Numerical Tables: KEVOS Sourcing Policy

Executive summary

Method is durable; catalogue data has a shelf life

KEVOS separates two kinds of knowledge. Method — definitions, algorithms, complexity results, decision criteria — is durable and belongs in the library. Catalogue data — tables of class numbers, fundamental units, curve coefficients — is a snapshot: it can be extended, corrected and superseded, and transcription introduces silent corruption. This library therefore documents how each table is computed and interpreted, and points to a live authoritative source for the values themselves.

Learning objectives

State the two-layer architecture and the reasoning behind it.
Identify the risks specific to transcribing numeric tables.
Locate the current authoritative source for each class of data.
Verify a value obtained from an external database.

Section 01The two-layer architecture

Layer 1Method — held in this library

Definitions, algorithms with their preconditions, complexity results, failure modes, verification procedures and selection criteria. These do not expire: the Round 2 algorithm works the same today as when it was published.

Layer 2Catalogue data — sourced externally

Class number tables, unit and regulator tables, curve tables, discriminant lists. These are extended and corrected continually, and any static copy begins going out of date immediately.

The rule

If a reader needs to know how a quantity is computed and what it means, the answer is here. If they need the value for a specific discriminant, the answer is a query against a live database — and this library says which one and how to interpret what comes back.

Section 02Why transcription is the wrong choice

Risk

Silent corruption

Digit substitution, dropped decimal points and collapsed columns produce values that are plausible and wrong. A corrupted class number carries no diagnostic signal — it is simply an integer of about the right size.

Risk

Data currency

Tables computed decades ago predate later corrections, extended ranges and improved certification. A value that was conditional then may be unconditional now, or may have been revised.

Risk

Loss of provenance

A transcribed number loses its conditionality, its precision and the algorithm that produced it. A regulator without its certified precision cannot be compared with anything.

Risk

Coverage

Any printed table stops somewhere. A live database has been extended enormously beyond the ranges that could be printed, and continues to grow.

Risk

Licensing

Substantial tables from published works are protected. Republishing them is neither necessary nor permissible.

Benefit

Linking instead

A pointer to a live source is always current, carries full provenance and metadata, and usually offers far more than the original table did.

The characteristic failure of table transcription

An error in a numeric table does not announce itself. It propagates into every downstream calculation, each of which succeeds and produces a well-formed result. This is the same failure mode that governs the rest of the subject, and it is the reason KEVOS treats catalogue data as a sourcing question rather than a content question.

Section 03Where to source each class of data

Authoritative sources by data type
Data	Source	Notes
Class numbers and class groups of quadratic fields	LMFDB; PARI/GP `quadclassunit`	Computed on demand for arbitrary discriminants, with conditionality reported
Number field tables by degree and signature	LMFDB number field database	Includes defining polynomial, discriminant, class group, unit rank and Galois group
Fundamental units and regulators	PARI/GP `bnfinit`; Magma	Report the precision and the proof flag with the value
Elliptic curves over ℚ	LMFDB; the Cremona database	Indexed by conductor; includes rank, torsion, Tamagawa numbers and generators
Modular forms and L-functions	LMFDB	Cross-referenced to the associated curves and fields
Factorisations of numbers of special form	The Cunningham Project tables	Maintained continuously; the reference for r^e ± s
Prime tables and records	The PrimePages and associated record lists	Includes certificates for record primes

Compute rather than look up, where possible

For a single discriminant it is usually faster and more reliable to compute the invariant directly in PARI/GP than to search a table — and the computation reports its own conditionality, which a table lookup may not.

Section 04Verifying an external value

Stage 01Record the queryNote the database, the version or access date, and the exact identifier used.
Stage 02Check the conventionsConfirm the discriminant convention, the equivalence convention for forms, and whether the regulator includes any index factor.
Stage 03Recompute independentlyRecompute the value in a system with an independent code path, and compare.
Stage 04Apply a structural checkVerify the fundamental identity for prime decomposition, the analytic class number formula for h and R, or Mazur's theorem for a torsion subgroup.

A structural check is worth more than a second lookup

Two databases may share an upstream source, so agreement between them is weaker evidence than it appears. A check against an independent mathematical identity — hR against the analytic formula, or ∑e_if_i = n — tests the value itself rather than its provenance.

ReferenceFrequently asked questions

Does this policy apply to worked examples?

No. Small illustrative examples computed and verified within a page are method, not catalogue data, and they belong here. The policy concerns systematic tables of values that a reader would otherwise treat as a reference dataset.

What if an external source disagrees with a computation?

Check conventions first, since most disagreements are convention mismatches rather than errors. If a genuine discrepancy survives, apply a structural check to determine which value is wrong, and report it to the database maintainers — corrections are how these resources stay reliable.

Why not cache a snapshot for offline use?

A cache is acceptable provided it records its retrieval date and is treated as a snapshot rather than as authority. The failure this policy guards against is a table that has lost its provenance and is subsequently trusted as though it were current.

NavigateContinue in this stream

Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Numerical Tables: KEVOS Sourcing Policy. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Numerical Tables: KEVOS Sourcing Policy as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—data, tables, number, numerical, policy—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Numerical Tables: KEVOS Sourcing Policy?