Chapter 17
Integration as Accumulated Coherence

Flow Algebra

flow "Integration as Accumulated Coherence"

# ---------- CONCEPT ----------
# Integration reconstructs whole coherence from infinitesimal change.
# It accumulates local contributions (densities/flows) into a global measure.
# As the inverse of differentiation (FTC), it reunites gradient with total form.

# ---------- STATES ----------
state IntegralSystem(I):
  Accumulator:
    description: "Sum of local contributions over a region — Riemann/Darboux perspective."
  Antiderivative:
    description: "Function F with F' = f — latent whole whose boundary change is f."
  MeasureSpace:
    description: "Domain (interval, area, volume) with measure for aggregation."
  KernelWeight:
    description: "Weighting function (e.g., u-substitution, integrating factor, convolution kernel) guiding accumulation."
  Path/Region:
    description: "Curve/surface/volume over which coherence is collected (line, surface, volume integrals)."

# ---------- PARAMETERS ----------
params:
  variable: "x (or vector x ∈ ℝⁿ)"
  bounds: "[a,b] for definite; open for indefinite"
  partition: "P = {x_0, …, x_n} with max|Δx_i| → 0"
  orientation: "Direction of accumulation; sign-sensitive"
  smoothness: "f integrable (Riemann/Lebesgue assumptions)"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Riemann limit: ∫_a^b f(x) dx = lim_{||P||→0} Σ f(ξ_i) Δx_i."
  - "Additivity on intervals: ∫_a^c f = ∫_a^b f + ∫_b^c f."
  - "Linearity: ∫(αf + βg) = α∫f + β∫g."
  - "Order reversal: ∫_a^b f = −∫_b^a f."
  - "FTC I: F(x) = ∫_a^x f(t) dt ⇒ F'(x) = f(x) (a.e.)."
  - "FTC II: If F' = f continuous on [a,b], then ∫_a^b f = F(b) − F(a)."
  - "Change of variables: ∫_a^b f(g(u)) g'(u) du = ∫_{g(a)}^{g(b)} f(x) dx."
  - "Integration by parts: ∫ u dv = uv − ∫ v du."

# ---------- TRANSFORMS ----------
transforms:
  Accumulate(f, [a,b]):
    rule: "Form Riemann sums and pass to limit — obtain definite integral."
  FindAntiderivative(f):
    rule: "Compute F with F' = f — reconstruct global coherence."
  Substitute(g):
    rule: "Use x = g(u) to realign density with simpler coordinates."
  ByParts(u,v):
    rule: "Redistribute coherence via product rule inversion: ∫ u dv = uv − ∫ v du."
  NumericalApprox(method):
    options: ["midpoint","trapezoid","Simpson","Gauss"]
    rule: "Approximate accumulation when closed form unavailable."
  ExtendToMultipleDims(f, region):
    rule: "Compute ∬, ∭ with Jacobian; apply Fubini/Tonelli when valid."
  PathIntegrate(F, C):
    rule: "∫_C F·dr — gather coherence along curves; conservative fields reduce to potentials."

# ---------- MONITORS ----------
monitors:
  ConvergenceCheck:
    measure: "Refinement error between successive partitions."
    meaning: "↓ to 0 ⇒ integral well-defined."
  AdditivityTest:
    check: "Split domain; sum partial integrals; compare with whole."
  OrientationFlag:
    output: "Sign change under reversed bounds/path direction."
  ErrorEstimator:
    methods: ["Romberg","Richardson","adaptive quadrature"]
    meaning: "Controls numerical accuracy."
  RegularityGuard:
    alert: "Detects improper integrals / singularities; tests for convergence."

# ---------- EQUATIONS ----------
equations:
  - "∫_a^b f(x)\,dx = \\lim_{\\|P\\|\\to 0} \\sum_i f(\\xi_i)\\,\\Delta x_i"
  - "F(x) = \\int_a^x f(t)\\,dt \\;\\Rightarrow\\; F'(x) = f(x)"
  - "\\int_a^b f(x)\\,dx = F(b) - F(a) \\quad (F' = f)"
  - "\\int u\\,dv = uv - \\int v\\,du"
  - "\\int_a^b f(g(u)) g'(u)\\,du = \\int_{g(a)}^{g(b)} f(x)\\,dx"
  - "\\iint_{\\Omega} f\\,dA,\\; \\iiint_{V} f\\,dV"
  - "\\int_{C} \\mathbf{F}\\cdot d\\mathbf{r} \\;\\,(=\\,\\phi(B)-\\phi(A)\\;\\text{if }\\mathbf{F}=\\nabla\\phi)"

# ---------- NOTES ----------
notes:
  - "Integration is coherence aggregation: it rebuilds the whole from infinitesimal structure."
  - "The Fundamental Theorem binds change and accumulation — differentiation and integration are reciprocal flows."
  - "Substitution and parts are alignment tools: they reshape the channel through which contributions are gathered."
  - "Multiple integrals require Jacobians — measuring how coordinates distort volume while preserving total coherence."
  - "Line/surface/volume integrals translate density into path/flux/total content — geometry of accumulation."
  - "In Flow Algebra, integration is the return path: local gradients fold back into a unified, closed form of truth."