realized_additive_closure
The realized hierarchy on a closed observable framework satisfies the additive relation levels(2) = levels(1) + levels(0) on its uniform scale ladder. Workers on the internal T5-T6 bridge cite this to obtain the Fibonacci recurrence directly from the realization structure without external composition hypotheses. The proof is a direct one-line projection of the additive_posting field.
claimLet $F$ be a closed observable framework and $H$ a realized hierarchy on $F$. Then the levels of the realized ladder satisfy $l(2) = l(1) + l(0)$, where $l(k) = r(T^k s_0)$ for base state $s_0$ and positive observable $r$.
background
ClosedObservableFramework is the structure with state space $S$, deterministic dynamics $T: S → S$, and positive observable $r: S → ℝ$ satisfying nontriviality (distinct values exist) and closure (no external input). RealizedHierarchy augments this with a base state whose iterates under $T$ define levels via $r$, plus the fields ratio_self_similar (constant inter-level ratio) and additive_posting (levels(2) = levels(1) + levels(0)). The module internalizes the hierarchy to replace prior external bridge hypotheses on sensitivity and additive composition with RS-native fields derived from J-cost self-similarity.
proof idea
The proof is the one-line term H.additive_posting, which directly supplies the equality from the structure definition of RealizedHierarchy.
why it matters in Recognition Science
This supplies the additive closure used by realized_hierarchy_forces_phi to conclude that the scale ratio equals φ, and by bridge_T5_T6_via_posting to route through posting extensivity. It closes the internal T5 J-uniqueness to T6 phi fixed point step by deriving the Fibonacci relation from the additive_posting field, eliminating external sensitivity and HasAdditiveComposition hypotheses.
scope and limits
- Does not establish the self-similar ratio property.
- Does not apply outside closed observable frameworks.
- Does not derive the value of the scale ratio itself.
- Does not address non-uniform ratios or continuous moduli.
Lean usage
hierarchy_emergence_forces_phi (realized_to_ladder F H) (realized_additive_closure F H)formal statement (Lean)
111theorem realized_additive_closure (F : ClosedObservableFramework)
112 (H : RealizedHierarchy F) :
113 (realized_to_ladder F H).levels 2 =
114 (realized_to_ladder F H).levels 1 + (realized_to_ladder F H).levels 0 :=
proof body
Term-mode proof.
115 H.additive_posting
116
117/-! ## Main Theorem: Realized Hierarchy Forces φ -/
118
119/-- **End-to-end theorem**: A realized hierarchy on a closed observable
120framework forces the scale ratio to be φ.
121
122This replaces the old bridge that required external `sensitivity` and
123`HasAdditiveComposition` hypotheses. Both are now derived from
124the realization's `ratio_self_similar` and `additive_posting` fields,
125which are RS-native physical principles. -/