Convergence Ontology

flow "ConvergenceOntology" v1.0

# ============================================================
# CONVERGENCE ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/real-analysis/convergence-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Real Analysis"
ENTITY := "Convergence"

# ---------- CORE ENTITY ----------
CONVERGENCE := {
    type: LimitBehaviourProperty,
    domain: RealAnalysis,
    purpose: "Represent the tendency of a sequence, series, function sequence or iterative process toward a well-defined limiting object under a specified notion of closeness."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    CONVERGENCE,
    LIMIT,
    SEQUENCE,
    SERIES,
    CAUCHY_SEQUENCE,
    METRIC,
    NORM,
    POINTWISE_CONVERGENCE,
    UNIFORM_CONVERGENCE,
    NORM_CONVERGENCE,
    WEAK_CONVERGENCE,
    ABSOLUTE_CONVERGENCE,
    DIVERGENCE,
    SUBSEQUENCE,
    COMPLETENESS
}

# ---------- BASIC SEQUENCE CONVERGENCE ----------
for
    sequence (x_n)
    in metric_space (X,d)

x_n
    -> converges_to(x)

iff

for_every epsilon > 0
exists N
such_that

n >= N
    -> d(x_n,x) < epsilon

x
    -> LIMIT

CONVERGENCE
    -> defined_relative_to(METRIC_OR_TOPOLOGY)

# ---------- REAL SEQUENCE ----------
for
    x_n in R

x_n -> L

iff

for_every epsilon > 0
exists N
such_that

n >= N
    -> abs(x_n-L) < epsilon

ABSOLUTE_VALUE
    -> provides(STANDARD_REAL_METRIC)

# ---------- UNIQUENESS ----------
in Hausdorff spaces:

if
    x_n -> x
    AND
    x_n -> y
then

x = y

METRIC_SPACE
    -> HAUSDORFF

therefore

METRIC_SEQUENCE_LIMIT
    -> UNIQUE

# ---------- BOUNDEDNESS ----------
in normed_or_real spaces:

CONVERGENT_SEQUENCE
    -> BOUNDED

BOUNDED_SEQUENCE
    -/> necessarily(CONVERGENT)

BOUNDEDNESS
    -> necessary_but_not_sufficient_in_many_standard_contexts

# ---------- CAUCHY CONVERGENCE ----------
x_n
    -> CAUCHY

iff

for_every epsilon > 0
exists N
such_that

m,n >= N
    -> d(x_m,x_n) < epsilon

CONVERGENT_SEQUENCE
    -> CAUCHY

CAUCHY
    -> CONVERGENT
       iff(SPACE_COMPLETE)

# ---------- COMPLETENESS ----------
COMPLETE_SPACE
    -> every_Cauchy_sequence_converges_within_space

COMPLETENESS
    -> converts(INTERNAL_CAUCHY_BEHAVIOUR_TO_EXISTENCE_OF_LIMIT)

R
    -> COMPLETE

BANACH_SPACE
    -> COMPLETE_IN_NORM

# ---------- SUBSEQUENCES ----------
if
    x_n -> x
then

every_subsequence
    x_(n_k)
        -> x

SUBSEQUENCE
    -> inherits(LIMIT_OF_CONVERGENT_SEQUENCE)

if
    two_subsequences_have_distinct_limits
then

original_sequence
    -> DIVERGENT

# ---------- MONOTONE CONVERGENCE ----------
for real sequence:

if
    x_n increasing
    AND bounded_above
then

x_n
    -> sup{x_n}

if
    x_n decreasing
    AND bounded_below
then

x_n
    -> inf{x_n}

MONOTONICITY
    + BOUNDEDNESS
    -> CONVERGENCE_IN_R

# ---------- SQUEEZE THEOREM ----------
if
    a_n <= b_n <= c_n
    eventually

and

a_n -> L
c_n -> L

then

b_n -> L

SQUEEZE_THEOREM
    -> transfers(CONVERGENCE_THROUGH_BOUNDS)

# ---------- ALGEBRA OF LIMITS ----------
if
    a_n -> A
    AND b_n -> B

then

a_n+b_n
    -> A+B

a_n*b_n
    -> A*B

alpha*a_n
    -> alpha*A

if
    B != 0
    AND b_n != 0 eventually
then

a_n/b_n
    -> A/B

CONVERGENCE
    -> compatible_with(STANDARD_ALGEBRAIC_OPERATIONS)

# ---------- SERIES CONVERGENCE ----------
for series

sum_from_n_1_to_infinity(a_n)

define

S_N
    :=
sum_from_n_1_to_N(a_n)

