M12 — Non-vacuity witness

{'text': '{\n "readback": "What the theorem asserts. There exists an object $D$ — a complete derivative-certificate package for one fixed, fully concrete training frame, taken at one fixed point — such that ten further conditions hold. The statement has no hypotheses and no quantifiers other than this leading existential.\n\nThe frame (fixed inline in $D$'s type). The parameter space is $W = \\\\mathbb{R}^1$ and the shared-state space is $V = \\\\mathbb{R}^1$, both with the Euclidean norm and inner product; a vector is written $(c)$ for the element of $\\\\mathbb{R}^1$ whose single coordinate is $c$. The occurrence index set has exactly two elements $\\\\{0, 1\\\\}$, and the occurrence set $I$ is all of it. The four frame components are:\n\n- shared computation: $h(w) = 6w$ (scalar multiplication by $6$ on $\\\\mathbb{R}^1$);\n- per-occurrence losses: $f_i(w, v) = \\\\tfrac{1}{2}\\\\lVert v - y_i \\\\rVert^2$ with the Euclidean norm, targets $y_0 = (1)$ and $y_1 = (4)$;\n- reduction weights $\\\\alpha_0 = \\\\tfrac{1}{4}$, $\\\\alpha_1 = \\\\tfrac{3}{4}$;\n- occurrence set $I = \\\\{0,1\\\\}$.\n\nAll certificates are taken at the pre-update point $w_0 = (2)$, so $h(w_0) = (12)$.\n\nWhat the existence of $D$ itself comprises. The package $D$ bundles four data fields — a continuous linear map h\' : \\\\mathbb{R}^1 \\\\to \\\\mathbb{R}^1, and three families indexed by occurrences: values $\\\\mathrm{val} : \\\\{0,1\\\\} \\\\to \\\\mathbb{R}$, direct gradients $\\\\mathrm{direct} : \\\\{0,1\\\\} \\\\to \\\\mathbb{R}^1$, shared cotangents $\\\\mathrm{shared} : \\\\{0,1\\\\} \\\\to \\\\mathbb{R}^1$ — together with five certificate fields that $D$ must satisfy merely by its type:\n\n- h\' really is the Fréchet derivative of $h$ at $w_0$;\n- $\\\\mathrm{val}_i = f_i\\\\big(w_0, h(w_0)\\\\big)$ for every $i \\\\in \\\\{0,1\\\\}$;\n- $\\\\mathrm{direct}_i$ is the Euclidean gradient of $w \\\\mapsto f_i\\\\big(w, h(w_0)\\\\big)$ at $w_0$, for every $i$;\n- $\\\\mathrm{shared}_i$ is the Euclidean gradient of $v \\\\mapsto f_i(w_0, v)$ at the point $h(w_0)$, for every $i$;\n- each $f_i$ is differentiable at the point $\\\\big(w_0, h(w_0)\\\\big)$.\n\nThus the existential claims that genuine mathematical values/derivatives, with certificates, exist for exactly the data pinned down below — not merely that some uninterpreted data satisfies the displayed equations.\n\nThe ten conjuncts, in the stated order.\n\n1. Tile partition. The two-element tile list $[\\\\{1\\\\}, \\\\{0\\\\}]$ is a valid tile partition of $\\\\{0,1\\\\}$ in the following sense: the union of the tiles, folded from the empty set, equals $\\\\{0,1\\\\}$, and distinct tiles of the list are disjoint — here $\\\\{1\\\\} \\\\cap \\\\{0\\\\} = \\\\varnothing$. (The definition would also tolerate empty or unevenly sized tiles; in this instance both tiles are singletons.)\n\n2. Shared derivative. $D$'s map h\' is exactly $6$ times the identity on $\\\\mathbb{R}^1$, i.e. $w \\\\mapsto 6w$.\n\n3. Direct gradients. $D$'s family $\\\\mathrm{direct}$ is the constant-zero family: $\\\\mathrm{direct}_i = (0)$ for $i = 0, 1$.\n\n4. Shared cotangents. $D$'s family $\\\\mathrm{shared}$ is $\\\\mathrm{shared}_i = h(w_0) - y_i$; numerically $\\\\mathrm{shared}_0 = (12) - (1) = (11)$ and $\\\\mathrm{shared}_1 = (12) - (4) = (8)$.\n\n5. Values. $D$'s family $\\\\mathrm{val}$ is $\\\\mathrm{val}_i = \\\\tfrac{1}{2}\\\\lVert h(w_0) - y_i\\\\rVert^2$; numerically $\\\\mathrm{val}_0 = \\\\tfrac{1}{2}\\\\cdot 11^2 = \\\\tfrac{121}{2}$ and $\\\\mathrm{val}_1 = \\\\tfrac{1}{2}\\\\cdot 8^2 = 32$.\n\n6. Tiled gradient equals $\\\\tfrac{105}{2}$. Unfolded: the tiled gradient first runs a streaming accumulator over the tile list $[\\\\{1\\\\}, \\\\{0\\\\}]$ and then outputs (accumulated direct) $+$ h\'^{\\\\top}(accumulated shared), where h\'^{\\\\top} is the adjoint (transpose) of h\'. The accumulator starts at $(\\\\mathrm{loss}, \\\\mathrm{direct}, \\\\mathrm{shared}) = (0, (0), (0))$ and, for each tile $B$ in list order (first $\\\\{1\\\\}$, then $\\\\{0\\\\}$), adds $\\\\sum_{i \\\\in B} \\\\alpha_i\\\\,\\\\mathrm{val}_i$, $\\\\sum_{i \\\\in B} \\\\alpha_i\\\\,\\\\mathrm{direct}_i$, and $\\\\sum_{i \\\\in B} \\\\alpha_i\\\\,\\\\mathrm{shared}_i$ to the three slots. Concretely: after tile $\\\\{1\\\\}$, $\\\\mathrm{shared} = \\\\tfrac{3}{4}\\\\cdot 8 = (6)$ and $\\\\mathrm{direct} = (0)$; after tile $\\\\{0\\\\}$, $\\\\mathrm{shared} = 6 + \\\\tfrac{1}{4}\\\\cdot 11 = (\\\\tfrac{35}{4})$ and $\\\\mathrm{direct} = (0)$ (the loss slot becomes $\\\\tfrac{3}{4}\\\\cdot 32 + \\\\tfrac{1}{4}\\\\cdot\\\\tfrac{121}{2} = \\\\tfrac{313}{8}$; it is not used in the output). Since the adjoint of $w \\\\mapsto 6w$ is again $w \\\\mapsto 6w$, the conjunct asserts\n

