url":"https://www.itl.nist.gov/div898/handbook/","establishes":"Methods for uncertainty analysis, calibration, time-series modeling, process monitoring, experimental design, reliability, and statistical comparison.","boundary":"Statistical procedures quantify evidence under a design and model; they do not repair biased sampling, outcome leakage, post-hoc hypotheses, or unmeasured confounding."},{"id":"nist-gum","title":"NIST Technical Note 1297: Guidelines for Evaluating and Expressing Measurement Uncertainty","publisher":"National Institute of Standards and Technology","url":"https://www.nist.gov/pml/nist-technical-note-1297","establishes":"A measurement framework for identifying uncertainty components, combining standard uncertainties, and reporting expanded uncertainty with declared coverage.","boundary":"Reported uncertainty describes the measurement model and included components. It is not a guarantee that all systematic errors or model inadequacies were found."},{"id":"iau-sofa","title":"Standards of Fundamental Astronomy","publisher":"International Astronomical Union","url":"https://www.iausofa.org/","establishes":"Authoritative algorithms and conventions for astronomical timescales, Earth orientation, reference systems, astrometry, and celestial-coordinate transformations.","boundary":"SOFA standardizes astronomical computation. It supplies no astrological symbols, meanings, auspiciousness judgments, or evidence of predictive validity."},{"id":"jpl-horizons","title":"JPL Horizons System","publisher":"NASA Jet Propulsion Laboratory","url":"https://ssd.jpl.nasa.gov/horizons/","establishes":"Ephemeris products and documented observer, target, timescale, coordinate, and output conventions for reproducible Solar System state and observable calculations.","boundary":"Ephemeris agreement validates positions under declared conventions; it does not validate downstream symbolic classifications or interpretations."},{"id":"nist-dads","title":"Dictionary of Algorithms and Data Structures","publisher":"National Institute of Standards and Technology","url":"https://xlinux.nist.gov/dads/","establishes":"Reference vocabulary for graphs, optimization, search, data structures, complexity, and computational methods used to make algorithms explicit.","boundary":"A formal data structure can represent domain relationships without establishing that the represented causal or interpretive relationships are true."},{"id":"fips-180-4","title":"FIPS PUB 180-4: Secure Hash Standard","publisher":"National Institute of Standards and Technology","url":"https://csrc.nist.gov/pubs/fips/180-4/upd1/final","establishes":"Standard secure hash algorithms that map messages to fixed-length digests and support integrity controls when used in an appropriate protocol.","boundary":"A digest can detect changed bytes relative to a committed value; it does not prove that the original input was truthful, complete, or independently observed."},{"id":"rfc-8785","title":"RFC 8785: JSON Canonicalization Scheme","publisher":"RFC Editor","url":"https://www.rfc-editor.org/rfc/rfc8785","establishes":"A deterministic JSON representation suitable for repeatable hashing and signing through constrained serialization and property ordering.","boundary":"Canonicalization makes equivalent data serialize consistently. It does not provide authentication, secrecy, timestamp authority, or semantic correctness by itself."}]}