Research · Papers · Symmetry, state encodings and search gauges · MF-050
Exact finite-horizon minimisation of natural SHA carry states
For the 22 natural SHA carry states, exact finite-horizon minimisation produces the identical partition across all 64 round constants: 22 classes through bit 29, 16 at bit 30, four
Published 2026-08-29
For everyone
Plain summary
When SHA passes carry states from bit to bit, 22 states start out distinct. Looking ahead to the remaining bit positions, states that behave identically under all allowed continuations merge into the same Nerode class. Exact minimisation leaves 22 classes through bit 29, 16 at bit 30, four at bit 31, and one terminal class across all 64 round constants. The 2,048 raw symbols project to 128 actions, with 16 original symbols per action and zero right-congruence violations. Ordinary non-injective recoding can therefore affect at most 128 terminal cases, ruling out a thousand-row route. The claim is limited to the natural deterministic interface and finite-horizon recoding; restricted alphabets, enlarged relations, and defect automata fall outside scope. The register states no prior-art position.
Result
For the 22 natural SHA carry states, exact finite-horizon minimisation produces the identical partition across all 64 round constants: 22 classes through bit 29, 16 at bit 30, four at bit 31, and one terminal class.
The 2,048 raw symbols project to 128 actions with 16 preimages each and zero right-congruence violations. Ordinary non-injective recoding of the full deterministic natural interface affects at most 128 terminal occurrences and cannot form a thousand-row route.
Setting and definitions
The natural interface evaluates 22 deterministic carry states over the remaining bit positions. A finite-horizon partition groups states into Nerode equivalence classes based on identical continuation behavior.
The pre-recoding alphabet consists of 2,048 raw symbols whose projection yields 128 actions, where each preimage is an original symbol mapped to an action. A right-congruence violation occurs if an allowed continuation distinguishes two merged states. A terminal occurrence denotes an affected instance at the terminal boundary under ordinary non-injective recoding of the full deterministic natural interface.
Method
The register records an exact finite-horizon minimisation and symbol projection verified by independent Fleet C and CONT verification receipts (the raw verification receipts and checksums are preserved in this paper's downloadable evidence pack). The register logs no solver, proof certificate, or separate written argument.
Discussion
The result is a structure theorem for the natural deterministic interface under finite-horizon recoding. The partition remains invariant across all 64 round constants, capping ordinary non-injective recoding of the full interface at 128 terminal occurrences and precluding any thousand-row route.
The analysis excludes restricted alphabets, enlarged relations, and defect automata, which alter the underlying state machine. The curation note limits the claim to the natural deterministic interface under finite-horizon recoding. The entry has no corrections, and the register records no prior-art position.
For everyone — the takeaway
What this means
When computing SHA bit by bit, 22 carry states start out distinct. As computation approaches the final bit, fewer states remain distinguishable: 16 possibilities at bit 30, four at bit 31, and one at the end. The 2,048 input symbols compress into 128 actions, with each action grouping 16 original symbols without breaking future consistency. This ordinary compression can touch at most 128 terminal cases, preventing any thousand-row route. The finding applies strictly to the natural interface and finite look-ahead, excluding changed alphabets, enlarged relations, or separate defect-tracking automata.
Register references
- Entry: MF-050
- Receipt: Fleet C
sha_carry_nerode_screen.json - Receipt: CONT
fleet-c-return7/fleet_c_independent_verification_receipt.json(SHA-2567043ea2c6155ec37c6013625cf1151fabe709842c04504c449268d7750c1f43d) - 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 2 of 2 receipt files bundled (5 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.