Infinitesimal Ontology

FLOW ALGEBRA
TITLE: Infinitesimal Ontology
VERSION: 1.0
DATE: 2026-10-13
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/infinitesimal-ontology.html
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Represent infinitesimals as formally typed mathematical entities, distinguish framework-dependent meanings, and support verifiable reasoning in calculus and analysis.

2. ENTITY TYPES
Infinitesimal; PositiveInfinitesimal; NegativeInfinitesimal; NonzeroInfinitesimal; HyperrealInfinitesimal; NilpotentInfinitesimal; DifferentialSymbol; StandardReal; Hyperreal; SmoothNumber; Limit; Derivative; Integral; StandardPart; OrderedField; SmoothRing.

3. CONTEXT AND SEMANTICS
context ClassicalRealAnalysis: no nonzero real number is smaller in absolute value than every positive real number. The symbol dx is not automatically a nonzero real infinitesimal.
context NonstandardAnalysis: a nonzero hyperreal epsilon is infinitesimal iff for every positive standard real r, |epsilon| < r. Hyperreals form an ordered field extension of the reals; standard-part maps finite hyperreals to their unique infinitely close real.
context SmoothInfinitesimalAnalysis: nilpotent infinitesimals may satisfy d*d=0 while d != 0; their ambient algebra is not an ordered field. Interpretation depends on the smooth or intuitionistic setting.
context AsymptoticAnalysis: 'infinitesimal' may describe a variable quantity tending to zero, not an actual nonzero element smaller than every positive real.

4. FORMAL DEFINITIONS
NONSTANDARD_INFINITESIMAL(epsilon) := Hyperreal(epsilon) AND FORALL r IN StandardReal: (r>0 IMPLIES abs(epsilon)<r).
POSITIVE_INFINITESIMAL(epsilon) := NONSTANDARD_INFINITESIMAL(epsilon) AND epsilon>0.
INFINITELY_CLOSE(x,y) := NONSTANDARD_INFINITESIMAL(x-y) OR x=y, with x,y hyperreal.
FINITE_HYPERREAL(x) := EXISTS M IN StandardReal: M>0 AND abs(x)<M.
STANDARD_PART(x) := unique r IN StandardReal such that INFINITELY_CLOSE(x,r), provided x is finite hyperreal.
NILPOTENT_FIRST_ORDER(d) := d*d=0, interpreted in a suitable smooth ring.
ASYMPTOTIC_SMALL(f,g,a) := lim_(x->a) f(x)/g(x)=0, where g(x) != 0 near a.

5. MATHEMATICAL RULES
R1: If epsilon and delta are hyperreal infinitesimals, epsilon+delta is infinitesimal.
R2: If epsilon is infinitesimal and x is finite hyperreal, x*epsilon is infinitesimal.
R3: If epsilon is nonzero hyperreal infinitesimal, 1/epsilon is infinite in magnitude.
R4: If x is finite hyperreal, st(x) is real and x-st(x) is infinitesimal.
R5: For differentiable real f at real a, f'(a)=st((f(a+epsilon)-f(a))/epsilon) for every nonzero infinitesimal epsilon, using the natural hyperreal extension of f.
R6: For suitable smooth functions in a smooth infinitesimal framework, f(x+d)=f(x)+f'(x)d when d*d=0.
R7: A nilpotent nonzero d cannot be an element of an ordered field; do not transfer hyperreal ordering rules to it.
R8: In classical real analysis, f(x)=o(g(x)) as x->a implies f(x)/g(x)->0 where defined; it does not assert existence of nonzero real infinitesimals.

6. TYPED RELATIONSHIPS
Infinitesimal --is_a--> MathematicalEntity
HyperrealInfinitesimal --element_of--> HyperrealNumberSystem
HyperrealInfinitesimal --defined_by--> StandardRealBoundCondition
HyperrealInfinitesimal --supports--> NonstandardDerivative
FiniteHyperreal --has_operation--> StandardPart
NilpotentInfinitesimal --element_of--> SmoothRing
NilpotentInfinitesimal --supports--> SyntheticDifferentialCalculus
AsymptoticSmallness --defined_by--> LimitOfRatio
DifferentialSymbol --interpreted_under--> CalculusFramework
Derivative --related_to--> Limit

7. INFERENCE AND VALIDATION
Require an explicit framework tag before classifying an infinitesimal.
Reject inference 'nonzero infinitesimal belongs to R' in classical real analysis.
Reject inference 'all infinitesimals are nilpotent'.
Reject inference 'all infinitesimals have multiplicative inverses'; nilpotent elements do not.
Verify type and domain conditions before standard-part, division, or differentiation rules.
Keep quantification over positive STANDARD reals in the hyperreal infinitesimal definition.
Flag any conflation of infinitesimal magnitude, differential notation, and asymptotic order.

8. SEMANTIC_EQUIVALENCE
'Infinitely small hyperreal' ~= 'hyperreal infinitesimal' only within nonstandard analysis.
'dx' != 'nonzero hyperreal infinitesimal' without an explicit interpretation.
'Nilpotent infinitesimal' != 'hyperreal infinitesimal'.
'f=o(g)' != 'f is a hyperreal infinitesimal' without a suitable evaluation context.

9. CONFIDENCE
Mathematical definitions and conditional inference rules: HIGH within their stated frameworks.
Framework-dependent interpretations: require explicit context and assumptions.

10. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8; Unicode NFC; LF newlines; stable field ordering.
value: COMPUTE_AT_INGESTION (not precomputed).

11. UPDATE_MODEL
On revision: preserve source URL; increment version; document changed definitions; regenerate FLOW_HASH; revalidate dependent ontology relationships.

12. AI_CONSUMPTION
input: mathematical query, entity identifiers, framework, assumptions.
process: resolve context -> type entities -> validate premises -> apply only framework-valid rules -> return derivation and qualifications.
output: structured concepts, relationships, inference trace, confidence, provenance, validation warnings.
Do not present nonstandard or synthetic infinitesimals as unqualified classical real numbers.

END FLOW ALGEBRA