M10 — Checkpoint round trip

{"text": "{\n "readback": "The theorem asserts, for every natural number $d$ and every training state $S$ of type $\\\\text{TrainState}(d)$, that\n\n

\n\nUnfolding the definitions: a training state $S$ over $d$ real coordinates is a four-field record \u2014 a parameter vector $w \\\\in \\\\mathbb{R}^d$ (as a Euclidean-space element), a first-moment vector $\\\\text{mom1} \\\\in \\\\mathbb{R}^d$, a second-moment vector $\\\\text{mom2} \\\\in \\\\mathbb{R}^d$, and a step counter $t \\\\in \\\\mathbb{N}$. The encoder $\\\\mathrm{snap}$ produces a snapshot record whose fields are the coordinate lists of the three vectors \u2014 each obtained by $\\\\mathrm{listOfVec}$, i.e. the list $[x_0, x_1, \\\\dots, x_{d-1}]$ of length exactly $d$ \u2014 together with the unchanged counter $t$. The decoder $\\\\mathrm{restore}$ turns each list back into a vector by $\\\\mathrm{vecOfList}$, which reads the $j$-th coordinate from the list and substitutes $0$ whenever the list has length less than $j+1$ (so a short list yields a vector whose trailing coordinates are $0$; extra list entries beyond $d$ are ignored) and copies $t$ verbatim. Since equality of structures is field-wise, the claim is exactly the conjunction:\n\n- $\\\\mathrm{vecOfList}(\\\\mathrm{listOfVec}(S.w)) = S.w$,\n- $\\\\mathrm{vecOfList}(\\\\mathrm{listOfVec}(S.\\\\text{mom1})) = S.\\\\text{mom1}$,\n- $\\\\mathrm{vecOfList}(\\\\mathrm{listOfVec}(S.\\\\text{mom2})) = S.\\\\text{mom2}$,\n- $S.t = S.t$,\n\nthe last being trivial and the first three holding because each encoded list has length exactly $d$, so the decoder's short-list default ($0$ for missing coordinates, via getD) never fires and every coordinate round-trips through the real-valued list unchanged. No hypothesis is imposed on $S$ beyond its type; the statement is universally quantified over all $d \\\\in \\\\mathbb{N}$ \u2014 including $d = 0$, where the vectors are zero-dimensional, the lists are empty, and the claim holds vacuously coordinate-wise \u2014 and over all real values of the three vector fields and all step counters. The statement asserts exact mathematical equality of real numbers, not agreement up to rounding or floating-point tolerance; any finite-precision codec behavior is outside what is claimed here. Note also that the theorem as given is stated with a sorry proof placeholder, so it is asserted but not yet established in the file."\n}", "details": {"resolvedPath": "/home/ajax/.omp/agent/sessions/-math/2026-09-22T19-31-08-510Z_01a0ca99-be5e-7000-93c6-dac7c74cc004/RB-M10.md", "contentType": "text/markdown", "totalLines": 3, "displayContent": {"text": "{\n "readback": "The theorem asserts, for every natural number $d$ and every training state $S$ of type $\\\\text{TrainState}(d)$, that\n\n

\n\nUnfolding the definitions: a training state $S$ over $d$ real coordinates is a four-field record \u2014 a parameter vector $w \\\\in \\\\mathbb{R}^d$ (as a Euclidean-space element), a first-moment vector $\\\\text{mom1} \\\\in \\\\mathbb{R}^d$, a second-moment vector $\\\\text{mom2} \\\\in \\\\mathbb{R}^d$, and a step counter $t \\\\in \\\\mathbb{N}$. The encoder $\\\\mathrm{snap}$ produces a snapshot record whose fields are the coordinate lists of the three vectors \u2014 each obtained by $\\\\mathrm{listOfVec}$, i.e. the list $[x_0, x_1, \\\\dots, x_{d-1}]$ of length exactly $d$ \u2014 together with the unchanged counter $t$. The decoder $\\\\mathrm{restore}$ turns each list back into a vector by $\\\\mathrm{vecOfList}$, which reads the $j$-th coordinate from the list and substitutes $0$ whenever the list has length less than $j+1$ (so a short list yields a vector whose trailing coordinates are $0$; extra list entries beyond $d$ are ignored) and copies $t$ verbatim. Since equality of structures is field-wise, the claim is exactly the conjunction:\n\n- $\\\\mathrm{vecOfList}(\\\\mathrm{listOfVec}(S.w)) = S.w$,\n- $\\\\mathrm{vecOfList}(\\\\mathrm{listOfVec}(S.\\\\text{mom1})) = S.\\\\text{mom1}$,\n- $\\\\mathrm{vecOfList}(\\\\mathrm{listOfVec}(S.\\\\text{mom2})) = S.\\\\text{mom2}$,\n- $S.t = S.t$,\n\nthe last being trivial and the first three holding because each encoded list has length exactly $d$, so the decoder's short-list default ($0$ for missing coordinates, via getD) never fires and every coordinate round-trips through the real-valued list unchanged. No hypothesis is imposed on $S$ beyond its type; the statement is universally quantified over all $d \\\\in \\\\mathbb{N}$ \u2014 including $d = 0$, where the vectors are zero-dimensional, the lists are empty, and the claim holds vacuously coordinate-wise \u2014 and over all real values of the three vector fields and all step counters. The statement asserts exact mathematical equality of real numbers, not agreement up to rounding or floating-point tolerance; any finite-precision codec behavior is outside what is claimed here. Note also that the theorem as given is stated with a sorry proof placeholder, so it is asserted but not yet established in the file."\n}", "startLine": 1, "lineNumbers": [1, 2, 3]}, "meta": {"source": {"type": "internal", "value": "agent://RB-M10"}}}}