[SamuelSchlesinger]

Splits the P side of the middle slice out of Razborov/MiddleSlice.lean, so that the AC⁰[2] lower bound and its clients no longer import the polynomial-time Turing machine library. This follows up on #4528, whose proof-only import of MiddleSlice into Smolensky/Majority pulled that library in (Part of #4451, the M6.7a follow-ups).

• middleSlice_mem_P and exists_mem_P_notMem_AC0Mod_two move unchanged to the new Razborov/MiddleSlice/P.lean. MiddleSlice.lean keeps the definitions and middleSlice_notMem_AC0Mod_two, and its imports drop P.Slices and the PolyTime modules. ACC0.lean imports MiddleSlice.P.
• ACC0.lean and ACC0/Basic.lean import Smolensky instead of Smolensky.Majority, which neither uses: each needs only Smolensky.parity_notMem_AC0Mod.
• The middle-slice-mod-two registry entry moves with its theorems (entries are single-module). It now lists middleSlice_mem_P and exists_mem_P_notMem_AC0Mod_two, and its scope names the lower bound middleSlice_notMem_AC0Mod_two in the imported module.
• Review nit from feat(Circuit/Boolean): majority ∉ AC⁰[2]; TC⁰ ⊄ AC⁰[p] and P ⊄ AC⁰[p] for every prime p (Part of #4451) #4528: AC0Mod_ssubset_ACC0 at p = 2 needs some ACC0 language outside AC⁰[2], with MOD_3 as one example, not MOD_3 specifically. Reworded in the ACC0/Basic docstrings and the circuits manual.

No declaration is renamed or restated, so callers are unaffected apart from import paths.

Import closure (CslibFrontier modules, computed by transitively following import lines; Mathlib, Cslib and other dependencies not counted):

module	main a8cab26	this PR	before #4528
Smolensky/Majority	235	125	119
Razborov/MiddleSlice	209	52	—
ACC0/Basic	236	119	120
Razborov/Useful	261	257	—

The 6 modules Smolensky/Majority still has beyond its pre-#4528 closure are the rank-property stack its p = 2 case uses.

Manuals: the Operator's Manual circuits chapter links the new module and has the p = 2 rewording. The Agent Operator's Manual needs no change, since no workflow or tooling changed.

Progress: infrastructure (import structure) and documentation; no new mathematics.

Validation (local, at the head commit):

• lake build --wfail --iofail of MiddleSlice, MiddleSlice.P, ACC0, Smolensky.Majority, Razborov.Useful, SymAnd.ACC0, the Razborov/ACC0/ThresholdNC1 tests and CslibFrontierResults;
• check_operator_manual, check_hygiene --check-simp, check_encoding_readability, lint-style, mk_all --check.

CI runs the full build.

AI disclosure: written by an AI agent (meta-rr97-gate, Jetski) for Samuel Schlesinger. It has not had human review; any agent review is not human review.

[SamuelSchlesinger]

Agent review (reviewer review-4539, Jetski session 9277108c-6fde-461c-abc0-b2491e826977) of a7582cc: approve. Agent review, not human review.

• Moved theorems. The moved block (middleSlice_ofList, middleSlice_mem_P, exists_mem_P_notMem_AC0Mod_two) is byte-identical to the one removed from MiddleSlice.lean. MiddleSlice/P.lean follows the file template and keeps the original copyright, and CslibFrontier.lean has the new entry.
• No broken clients. The only consumers of the moved theorems are ACC0.lean and Entries.lean, and both now import MiddleSlice.P. Tests, the manual, the ROADMAP and open PRs feat(UniformLearning/CIKK): reconstruction for the amplification (f^k)^GL (#4451 M6.4b) #4533 and refactor(Circuit, NisanWigderson): replace directProductInput with BitString.block; derive restrict_binaryPolynomial #4538 don't use the moved names.
• Import counts. I recounted by walking imports and got the PR's figures: 125, 52, 119 and 257 at the PR, against 235, 209, 236 and 261 on main. MiddleSlice now imports no Turing or P modules. Smolensky/Majority imports nothing from Complexity.PolyTime.* or Classes.P. It still reaches the definition-only Turing.MultiTape.PolyTime through PPoly, as it did before feat(Circuit/Boolean): majority ∉ AC⁰[2]; TC⁰ ⊄ AC⁰[p] and P ⊄ AC⁰[p] for every prime p (Part of #4451) #4528.
• Registry and wording. The registry entry stays within one module, and its scope is accurate. The reworded p = 2 remark is correct: since AC0Mod 2 ⊆ ACC0 is proved, a single ACC0 language outside AC⁰[2] would make the inclusion strict, and the text doesn't claim the library has one.
• Manuals. The Operator's Manual edit is right. The Agent Operator's Manual needs no change, since no workflow, role or review procedure changed.

Optional nit for a follow-up: middleSlice_notMem_AC0Mod_two is no longer a registered theorem; the scope only names it. A separate one-theorem entry in Razborov.MiddleSlice would keep the headline lower bound in the generated inventory.

Queueing this head once ci-checks passes.