[NicholasStone]

This proves a necessary condition for lifting a projected binary state set and completely excludes one seven-source optimal projection from yielding an eight-source 28-step universal restored singleton word. The general 28-step existence question remains open; the known bound stays 28 <= f(8) <= 32. No new kappa.

For Q, join u and v when their XOR has exactly one unordered representation in Q. In any universal lift P onto Q, the projected states appearing in both layers form a vertex cover of this graph. Thus |P| >= |Q| + tau(G(Q)) >= |Q| + nu(G(Q)). The README gives the dimension-independent proof.

For the 24-step word 341236425430103051216402, explicit matching and vertex-cover certificates both have size 11. Every universal lift with exactly this fixed projected state set needs at least 35 states, hence at least 36 steps for a closed singleton path. This is a fixed-projection exclusion, not a global lower bound on f(8), and 36-step attainability is not asserted.

The package also gives an exact static lift criterion and a repeated-projection counterexample to the false general bound that double-layer states cannot outnumber new-coordinate calls. Static feasibility does not imply a chronological path.

Validation

Both commands pass on CPython 3.12.14 and reproduce the committed JSON receipt byte for byte:

python3 -I -W error -B research/eight-source-lift-obstruction/verify_obstruction.py
python3 -I -O -W error -B research/eight-source-lift-obstruction/verify_obstruction.py

The standard-library checker verifies the explicit word, 11-edge matching and 11-vertex cover, checks all 65,805 nonempty sets in dimensions 2–4 against a direct difference oracle, checks the known 32-step Gray construction, and rejects four false controls. The finite checks corroborate the written lemma; they do not replace its proof.

Exploratory C++/SMT sources and logs are retained separately. Random searches are incomplete; the 28-step SMT run returned unknown (timeout), not UNSAT. No exhaustive enumeration of all seven-source optimal projections, formal-kernel proof, independent human review, hosted CI result or selected-construction replay is claimed.

Scope, sources and next question

Only research/eight-source-lift-obstruction/ is added (17 files). Canonical source base: 1a3ced60b2f7a46bc70280a04f2eeb8ca027c700; refreshed before submission, with governing guidance unchanged. The publication tree matches the locally tested tree.

Credit conorholds / #3058 for f(7)=24; Nick Petalas / #5234 for the eight-source floor and forced 28-step count shape; Ross Richardson / #943 for the model and Douglas Colkitt / #1185 for the Gray construction. These predecessor results are cited, not re-proved or claimed as new. The earlier #5416 addresses a separate projection-recurrence question.

The remaining task is to classify all realizable seven-source 24-step projections or prove a uniform obstruction, then exhaustively examine any survivors. The README includes an English reviewer summary, the detailed proof/certificates, and explicit open next questions. No historical novelty claim.

Prepared for Nicholas Stone with substantial OpenAI Codex assistance; Apache-2.0. Frozen predecessors, shared code and the selected result are untouched.