SERIES
    -> CONVERGES

iff

S_N
    -> finite_limit

SERIES_CONVERGENCE
    -> convergence_of(PARTIAL_SUM_SEQUENCE)

# ---------- ABSOLUTE CONVERGENCE ----------
sum(a_n)
    -> ABSOLUTELY_CONVERGENT

if

sum(abs(a_n))
    -> converges

ABSOLUTE_CONVERGENCE
    -> implies(SERIES_CONVERGENCE)

CONVERSE
    -/> always_true

# ---------- FUNCTION SEQUENCE ----------
f_n
    -> sequence_of_functions

CONVERGENCE_OF_FUNCTIONS
    -> depends_on(CHOSEN_MODE)

MODES := {
    POINTWISE,
    UNIFORM,
    NORM,
    WEAK,
    WEAK_STAR,
    ALMOST_EVERYWHERE,
    IN_MEASURE
}

# ---------- POINTWISE CONVERGENCE ----------
f_n
    -> pointwise_converges_to(f)

iff

for_every x in D

f_n(x)
    -> f(x)

N
    -> may_depend_on(x,epsilon)

# ---------- UNIFORM CONVERGENCE ----------
f_n
    -> uniformly_converges_to(f)

iff

for_every epsilon > 0
exists N
such_that

for_all n >= N
and
for_all x in D:

abs(f_n(x)-f(x))
    < epsilon

equivalently:

sup_x(
    abs(f_n(x)-f(x))
)
    -> 0

UNIFORM_CONVERGENCE
    -> implies(POINTWISE_CONVERGENCE)

# ---------- UNIFORM LIMIT THEOREM ----------
if
    each f_n continuous
    AND
    f_n -> f uniformly
then

f
    -> continuous

UNIFORM_CONVERGENCE
    -> preserves(CONTINUITY)

# ---------- NORM CONVERGENCE ----------
in normed space:

x_n
    -> norm_converges_to(x)

iff

||x_n-x||
    -> 0

NORM_CONVERGENCE
    -> metric_convergence_under(NORM_METRIC)

# ---------- Lp CONVERGENCE ----------
f_n
    -> converges_to(f) in Lp

iff

||f_n-f||_p
    -> 0

LP_CONVERGENCE
    -> NORM_CONVERGENCE_IN(Lp_SPACE)

LP_CONVERGENCE
    -/> generally_implies(POINTWISE_CONVERGENCE_EVERYWHERE)

# ---------- WEAK CONVERGENCE ----------
in normed space X:

x_n
    -> weakly_converges_to(x)

iff

phi(x_n)
    -> phi(x)

for_all
    phi in X_star

NORM_CONVERGENCE
    -> implies(WEAK_CONVERGENCE)

WEAK_CONVERGENCE
    -/> generally_implies(NORM_CONVERGENCE)

# ---------- WEAK STAR CONVERGENCE ----------
for
    phi_n in X_star

phi_n
    -> weak_star_converges_to(phi)

iff

phi_n(x)
    -> phi(x)

for_all
    x in X

WEAK_STAR_CONVERGENCE
    -> pointwise_convergence_on(PRIMAL_SPACE)

# ---------- ALMOST EVERYWHERE ----------
f_n
    -> converges_almost_everywhere_to(f)

if

f_n(x)
    -> f(x)

for_all x
except
    set_of_measure_zero

AE_CONVERGENCE
    -> measure_theoretic_mode

# ---------- CONVERGENCE IN MEASURE ----------
f_n
    -> converges_in_measure_to(f)

if

for_every epsilon > 0

measure(
    {
        x :
        abs(f_n(x)-f(x)) > epsilon
    }
)
    -> 0

CONVERGENCE_IN_MEASURE
    -> weaker_than_many_pointwise_like_modes
       depending_on_context

# ---------- ITERATIVE CONVERGENCE ----------
ITERATION
    := x_(n+1) = F(x_n)

ITERATION
    -> converges_to(x_star)

if

x_n -> x_star

and often

F(x_star)
    = x_star

FIXED_POINT
    -> possible_iteration_limit

# ---------- RATE OF CONVERGENCE ----------
if
    errors
        e_n := ||x_n-x_star||

and

e_(n+1)
    ~= C*(e_n)^p

then

p
    -> ORDER_OF_CONVERGENCE

p = 1
    -> LINEAR_CONVERGENCE

p = 2
    -> QUADRATIC_CONVERGENCE

RATE
    -> describes(SPEED_OF_APPROACH_TO_LIMIT)

