Exact A3/E3 modular relations
The experiment proved and tested exact relations A3 = const_A + W3, E3 = const_E + W3 and E3 = A3 + Δheader modulo 2^32. All 1,000,000 checks agreed. This is a structural identity, not an end-to-end mining shortcut.
- Published
- 2026-08-14
- Updated
- 2026-08-14
- Authors
- BTC PoW Lab
- Replication
- Deterministically validated
Local structure, statistical signal and operational advantage are evaluated separately.
Question
Can the combined A3/E3 state be represented by one shared 32-bit source and fixed job constants?
Hypothesis
Both words are exact affine translations of W3 for a fixed header context.
Why this matters
Exact dimensional reduction can simplify later reasoning even when it does not reduce total mining work.
Method
Derive the modular identities, evaluate them across one million nonce-dependent inputs and compare against full-round computation.
Experimental setup
- Baseline
- Direct SHA-256 state computation.
- Setup
- Fixed header context with W3 varied; exact 32-bit modular arithmetic.
- Hardware
- Classical deterministic verifier.
- Dataset
- One million registered checks.
Results
- 1,000,000 verifications; 0 discrepancies.
- A valid A3/E3 pair occupies a 2^-32 fraction of the unconstrained combined space.
- Fixing 16 bits yielded 65,536 corresponding solutions in the registered experiment.
Scientific analysis
For a fixed Bitcoin header context, the only changing message-schedule word at this point is W3. SHA-256 uses 32-bit modular addition, so the round-3 state words can be separated into job constants plus one varying source.
The identity reduces representation dimension: instead of treating A3 and E3 as two independent 32-bit words, a valid pair lies on a 32-bit affine subset of the nominal 64-bit product space. This is exact algebra, not a probabilistic correlation.
Dimensional reduction can simplify a verifier or a later symbolic model, but computing the constants, transporting the state and completing both SHA-256 passes remain charged work. The identity alone therefore cannot establish mining advantage.
A₃ = W₃ + C_AE₃ = W₃ + C_EMathematical formulation
A₃ = (C_A + W₃) mod 2³²C_A contains the fixed job contribution to A after round 3.
E₃ = (C_E + W₃) mod 2³²E has the same unit coefficient on W3 under the fixed context.
E₃ − A₃ = C_E − C_A = Δheader (mod 2³²)Subtracting cancels W3 exactly; the offset is constant for that job.
|valid pairs| / |all pairs| = 2³² / 2⁶⁴ = 2⁻³²Only one affine 32-bit lane is reachable within the unconstrained 64-bit pair space.
Statistical reading
One million registered evaluations produced zero discrepancies; this is a test of an algebraically derived identity, not an estimate of a success rate.
Empirical agreement checks implementation and endianness. The derivation supplies the general fixed-context claim.
When 16 bits are fixed, the remaining free dimension yields 65,536 corresponding solutions in the registered construction.
Validity and scope
- Exact for the stated round convention and fixed header constants.
- Constants must be recomputed when the job context changes.
- Downstream carries, rotations, Ch/Maj and schedule expansion remain nonlinear work.
Methodological references
Interpretation
The relation is exact and useful for representation. No complete mining advantage follows from the identity alone.
Limitations
- Fixed-context constants change with the header job.
- Downstream SHA-256 nonlinearity and implementation cost remain.
Reproduction
Recompute constants from an independent fixed header, enumerate W3 values and compare both identities with direct state evaluation.
Artifacts
Deterministic verifier publication package pending review.
Experimental content is provided for research and educational use, without warranty. Validate independently before relying on it. Read the full disclaimer.