possibility_space_connected
plain-language theorem explainer
Every possibility configuration reaches the identity state (value 1) along a discrete path indexed by natural numbers. Modal geometry in Recognition Science cites this to equip possibility space with a star topology centered at identity. The proof supplies an explicit path function that equals the input only at step zero and thereafter maps to identity_config with time offset by multiples of 8.
Claim. For any possibility configuration $c$, there exists a sequence of configurations $(p_n)_{n=0}^∞$ such that $p_0 = c$ and, for every $n$, there is an index $m > n$ with $p_m$ satisfying value 1.
background
Possibility space is the collection of configurations each equipped with a value field (positive real) and a time parameter; identity sits at value 1, the unique J-cost minimum. The theorem lives in ModalGeometry, which imports Possibility and Actualization to define paths and the identity_config constructor. Upstream, ObserverForcing.identity states that the canonical identity event has state 1 and zero cost; PhiForcingDerived.of and DAlembert.LedgerFactorization.of calibrate the underlying J function whose second derivative later appears in curvature statements.
proof idea
Term proof by explicit construction. Define the path to equal the input configuration at index 0 and identity_config(c.time + 8*n) thereafter. Verify the base case by simp. For the tail condition, pick index n+1, apply omega to obtain the strict inequality, then simp on the identity_config definition to obtain value 1.
why it matters
The result supplies the connectivity axiom for the star topology of possibility space, directly filling the connectivity claim in the modal layer. It references the eight-tick octave through the explicit 8*n time increments. No downstream theorems are recorded, yet the statement underpins the curvature and period lemmas that follow in the same module.
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