OCT Typed Formal Spec v0.1

Scope:fix a unified typed scheme for Coh, Phi, Delta, Omegacompatible with classical category theory and usable in OCT theorems.

1) Formal layers

The specification separates three layers:1. Syntactic: classical category.2. Ordering: consistency and validity thresholds.3. Emergent: emergent function and degeneration.

2) Typed basic data

Is:- O a locally small category.- Diag_fin(O) the class of finished diagrams in O.- ObsCtx a category (or preorder) of observational contexts.- Omega in Ob(ObsCtx) an observational context. For each Omega we assign:

1. Local consistency on morphisms- coh_Omega^1 : Mor(O) -> [0,1]
2. Diagrammatic coherence- Coh_Omega : Diag_fin(O) -> [0,1]
3. Emergent space- E_Omega = (E_Omega, +, 0_E, <=_E)- E_Omega and a preordained commutative monoid with null element 0_E
4. Emergent function- Phi_Omega : Diag_fin(O) -> E_Omega
5. Threshold of reality- tau_Omega in (0,1]
6. Degeneration- Delta_Omega : Diag_fin(O) -> [0,1]- defined by Delta_Omega(D) = 1 - Coh_Omega(D)

3) Fundamental predicates

3.1 OCT-valid morphism

For f in Mor(O): OCTVal_Omega(f) := coh_Omega^1(f) >= tau_Omega

3.2 OCT-real diagram

For D in Diag_fin(O): Real_Omega(D) := (Coh_Omega(D) >= tau_Omega) and (Phi_Omega(D) != 0_E)

3.3 OCT-degenerate diagram

Deg_Omega(D) := (D != vuoto) and (Phi_Omega(D) = 0_E)

4) Minimal typed axioms (TS1-TS8)

TS1 - Classic compatibility

The classic definitions of identity and composition in O remain unchanged.

TS2 - Consistency normalization

coh_Omega^1(f) in [0,1] and Coh_Omega(D) in [0,1].

TS3 - Identity coherence

For each A object: coh_Omega^1(id_A) = 1.

TS4 - Local compositional stability

There exists a t-norm T : [0,1] x [0,1] -> [0,1] such that,for each composite pair f: A->B, g: B->C: coh_Omega^1(g o f) >= T(coh_Omega^1(g), coh_Omega^1(f)).

TS5 - Local/global compatibility

For each finished diagram D: Coh_Omega(D) <= inf { coh_Omega^1(f) : f e freccia di D }.

TS6 - Canonical emerging nullity

If D is empty or trivially devoid of dynamics, thenPhi_Omega(D) = 0_E.

TS7 - Strong anti-degeneration

If Real_Omega(D) then Deg_Omega(D) is false.

TS8 - Observational reindexing

For each context morphism u: Omega -> Omega' in ObsCtx,There are transport maps:- R_u^Coh : [0,1] -> [0,1]- R_u^Phi : E_Omega -> E_Omega' with compatibility:- Coh_Omega'(D) = R_u^Coh(Coh_Omega(D))- Phi_Omega'(D) = R_u^Phi(Phi_Omega(D)) in a declared regime (exact or approximate with controlled error).

5) Immediate consequences

1. F03 and well typed via TS4.2. F08 and well-typed via 0_E and Real_Omega.3. F10 and can be formalized by imposing limit regime:- coh_Omega^1 = 1,- Coh_Omega = 1,- Phi filter disabled.

6) Choices open for v0.2 of the specification

1. explicit choice of the t-norm T (min, product, Lukasiewicz, other).2. choice of space E_Omega (ordered monoid vs semiring vs lattice).3. threshold policy tau_Omega (fixed, adaptive, domain dependent).4. formalization of transport error between contexts (epsilon-consistency).

7) Criterion for adoption in the work

The specification is considered adopted when:1. all foundation theorems use the same typed signatures;2. untyped uses of Phi, Coh, Omega no longer appear;3. the empirical protocol (Volume IV) uses these same quantities without redefinitions.