\n\n7. Monolithic gradient. The monolithic objective, unfolded, is\n

\nand the conjunct asserts that $L$ has Euclidean gradient $\\\\big(\\\\tfrac{105}{2}\\\\big)$ at $w_0 = (2)$ — the Riesz representation of its derivative there. (Arithmetically, $L\'(w) = \\\\tfrac{3}{2}(6w-1) + \\\\tfrac{9}{2}(6w-4)$, which at $w = 2$ gives $\\\\tfrac{33}{2} + 36 = \\\\tfrac{105}{2}$.) This pins the same numerical vector as conjunct 6, but asserted independently; no general equivalence between the two is invoked.\n\n8. Nonvanishing. The vector $\\\\big(\\\\tfrac{105}{2}\\\\big)$ is not the zero vector of $\\\\mathbb{R}^1$.\n\n9. One explicit update step. Subtracting one-hundredth of the gradient vector from $w_0$ gives\n

\nan equality of vectors in $\\\\mathbb{R}^1$ (note $\\\\tfrac{1}{100}\\\\cdot\\\\tfrac{105}{2} = \\\\tfrac{105}{200} = \\\\tfrac{21}{40}$ and $2 = \\\\tfrac{80}{40}$).\n\n10. Direct accumulator is zero. The direct slot of the very same folded accumulator as in conjunct 6 (over the tiles $[\\\\{1\\\\}, \\\\{0\\\\}]$, with the same $\\\\alpha$, $\\\\mathrm{val}$, $\\\\mathrm{direct}$, $\\\\mathrm{shared}$) equals the zero vector $(0)$ — that is, the total weighted direct contributions alone, with no reverse pass through h\', sum to zero.\n\nFine print. All norms are the Euclidean norm on $\\\\mathbb{R}^1$, i.e. the absolute value of the single coordinate. Every vector displayed as $(c)$ is the element of $\\\\mathbb{R}^1$ whose single coordinate is $c$; conjuncts 6–9 are equalities/inequalities between such concrete vectors. The only binding construct in the whole statement is the leading existential over $D$; every certificate condition listed above rides along with that existential, and conjuncts 2–5 additionally force $D$'s data fields to the specific values shown, so the statement is the claim that certified genuine derivatives exist for exactly these numbers."\n}', 'details': {'resolvedPath': '/home/ajax/.omp/agent/sessions/-math/2026-09-22T19-31-08-510Z_01a0ca99-be5e-7000-93c6-dac7c74cc004/RB2-M12.md', 'contentType': 'text/markdown', 'totalLines': 3, 'displayContent': {'text': '{\n "readback": "What the theorem asserts. There exists an object $D$ — a complete derivative-certificate package for one fixed, fully concrete training frame, taken at one fixed point — such that ten further conditions hold. The statement has no hypotheses and no quantifiers other than this leading existential.\n\nThe frame (fixed inline in $D$'s type). The parameter space is $W = \\\\mathbb{R}^1$ and the shared-state space is $V = \\\\mathbb{R}^1$, both with the Euclidean norm and inner product; a vector is written $(c)$ for the element of $\\\\mathbb{R}^1$ whose single coordinate is $c$. The occurrence index set has exactly two elements $\\\\{0, 1\\\\}$, and the occurrence set $I$ is all of it. The four frame components are:\n\n- shared computation: $h(w) = 6w$ (scalar multiplication by $6$ on $\\\\mathbb{R}^1$);\n- per-occurrence losses: $f_i(w, v) = \\\\tfrac{1}{2}\\\\lVert v - y_i \\\\rVert^2$ with the Euclidean norm, targets $y_0 = (1)$ and $y_1 = (4)$;\n- reduction weights $\\\\alpha_0 = \\\\tfrac{1}{4}$, $\\\\alpha_1 = \\\\tfrac{3}{4}$;\n- occurrence set $I = \\\\{0,1\\\\}$.\n\nAll certificates are taken at the pre-update point $w_0 = (2)$, so $h(w_0) = (12)$.