identity_always_possible
plain-language theorem explainer
Any configuration reaches the identity state after exactly eight ticks while preserving non-increasing total J-cost. Modal logicians in the Recognition framework cite this to ground reachability of the fixed point. The tactic proof splits the Possibility membership into a trivial time step and a cost inequality reduced via J_at_one, the explicit J_stasis form, and nlinarith on the log_bound hypothesis.
Claim. For any configuration $c$ with value $v>0$, time $t$, and $|log v|≤16$, the identity configuration at time $t+8$ belongs to the possibility set $P(c)$, i.e., satisfies $y.time=t+8$ and $J(v)+J_{transition}(v,1,...) + J(1) ≤ J(v) + J_{stasis}(v)$.
background
A configuration is a point in recognition state space carrying a positive real value, positivity witness, natural-number time coordinate, and the structural bound $|log(value)|≤16$ that keeps all admissible states inside the octave range. The possibility operator collects configurations reachable in one eight-tick step whose transition cost does not exceed the stasis cost at the source. Upstream, J_at_one states that the multiplicative identity carries zero cost; this is the algebraic normalization used to simplify every cost comparison.
proof idea
The tactic proof unfolds Possibility, splits the conjunction, and dispatches the time component by reflexivity. The cost component invokes J_at_one to replace J(1) by zero, substitutes the explicit equality J_stasis = 8·J, unfolds J_transition, rewrites the absolute log via log_inv, and closes the resulting inequality by linarith on the coefficient bound |log v|/2 ≤8 together with non-negativity of J and the absolute value.
why it matters
This lemma is invoked directly by every_config_actualizes to witness that every configuration possesses an actualization path and by modal_T_weak to obtain the weak reflexivity axiom for the necessity operator. It supplies the concrete reachability step required by the eight-tick octave (T7) in the forcing chain and by the adjointness theorems that relate possibility and actualization. The result therefore anchors the modal layer on top of the cost algebra.
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