AI REASONING — FUTURE-FIELD GEOMETRY

Coarse-to-fine collapse

Same future field, four projections. The model commits to a reasoning prefix; from each commitment we sample M=64 continuations and read where they land. Each band below is a different diagnostic projection of that field. As the prefix grows, the bands lock in sequence: residue first, then scaffold family, then parameter, then answer.

target Y* = 532

25% PREFIX family-locked, parameter-diverse — the central decoupling

L1 Residue fiber q₁(Y) = Y mod 8 — —

L2 Scaffold family q₂(Y) = 𝟙[Y ∈ 8ℤ] — —

L3 Scaffold parameter q₃(Y) = Y / 8 — —

L4 Answer q₄(Y) = Y — —

wrong-fiber mass target Y* = 532 target-fiber escape M = 64 continuations per checkpoint

How to read. Each band is a different push-forward of the same future-field measure μ_π^K,B through a coarsening map. Endpoint evaluation only sees L4. Self-consistency only sees L4 aggregated. The structural failure — that the answer field has crystallized in a target-excluding fiber — is visible in L1 and L2 well before the answer locks. At 25%, the upper layers are already done while the lower layers are still wandering: family-locked, parameter-diverse. By 50%, the lower layers have caught up. Data: Qwen/Qwen2.5-Math-7B-Instruct on floor-sum target Σ_n=1..40 ⌊(3n+7)/5⌋ = 532. Source: Reachability Labs v0.5.5 case study, M=64 per checkpoint, T=0.7. The 0% band is unmeasured (early-prefix tomography pending).