Exact mathematics
CDERE
Research Public research sandbox at indivisa.science.
Many equations in physics and engineering are classical equations in disguise: Bessel, Airy, hypergeometric, Heun. CDERE takes an equation with rational coefficients, sees through the disguise using exact arithmetic, and tells you which family it belongs to, with a certificate you can check.
- Status
- Research: public sandbox, open to try
- Try it
- indivisa.science
- Input
- y″ + P(z) y′ + Q(z) y = 0, with P and Q rational in z
- Engine
- Julia, exact rational arithmetic, no floating point in decisions
Known answer
Input P(z) = 0 Q(z) = −z
y″ − z·y = 0
Result Airy equation · certificate issued
What it does
You give CDERE the two coefficients P(z) and Q(z) of a second-order linear equation. It locates every singular point, computes its local data exactly, and matches the result against classical families. When the family is fully supported, it issues an algebraic certificate and closed-form solutions in terms of that family's special functions.
Recognized families include Gauss hypergeometric (₂F₁), Kummer, Heun and confluent Heun, Airy, Bessel, Hermite and Euler–Cauchy. Doubly-, bi- and triconfluent Heun equations are recognized at the classification level only.
Four verdicts, used consistently
- Verified
- A certificate was issued and every check it attempted passed. This is a machine-checkable claim.
- Classified
- The family was identified, but certification for it is outside the current scope. This is not a failure.
- Gap
- A capability that should exist and does not yet. Gaps are tracked and meant to shrink to zero.
- Out of scope
- The equation lies mathematically outside what rational-coefficient methods can reach. This is permanent and is not a defect.
CDERE says which of the four applies to every result. It does not round a gap up into a success.
Who it helps
- Researchers who meet an unfamiliar equation and want to know whether it is a known one in disguise.
- Students and teachers working through special functions, with exact singularity data to check by hand.
- Anyone cross-checking a computer-algebra result against an independent, exact answer.
Why exact arithmetic matters
Classification turns on equalities: whether an exponent difference is an integer, whether a resonance term vanishes. Floating-point arithmetic can only say that a number is close to zero, and close is the wrong answer to an exact question. CDERE keeps every decision in exact fields (the rationals, extended by quadratic surds or by i where needed), so the same input always gives the same verdict.
Technical detail
- Local analysis
- Frobenius exponents at each regular singular point, irregular rank at irregular points, and the resulting singularity signature used for dispatch.
- Heun class
- Exact detection of polynomial solutions and of the logarithmic case at each regular singular point, through a resonance test evaluated in exact arithmetic. Accessory-parameter conditions that clear a logarithm are reported as findings.
- Validation
- Equations from black-hole perturbation theory, including the Teukolsky (s = 0) and Regge–Wheeler (s = 2) equations, are recognized as confluent Heun with the singularity data predicted by hand.
- Known limits
- Some equations that reduce to a supported family only through an exponential change of gauge do not yet receive a certificate. The sandbox reports those cases as not verified.
Research: Fuchs-11 and p-curvature
Alongside the engine, Indivisa is studying invariants of the p-curvature of Fuchsian differential operators: an object that appears when an operator's coefficients are reduced modulo a prime p. The working example is an eleven-point Fuchsian family (Fuchs-11). The question is whether the characteristic polynomial of the p-curvature gives a useful, gauge-invariant algebraic fingerprint of an equation.
This is research. It is not a product, not a hash function and not a cryptographic primitive. An earlier proposal to use these invariants as a hash was reviewed and rejected, and the work now treats them only as objects of mathematical interest.