\n\nWhat the existence of $D$ itself comprises. The package $D$ bundles four data fields — a continuous linear map h\' : \\\\mathbb{R}^1 \\\\to \\\\mathbb{R}^1, and three families indexed by occurrences: values $\\\\mathrm{val} : \\\\{0,1\\\\} \\\\to \\\\mathbb{R}$, direct gradients $\\\\mathrm{direct} : \\\\{0,1\\\\} \\\\to \\\\mathbb{R}^1$, shared cotangents $\\\\mathrm{shared} : \\\\{0,1\\\\} \\\\to \\\\mathbb{R}^1$ — together with five certificate fields that $D$ must satisfy merely by its type:\n\n- h\' really is the Fréchet derivative of $h$ at $w_0$;\n- $\\\\mathrm{val}_i = f_i\\\\big(w_0, h(w_0)\\\\big)$ for every $i \\\\in \\\\{0,1\\\\}$;\n- $\\\\mathrm{direct}_i$ is the Euclidean gradient of $w \\\\mapsto f_i\\\\big(w, h(w_0)\\\\big)$ at $w_0$, for every $i$;\n- $\\\\mathrm{shared}_i$ is the Euclidean gradient of $v \\\\mapsto f_i(w_0, v)$ at the point $h(w_0)$, for every $i$;\n- each $f_i$ is differentiable at the point $\\\\big(w_0, h(w_0)\\\\big)$.\n\nThus the existential claims that genuine mathematical values/derivatives, with certificates, exist for exactly the data pinned down below — not merely that some uninterpreted data satisfies the displayed equations.\n\nThe ten conjuncts, in the stated order.\n\n1. Tile partition. The two-element tile list $[\\\\{1\\\\}, \\\\{0\\\\}]$ is a valid tile partition of $\\\\{0,1\\\\}$ in the following sense: the union of the tiles, folded from the empty set, equals $\\\\{0,1\\\\}$, and distinct tiles of the list are disjoint — here $\\\\{1\\\\} \\\\cap \\\\{0\\\\} = \\\\varnothing$. (The definition would also tolerate empty or unevenly sized tiles; in this instance both tiles are singletons.)\n\n2. Shared derivative. $D$'s map h\' is exactly $6$ times the identity on $\\\\mathbb{R}^1$, i.e. $w \\\\mapsto 6w$.\n\n3. Direct gradients. $D$'s family $\\\\mathrm{direct}$ is the constant-zero family: $\\\\mathrm{direct}_i = (0)$ for $i = 0, 1$.\n\n4. Shared cotangents. $D$'s family $\\\\mathrm{shared}$ is $\\\\mathrm{shared}_i = h(w_0) - y_i$; numerically $\\\\mathrm{shared}_0 = (12) - (1) = (11)$ and $\\\\mathrm{shared}_1 = (12) - (4) = (8)$.\n\n5. Values. $D$'s family $\\\\mathrm{val}$ is $\\\\mathrm{val}_i = \\\\tfrac{1}{2}\\\\lVert h(w_0) - y_i\\\\rVert^2$; numerically $\\\\mathrm{val}_0 = \\\\tfrac{1}{2}\\\\cdot 11^2 = \\\\tfrac{121}{2}$ and $\\\\mathrm{val}_1 = \\\\tfrac{1}{2}\\\\cdot 8^2 = 32$.\n\n6. Tiled gradient equals $\\\\tfrac{105}{2}$. Unfolded: the tiled gradient first runs a streaming accumulator over the tile list $[\\\\{1\\\\}, \\\\{0\\\\}]$ and then outputs (accumulated direct) $+$ h\'^{\\\\top}(accumulated shared), where h\'^{\\\\top} is the adjoint (transpose) of h\'. The accumulator starts at $(\\\\mathrm{loss}, \\\\mathrm{direct}, \\\\mathrm{shared}) = (0, (0), (0))$ and, for each tile $B$ in list order (first $\\\\{1\\\\}$, then $\\\\{0\\\\}$), adds $\\\\sum_{i \\\\in B} \\\\alpha_i\\\\,\\\\mathrm{val}_i$, $\\\\sum_{i \\\\in B} \\\\alpha_i\\\\,\\\\mathrm{direct}_i$, and $\\\\sum_{i \\\\in B} \\\\alpha_i\\\\,\\\\mathrm{shared}_i$ to the three slots. Concretely: after tile $\\\\{1\\\\}$, $\\\\mathrm{shared} = \\\\tfrac{3}{4}\\\\cdot 8 = (6)$ and $\\\\mathrm{direct} = (0)$; after tile $\\\\{0\\\\}$, $\\\\mathrm{shared} = 6 + \\\\tfrac{1}{4}\\\\cdot 11 = (\\\\tfrac{35}{4})$ and $\\\\mathrm{direct} = (0)$ (the loss slot becomes $\\\\tfrac{3}{4}\\\\cdot 32 + \\\\tfrac{1}{4}\\\\cdot\\\\tfrac{121}{2} = \\\\tfrac{313}{8}$; it is not used in the output). Since the adjoint of $w \\\\mapsto 6w$ is again $w \\\\mapsto 6w$, the conjunct asserts\n