# ---------- DIVERGENCE ----------
DIVERGENCE
    -> failure_to_converge_under(SPECIFIED_MODE)

DIVERGENCE
    -> mode_dependent_for(FUNCTION_SEQUENCES)

not_uniformly_convergent
    -/> necessarily_not_pointwise_convergent

CONVERGENCE_MODE
    -> must_be_explicit

# ---------- APPLICATIONS ----------
CONVERGENCE
    -> applied_to(SEQUENCES)
    -> applied_to(SERIES)
    -> applied_to(FUNCTION_SEQUENCES)
    -> applied_to(NUMERICAL_METHODS)
    -> applied_to(FUNCTIONAL_ANALYSIS)
    -> applied_to(PROBABILITY)
    -> applied_to(OPTIMISATION)
    -> applied_to(DIFFERENTIAL_EQUATIONS)

# ---------- RELATION GRAPH ----------
SEQUENCE
    -> may(CONVERGE_TO_LIMIT)

CONVERGENCE
    -> implies(CAUCHY_IN_METRIC_SPACE)

COMPLETENESS
    -> makes(CAUCHY_IMPLY_CONVERGENCE)

SERIES
    -> converges_via(PARTIAL_SUM_SEQUENCE)

UNIFORM_CONVERGENCE
    -> implies(POINTWISE_CONVERGENCE)

NORM_CONVERGENCE
    -> implies(WEAK_CONVERGENCE)

CONVERGENCE_MODE
    -> determines(PRESERVED_PROPERTIES)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "convergence" ~= "approach to a limiting object under a specified notion of closeness",
    "sequence convergence" ~= "eventual arbitrary closeness of sequence terms to a limit",
    "series convergence" ~= "convergence of the partial-sum sequence",
    "uniform convergence" ~= "function convergence with one index threshold valid across the whole domain",
    "norm convergence" ~= "convergence measured by norm distance",
    "weak convergence" ~= "convergence tested by all continuous linear functionals"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    convergence_requires_specified_space_or_mode: true,
    metric_sequence_limits_are_unique: true,
    convergent_sequence_is_cauchy: true,
    cauchy_implies_convergence_only_in_complete_space: true,
    uniform_convergence_implies_pointwise_but_not_conversely: true,
    norm_convergence_implies_weak_but_not_conversely_in_general: true,
    convergence_implications_depend_on_mathematical_context: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    metric_sequence_relation: 0.99,
    cauchy_completeness_relation: 0.99,
    series_relation: 0.99,
    pointwise_uniform_relation: 0.99,
    norm_weak_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(object_sequence_series_or_function_family)
    -> identify(underlying_space)
    -> identify(convergence_mode)
    -> identify(candidate_limit)
    -> apply(mode_specific_definition)
    -> test(cauchy_or_boundedness_conditions_if_relevant)
    -> use(completeness_compactness_or_monotonicity_if_available)
    -> compare(stronger_and_weaker_modes_if_requested)
    -> analyse(rate_of_convergence_if_iterative)
    -> verify(theorem_hypotheses)
    -> classify(convergent_or_divergent)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed convergence semantic graph",
    primary_entity: CONVERGENCE,
    preserve: [
        object_type,
        underlying_space,
        convergence_mode,
        candidate_limit,
        metric_or_norm,
        completeness,
        domain,
        quantifier_order
    ],
    infer_through: [
        sequence_limits,
        cauchy_sequences,
        series,
        pointwise_convergence,
        uniform_convergence,
        norm_convergence,
        weak_convergence,
        iterative_convergence
    ],
    avoid: [
        discussing_convergence_without mode_or_space,
        assuming_boundedness_implies convergence,
        assuming_cauchy_implies convergence_without completeness,
        confusing_pointwise_and_uniform convergence,
        reversing_norm_to_weak implication,
        changing_quantifier_order_in convergence definitions
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(convergence_context)
    -> compare(existing_relations)
    -> preserve(space_mode_limit_and_quantifier_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + CONVERGENCE
    + LIMIT
    + CAUCHY_SEQUENCE
    + COMPLETENESS
    + POINTWISE_CONVERGENCE
    + UNIFORM_CONVERGENCE
    + NORM_CONVERGENCE
    + WEAK_CONVERGENCE
    + ABSOLUTE_CONVERGENCE
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> CONVERGENCE
    -> OBJECT_IDENTIFICATION
    -> SPACE_IDENTIFICATION
    -> CONVERGENCE_MODE_IDENTIFICATION
    -> LIMIT_IDENTIFICATION
    -> MODE_SPECIFIC_DEFINITION_ANALYSIS
    -> CAUCHY_BOUNDEDNESS_ANALYSIS
    -> COMPLETENESS_COMPACTNESS_ANALYSIS
    -> MODE_IMPLICATION_ANALYSIS
    -> RATE_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> CONVERGENCE_CLASSIFICATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW