# Verification report

**Result:** Complete proof of the adjacent-power realization theorem.
For fixed k and i, every nonnegative integer sequence is realized as
length H^i_m(R/J_n), with I^(n+1) contained in J_n contained in I^n in a
fixed complete regular local ring. Lower cohomology and lower transition
maps vanish. No multiplicativity is asserted.

**Proof basis:** Exact one-column cokernel, a top-Cech annihilator basis,
and a degree-layer presentation of the quotient. The detailed self-audit
checks quantifiers, i=0, a_n=0, n=1, support, finite length, transition maps,
and the distinction between A-linear and R-linear decompositions.

**Finite evidence:** 84 exact kernel checks and 120 independent truncated
quotient Groebner-basis checks, plus a rational nonmembership witness for
failure of multiplicativity. Normal and optimized Python runs agree.
Python 3 and SymPy 1.14.0 suffice to replay the included checker.

**Source comparison:** The original Oberwolfach question and its context
were inspected, including the actual page image. Dao-Montano's structured
power results and Bhatt-Blickle-Lyubeznik-Singh-Zhang's arbitrary-thickening
examples are credited. The search is targeted, not exhaustive.

**Scope accounting:** The theorem is complete. The broad source programme
remains PARTIALLY_SOLVED; this paper adds zero whole-source closures.
Publication is not proof verification or certification of novelty.

**Review status:** Originating researcher self-audit; no independent
review or formal proof-assistant verification claimed. AI assistance is
disclosed in the prose. Author: Alper Ferudun, Mercury Software GmbH.

**Artifact checks:** Five pages inspected visually, desktop compilation
successful. The sealed package separately records the portable compilation,
exact replay checks, source hashes, and upload hashes.