RealizedHierarchy
plain-language theorem explainer
A hierarchy of observed levels is generated by iterating the dynamics T on a base carrier state inside a closed observable framework, with the observable r producing positive values that obey uniform adjacent ratios and additive posting at the second level. Researchers closing the T5 to T6 bridge in the forcing chain would cite this structure to eliminate external sensitivity and composition hypotheses. The definition is a direct structure declaration whose five fields encode the physical requirements without further derivation.
Claim. A structure over a closed observable framework $F$ consists of a base state $s_0$ in the state space of $F$ together with level values $l_k = r(T^k s_0)$ satisfying $l_k > 0$ for all $k$, $l_1/l_0 > 1$, the self-similar ratio condition $l_{k+2}/l_{k+1} = l_{k+1}/l_k$ for every $k$, and the additive posting relation $l_2 = l_1 + l_0$.
background
A closed observable framework supplies a state space $S$, deterministic dynamics $T: S → S$, and positive observable $r: S → ℝ$ such that $r$ distinguishes at least two states; the framework also enforces closure and the absence of continuous moduli. The module internalizes earlier external bridge hypotheses by making self-similar scaling and additive posting native fields of the hierarchy structure rather than separate assumptions on a zero-parameter ledger. Upstream, the Breath1024 period and Gap45 triangular numbers supply the discrete time and counting conventions that label the iterates, while the obstruction module shows that the framework primitives alone do not force the required fields.
proof idea
This is a structure definition whose fields directly record the base state, the level map defined by iteration of $T$ and application of $r$, positivity, the growth inequality, the self-similar ratio equality, and the additive posting equality at level two. No separate proof term or tactic sequence is required beyond the field declarations themselves.
why it matters
The structure supplies the input to the downstream bridge theorems bridge_T5_T6_internal and realized_hierarchy_forces_phi, which derive that the common ratio equals φ using only the self-similar ratio field and the additive posting field. It thereby completes the RS-internal route from T5 J-uniqueness through T6 phi as the self-similar fixed point without invoking external hypotheses. The construction touches the open question whether a closed observable framework by itself suffices or still requires an earlier geometric scale sequence witness.
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