The 399 rational-floor example

The 399 record above is one specialization. It is not the general statement, and the scalar $\varphi+\pi$ is not the geometric object those marks come from. Let $Q$ and $S$ be positive integers and let $d$be an integer. Set $d_{\mathrm{next}}=\lfloor Qd/S\rfloor$ and $\rho=Qd-Sd_{\mathrm{next}}$. Then $0\le\rho<S$, $Qd=Sd_{\mathrm{next}}+\rho$, and $d=(Sd_{\mathrm{next}}+\rho)/Q$. The hypothesis $Q<S$ is not used. That inequality belongs to a separate contraction statement.

The floor definition says $d_{\mathrm{next}}$ is the unique integer with $d_{\mathrm{next}}\le Qd/S<d_{\mathrm{next}}+1$. Multiplying by the positive scale $S$ gives the residual bounds and the division identity. Dividing that identity by $Q$ recovers $d$.

An integer pair $(d_{\mathrm{next}},\rho)$ with $0\le\rho<S$ is an exact predecessor record when $Q$ divides $Sd_{\mathrm{next}}+\rho$. If the declared source domain is the nonnegative integers, the predecessor must also be nonnegative. The residual bounds do not imply that extra condition. On all integers, the predecessor $d=-1$ of $(Q,S,d_{\mathrm{next}},\rho)=(5,2,-3,1)$ is admissible.

Two reconstructions are not the same map. The SRSD decoder $D_S(d_{\mathrm{next}},\rho)=Sd_{\mathrm{next}}+\rho$ returns the observation $Qd$. The source state is the further quotient $d=(Sd_{\mathrm{next}}+\rho)/Q$. For integer records the reachable pairs are those $(k,\rho)$ with $0\le\rho<S$ and $Q$ dividing $Sk+\rho$. A nonnegative source domain also requires $Sk+\rho\ge 0$. The feedback $d\mapsto d_{\mathrm{next}}$ is application data. Its injectivity is not decided here. A completed application record is not SRSD conformance.

The argument above is supplied. It has not been given a separate review. A reading produced in this archive is not an independent human review. The absence of an ASD-STE100 review does not by itself show a mathematical defect, and acceptance of this one record would not verify the archive.

The canonical specialization keeps $Q=339949771344778$ and $S=10^{15}$. Its statement text was not revised. The 24 September checker pass remains that checker’s declared result. It was not reused as the result below.

Checker: verify_floor_general.py (3,683 bytes, SHA-256 c4bb196ae7ef360e11eaa4e356678286f42b87bf33d3c9f94ab831a9f985e564). The 7 October logs remain. The 8 October record is execution.json. Its stdout hashes match the earlier logs because the checker prints no date. The checker rejects a zero or negative scale, a residual outside $[0,S)$, a numerator not divisible by $Q$, and a negative predecessor when the domain is nonnegative. It has no Python assert statements.

Python 3.10.21 and Node v22.23.2, 2026-09-24T00:50:51Z. Every listed command exited 0. Orbit 399, 135, 45, 15, 5, 1, 0. Recovered seed 399. All 32768 seeds. 300871 unpadded transitions. Maximum depth 10. Domain checksum 804ac6ba40a63a0596bc577d0721e98000c30e2a00e4a9173d3749e8d7adf059.

python3 floor_kernel.py 399 --verify --site-reference evidence/omega_reference_checker.py and the same command under -O reproduce verification_python.txt. node verify_floor_kernel.mjs --site-bridge evidence/bridge.js reproduces verification_node.txt. The commands without those optional comparisons also pass. Their stdout omits the downloaded-site line, so it is a different file.

A separate module run exported the production 399 record and imported that text back to seed 399. A separate browser check of the same control cleared the seed field and recovered 399 from the imported record. The checker runs themselves were not a browser session.