State Σ
Σ holds fields, corridors, and habitat. Π publishes the layer, the stamp, and the residual disagreement — not the unshown survey.
Ontology
Primitives, then the sextuple
Strategy, Flow, and the permutohedron of Flow sit beside Mathematical Organization Theory. They do not replace it. Organizational state remains Σ. Strategy is a set of directions, and inside the five-term update that signal is S(t).
Strategy
Strategy := { dᵢ, dⱼ, dₖ }
Admissible directions of change. Inside PFISD this signal is S(t). It is not the MOT state Σ.
Flow
Flow := Set of { i, j, k }
What may propagate. In the sextuple the same role is F: connectivity, not a person and not a topology of boxes.
Permutohedron
Pₘ : permutohedron of Flow, order m
The permutohedron of order m is the convex hull of the permutations of (1,…,m): an (m−1)-polytope with m! vertices. Here it orders Flow. Left-to-right, right-to-left, and center-out are different evaluations.
PFISD
PFISDₜ₊₁ := Kₚ×P(t) + Kf×F(t) + Kᵢ×I(t) + Kₛ×S(t) + Kd×D(t)
Five-term update beside a PID law: proportional, flow, integral, stress, derivative. Kₛ := (1 − ||Kf||) · Kf† = (1 − ||Kf||) / Kf.
O = (Σ, F, D, P, U, Π)
Definition 5.1, from the monograph. Nothing in the sextuple is optional. Σ here is microscopic state, not Strategy.
Σₜ₊₁ := Update(Σₜ, Iₜ, F, D, P) / ε
U is Update. ε is the error on that step. Observation stays separate: M = Π(Σ), and I(M) ≤ I(Σ). PFISD does not replace Update.
Volume I
Foundations
Everything is derived from six primitives. An organization is the ordered sextuple, and nothing else is primitive.
Volume II
Dynamics
Mathematical Organizational Dynamics. Trajectories, stability, compression, and macro-state determinism.
Volume III
Intelligence
Mathematical Organizational Intelligence. Inference, decision, control, and action. Evolution alone is not yet intelligence.
Volume IV
Refinements
Mathematical Organizational Refinements. Every sufficiently organized system admits the same representation. Only the interpretation changes.
Volume II · Organizational trajectories
Filed, watched, verified, sealed, or returned. Volume II leaves Σ abstract and studies the path. A ledger that keeps only the present has thrown the dynamics away.
γ : ℝ → Σ
Volume IV refinements
A refinement is a homomorphism R : O → O_D. The theorems of Volumes I–III stay. Only the reading of state and of decision changes.
Decision A
Decisions are routing, land use, and response. They amend parameters. They do not quietly redraw the ground.
The ledger that carries these readings is Decentralis Repositorium.
The Lₘ residual
Lₘ(N, Θ, Φ) = exp(1/N! ∘ Pₘ ∘ exp(Θ + Φ i))
Creativity synthesis is a bounded composition. Pₘ is the permutohedron of Flow of order m. N is taken modulo m. Θ and Φ are the real phase pair named in the quantum-gravity note. Left-to-right, right-to-left, and center-out evaluation are different Representables.
∞ₙᵢⱼₖ := Σ_N^∞ (± Π_Θ^N (± E_Φ^Θ ({CD(ℝ) ∧ R[N; σ, δ] ∧ e ∧ ω ∧ ε ∧ N}_π(Θ)^⟨π(Φ)ι⟩)))
That display is the note’s stated form, kept as notation. It is not a recognized quantization of gravity. Bose–Einstein, Einstein–Rosen, Boltzmann, Feynman, Turing, and Schwarzschild are separate theories. P versus NP is an open problem. None of them is a device this ledger operates, and none of them amends the charter.
Lₘ is also not the hidden organizational information I(Σ) − I(M). Any study of the residual needs consent, data minimization, and a published evaluation order. This ledger does not run that study on a person.