[Lean] Compose prepared-input counting within 18n+12

[jamesyc]

Change and mathematical scope

One new source, BinaryPreparedInputCount.lean, composes #488 input preparation with #519 automatic-width counting and #526's amortized bound. The actual 24-state, five-tape machine reaches its exact halted endpoint in some elapsed t <= 18*n+12, where n = max(length x, length y).

The initial tapes are canonical finite Boolean inputs x and y plus three pristine blank tapes. Empty and unequal-length inputs are included. States 0–13 preserve preparation's live state-13 seek loop; one tape-preserving handoff enters states 14–23 for counting. The bound is 6*n+7 + 1 + 12*n+4. Final state 23 is halted, and the width-n counter has value n. All five represented tape footprints and boundary blanks are explicit in README and the theorem. This is an elapsed upper bound, not a first-completion or exact-runtime claim.

Author-checked conditional project context at base 12a4934d15006e8c10f9b2264f13f8f341be802b:
κ = 90446448750933/125000000000000000 = 0.000723571590007464.
The selected-result/certificate linkage and rational/decimal equality were checked. This module proves no new κ or multiplication theorem.

Coverage

• Finite-machine coverage: one concrete 24-state transition table, exact preparation/count handoff and halted endpoints for arbitrary finite inputs. Three work tapes are initially pristine; no additional spectator tape is assumed.
• Conditional interface: caller initialization and representation premises are explicit. Output remains a width-n marked record; compact copying, selected-source caller, radix/carry Plan and complete Gaussian represented-list Budget remain separate.
• Full theorem: all-size compiler, chart/routing, prime supply, precision/recovery, complex correctness and analytic-transfer obligations remain open.

Reuse and duplicate comparison

Only one 17,069-byte Lean source is added. Fourteen exact external sources are reused:

• [Lean] Bound automatic-width counting by 12n+4 #526 e7d6dc6e474f8690d72ef8b841290948028f66f3: public list/arithmetic extraction and receipt transport.
• [Lean] Add automatic-width binary counting #519 ffd3c48a80ba8b579dc300e785e387fe8acddfef: three counting/allocation modules.
• [Lean] Add checked binary count dispatcher #511 79f7c03c7f820ab89955b75ef239cbaccc627e6e: four unchanged dispatcher/carry sources; this source pin predates the independent CI provenance repair.
• [Lean] Verification of binary input preparation, κ=0.000661885549259598 #488 edd3102b415df5b9d6f633ac0a4dedcf69c58525: four machine/input-preparation sources.
  PR64's potential proof and predecessor authorship remain credited. No claim of a new generic counting algorithm or repeated induction is made. Existing [Lean] Verification of binary input preparation, κ=0.000661885549259598 #488/[Lean] Add checked binary count dispatcher #511/[Lean] Add automatic-width binary counting #519/[Lean] Bound automatic-width counting by 12n+4 #526 and relevant PR64 scope were compared; the contribution here is this concrete combined machine and exact handoff. No existing PR is closed.

Validation

Lean 4.21.0 and Mathlib 308445d7985027f538e281e18df29ca16ede2ba3.
Accepted research root: 4.6331 compiler seconds, 1,775,390,720 bytes peak VM.
Public reproduction: twelve exact verified retained #526 outputs plus three fresh roots; 10.0393 fresh compiler seconds, 0.00221 FIFO seconds, 3,407,740,928 bytes peak VM. All 102 complete standard-only probes and all fifteen reference object hashes match. Fourteen logs match exactly; one differs only in warning source-path presentation. This is explicitly a guarded retained-prefix reproduction, not an uninterrupted fresh fifteen-source run. Rejected attempts remain separate and are not counted as passes.

Fifteen changed files, all under research/lean-prepared-input-count. No root, scientific-check, runtime-limit or workflow change.

Offline reproduction

With exact source worktrees and an already-built pinned runtime:

python3 research/lean-prepared-input-count/check.py --lean /path/to/lean-4.21.0-linux/bin/lean --mathlib-root /path/to/mathlib4 --input-foundation-root /path/to/PR488/research/lean-binary-input-preparation --dispatcher-foundation-root /path/to/PR511/research/lean-binary-count-dispatcher --automatic-width-root /path/to/PR519/research/lean-automatic-width-count --amortization-root /path/to/PR526/research/lean-count-receipt-amortization --build-dir /path/to/new-empty-reproduction-directory

This supported command freshly checks fifteen sources; the recorded replay separately identifies the twelve retained and three fresh outputs. No dependency download or Mathlib rebuild occurs.

Kernel checks are distinct from fork/upstream CI; no exact-head CI pass is claimed. The inherited joint-certificate root-file provenance failure is outside this chunk.

Prepared by James Chang with OpenAI dot and GPT Astra assistance. Upstream attribution and assistance disclosures are preserved.