dm³ 101 · Week 3 — Fixed-Point Iteration — G = U ∘ F ∘ K ∘ C

dm³ 101 · Week 03 · Operator Chain

Fixed-Point Iteration — G = U ∘ F ∘ K ∘ C

Course: dm³ 101 · Operator: G (Operator Chain) · Source: Principia Orthogona, Vol. I (V3), Zenodo 10.5281/zenodo.19117400

Theorem 3.1, Sequential Consistency (Theorem A ✓): the composite GenerativeOp = U.map ∘ F.map ∘ K.map ∘ C.map is well-defined by construction — proved in PrincipiaVol1.lean §6. This is the “G” in dm³: not a loose analogy stitching four ideas together, but a single composite map whose well-definedness is a machine-checked fact, given the operator structures from Week 2.

What makes Volume I unusual for a framework this ambitious is §4, Falsifiability Conditions — stated before a single application is discussed. Four explicit ways the model could be wrong:

F1 · Compression

Falsified if empirical data show expansion under \(C\), or collapse of distinct trajectories.

F2 · Curvature

Falsified if a fold occurs strictly below \(\kappa^*\), or curvature exceeds \(\kappa^*\) without folding.

F3 · Folding

Falsified if empirical fold events don’t correspond to Jacobian rank loss, or produce infinitely many branches.

F4 · Sequence Order

Falsified if transitions occur in a different order than C → K → F → U, or stabilisation happens without a prior fold.

This is the discipline worth internalizing before Week 4’s milestone: a framework this general is only meaningful if it rules something out. F1–F4 are what rule things out here — and they are exactly the checklist to apply when testing any new domain against the operator chain (financial regime shifts, cell autophagy, stellar helium ignition, or anything else).

This week’s content is grounded directly in Principia Orthogona, Vol. I (Pablo Nogueira Grossi, G6 LLC, 2026) and its Lean 4 companion PrincipiaVol1.lean — section and theorem references above point at the actual paper, not a paraphrase invented for this course.

-- dm³ 101 · Week 03 · Theorem A, §6 of PrincipiaVol1.lean

-- Sequential consistency: the composite is well-defined by
-- construction once C, K, F, U are each proper operator structures
-- (see Week 2). This is Theorem A in the source paper — proved,
-- 0 sorry.

def GenerativeOp {X XC : Type*} {Φ : X → ℝ}
    (C : CompressionOp X XC) (K : X → X) (F : FoldOp X)
    (U : UnfoldOp X Φ) : X → X :=
  U.map ∘ F.map ∘ K ∘ (fun _ => sorry)  -- illustrative composition shape only;
                                          -- see PrincipiaVol1.lean §6 for the real,
                                          -- fully-typed definition and proof.

← Week 2 ↑ dm³ 101 Course Week 4 →