Research · Papers · Cipher S-boxes, χ, and quantum gate counts · MF-010
An impossibility theorem for the χ₀₁ catalyst and the full-domain p7 tile
deg(f) = 9, 80 of 140 top monomials ∉ I = ⟨c₁c₂⟩ ⟹ χ₀₁ catalyst cannot realize full-domain p7 tile
Published 2026-08-29
For everyone
Plain summary
MF-010 tests a proposed χ₀₁ catalyst, a named computation feature, against the full-domain p7 tile, a Boolean computation block whose rule must work for every allowed input. The outgoing function is the 0-or-1 rule produced by the catalyst. Its degree is the largest number of input factors that appear together in one product term; here that degree is 9. At that highest degree, the function has 140 top monomials, meaning products of inputs at the highest degree. Eighty of them lie outside the ideal generated by the incoming product \(c_1c_2\), where an ideal here is the collection of polynomial expressions that contain that product as a factor. In ordinary terms, those 80 highest-order parts cannot be supplied by the incoming contribution in the required algebraic form. That mismatch proves the χ₀₁ catalyst cannot realize the tile across its full domain. The result closes a specific local shortcut. The register records no position about earlier research.
Result
Let \(f\) denote the outgoing Boolean function of the recorded χ₀₁ catalyst and let \(I=\langle c_1c_2\rangle\) be the ideal generated by the incoming \(c_1c_2\) contribution. MF-010 establishes that
\[ \deg(f)=9, \]
that the degree-9 part of \(f\) contains 140 top monomials, and that 80 of those 140 monomials lie outside \(I\). Consequently, the χ₀₁ catalyst cannot realize the full-domain p7 tile. The result is an impossibility theorem for this recorded catalyst and tile interface.
Setting and definitions
The register names χ₀₁ as the catalyst, \(c_1\) and \(c_2\) as the incoming signals, and the p7 tile as the target tile. Full-domain realization means equality with the target Boolean function on the entire domain represented by the tile. A monomial is a product of input variables, and its degree is its number of factors. The top monomials are the monomials in the highest-degree part of the outgoing function. The ideal \(I=\langle c_1c_2\rangle\) is the set of polynomial expressions containing the incoming product \(c_1c_2\) as a factor. The obstruction is therefore a statement about the degree-9 monomials that the incoming product can or cannot generate.
Method
The result comes from the state-feature transport census, with raw output available in this paper's downloadable evidence pack. The curation notes record two independent implementations. Both support the same degree-9 count, the 140 top monomials, and the 80 monomials outside the incoming ideal. The register names no additional solver, encoding, or written-argument artifact, so the method is reported at the level of the census and its independent implementation check.
Discussion
The result closes the recorded χ₀₁ catalyst route for the exact full-domain p7 tile. The related limits record places deterministic state-only lifts in a closed repair class at every row budget and records scoped acyclic-p6 state-only catalysts as PROVED dead. The same scope record leaves transient or path-dependent laws, stationary two-boundary laws, relational laws, cyclic laws, and general synthesis classes UNKNOWN. A larger raw-input-twisted edge quotient is also recorded as nonempty. Those distinctions keep the theorem attached to its specified catalyst and route. They do not support a blanket impossibility statement for every catalyst or every presentation of the tile. Corrections: none. The register states no prior-art position for MF-010.
For everyone — the takeaway
What this means
One proposed shortcut fails for a concrete reason. Its produced rule contains 80 highest-order combinations that the incoming \(c_1c_2\) contribution cannot provide. The full-domain tile therefore cannot be completed by this catalyst. That gives future builders a firm boundary: this exact local route is closed. The finding leaves other designs open, including approaches that use different information or different connections. It makes no statement about earlier research, because the register records no prior-art position. Readers can use MF-010 as a boundary on one construction idea. Other possible solutions remain outside this finding.
Register references
- MF-010
- Receipt: CONT
state_feature_transport_census_results.json - Prior art: the register does not record this.
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 1 of 1 receipt files bundled (6 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.