\n\n7. Monolithic gradient. The monolithic objective, unfolded, is\n

\nand the conjunct asserts that $L$ has Euclidean gradient $\\\\big(\\\\tfrac{105}{2}\\\\big)$ at $w_0 = (2)$ — the Riesz representation of its derivative there. (Arithmetically, $L\'(w) = \\\\tfrac{3}{2}(6w-1) + \\\\tfrac{9}{2}(6w-4)$, which at $w = 2$ gives $\\\\tfrac{33}{2} + 36 = \\\\tfrac{105}{2}$.) This pins the same numerical vector as conjunct 6, but asserted independently; no general equivalence between the two is invoked.\n\n8. Nonvanishing. The vector $\\\\big(\\\\tfrac{105}{2}\\\\big)$ is not the zero vector of $\\\\mathbb{R}^1$.\n\n9. One explicit update step. Subtracting one-hundredth of the gradient vector from $w_0$ gives\n

\nan equality of vectors in $\\\\mathbb{R}^1$ (note $\\\\tfrac{1}{100}\\\\cdot\\\\tfrac{105}{2} = \\\\tfrac{105}{200} = \\\\tfrac{21}{40}$ and $2 = \\\\tfrac{80}{40}$).\n\n10. Direct accumulator is zero. The direct slot of the very same folded accumulator as in conjunct 6 (over the tiles $[\\\\{1\\\\}, \\\\{0\\\\}]$, with the same $\\\\alpha$, $\\\\mathrm{val}$, $\\\\mathrm{direct}$, $\\\\mathrm{shared}$) equals the zero vector $(0)$ — that is, the total weighted direct contributions alone, with no reverse pass through h\', sum to zero.\n\nFine print. All norms are the Euclidean norm on $\\\\mathbb{R}^1$, i.e. the absolute value of the single coordinate. Every vector displayed as $(c)$ is the element of $\\\\mathbb{R}^1$ whose single coordinate is $c$; conjuncts 6–9 are equalities/inequalities between such concrete vectors. The only binding construct in the whole statement is the leading existential over $D$; every certificate condition listed above rides along with that existential, and conjuncts 2–5 additionally force $D$'s data fields to the specific values shown, so the statement is the claim that certified genuine derivatives exist for exactly these numbers."\n}', 'startLine': 1, 'lineNumbers': [1, 2, 3]}, 'meta': {'source': {'type': 'internal', 'value': 'agent://RB2-M12'}}}}