1\section{Implications for Cryptographic Security}2\label{sec:implications}34\paragraph{For secp256k1.} Nothing in this study weakens the curve. The three structural features that make5\secp{} ``special''---the generalised-Mersenne prime, $a=0$, and complex multiplication by $\Z[\omega]$---translate into6exactly the constant factors the literature predicts and we measured: faster field reduction, cheaper doubling, a7$2\times$ reduction in doublings for scalar multiplication through GLV, and a $\sqrt6$ reduction in the expected8number of rho iterations (Section~\ref{sec:results}). Proposition~\ref{prop:end} shows that the endomorphism ring is as9small as it can be given $j=0$, so there is no further endomorphism to exploit; Proposition~\ref{prop:noquotient}10shows that the automorphism orbits cannot be turned into a smaller group; and the exact exclusion table11(Section~\ref{sec:secp}) confirms that every known algebraic transfer is inapplicable. The generic bound therefore12stands: solving one \secp{} instance costs about $\sqrt{\pi n/12}\approx 2^{126.7}$ group operations with the full13automorphism trick, roughly $2^{70}$ times the largest ECDLP computation ever completed. Preprocessing does not14change this conclusion: the per-instance cost $\sqrt{n/T}$ after a table of $T$ points requires $\sqrt{nT}$ work up15front, and a table reducing each instance to $2^{80}$ operations would itself cost $2^{174}$ operations and $2^{93}$16entries. Our measurements (Hypothesis~\Hname{H12}) put the constant at about $1.5$, which does not alter the17exponent.1819\paragraph{For the signature schemes.} The security difference between ECDSA and BIP340 is structural: Schnorr's20linear response $s=k+ed$ admits a clean reduction and enables batch verification, aggregation and threshold21protocols; ECDSA's $k^{-1}$ obstructs all three and forces the proofs into stronger idealisations. Operationally22the two schemes cost the same to verify in \libsecp{} (within $5\%$), so the choice is about provable security and23protocol features, not speed. Batch verification is the one verification-side improvement that is both real and24consensus-neutral; its gain is bounded by the non-linear-algebra work per signature (square root, hashing) and25saturates near $3\times$ in our measurements. The quantities that actually decide security in practice are26nonce generation and side-channel resistance, which deterministic nonces and constant-time scalar multiplication27address; none of the algebra in this paper changes that assessment.2829\paragraph{For SHA-256.} Our reduced-step experiments reproduce the easy end of the known picture: full diffusion30within a few steps, no detectable bias beyond a handful of steps, SAT preimages and collisions feasible only far31below the published dedicated-tool records. The step function being a bijection and the schedule being invertible32are reminders that the compression function alone offers no preimage resistance---the Merkle--Damg\aa rd33feed-forward does---and that ``free-start'' results must not be confused with attacks on the hash. The security34margin of SHA-256 is the distance between 31 (practical collision), 37 (theoretical collision), 45 (theoretical35preimage) and 64 steps; nothing here narrows it.3637\paragraph{For automated discovery.} The pipeline's value was not a new attack but the discipline it imposed: every38candidate shortcut was either explained by existing theory, measured to be a constant, or refuted. An autonomous39agent working on cryptographic primitives is exposed to a specific failure mode---mistaking a constant-factor or40representation improvement for a complexity improvement---and the classification rubric, the operation counts and41the formal layer are the controls against it. We regard the fact that the agent produced zero claims of a break, while42producing a closed-form explanation of a set of implementation constants and a dozen quantified reproductions, as43the correct outcome.4445\paragraph{Responsible-disclosure assessment.} No security-relevant result emerged that would require disclosure.46All instances attacked were synthetic; no real key, address or blockchain datum was used or produced; the code47released with the paper attacks only the reduced-size curves it generates itself.4849\paragraph{Quantum computing.} The analysis above is classical. Shor's algorithm \citep{shor1997polynomial} solves the50ECDLP in polynomial time on a fault-tolerant quantum computer, with concrete resource estimates for 256-bit curves51in the range of a few thousand logical qubits \citep{roetteler2017quantum}; this, not any classical shortcut, is the52known threat to \secp{}, and it is independent of the curve's special structure.53