Canonical rational-base arithmetic and frozen SRSD audit instrument

Frozen SRSD (Paper III) is the external audit instrument and is not enlarged here. Omega/JPRV is not identified with radial icosahedral cubature. Those claims live at /srsd and /cubature. The July 2026 Omega-Alpha synthesis remains a historical consolidation at /dynamics/omega-alpha and is not a source paper.

Never identify $\lambda_Q$ with $\lambda$. Never recompute the canonical integer $Q=339949771344778$ from binary floating-point evaluations of $\phi$ and $\pi$. The reduced integers are $P=S/2=5\times10^{14}$ and $R=Q/2=169974885672389$, with $\gcd(P,R)=1$.

Equivalently $R d_k = P d_{k+1}+a_k$. The residual is even because $Q=2R$ and $S=2P$. The normalized digit $u_k=\rho_k/S=a_k/P$ satisfies the exact unit-scale SRSD identity

That identity is representational. It does not make SRSD and rational-base numeration the same native dynamical system.

$\rho(d)=\rho(e)$ if and only if $d\equiv e\pmod{P}$. The inverse of $R$ modulo $P$ is $281419755786509$. Because $2^{15}<P$, the first residual is injective on $D_{15}=\{0,\ldots,32767\}$.

Worked check: $d_0=399$ has first residual $\rho_0=639958766566422$ and recovers as

Every state in $D_{15}$ reaches $0$ in at most ten quotient steps. The trajectory from $17697$ uses ten steps, so ten is the exact uniform bound. The configured length $N=23$ is protocol padding after extinction, not the termination theorem. Finite-domain successor facts remain at /dynamics/successors.

Retired as a canonical statement: the complete 23-step ledger is necessary to reconstruct a canonical 15-bit limb. Replacement: the ledger reconstructs exactly and records the execution path, but the first residual already identifies the limb.