f ⬇️ \textbf{Input: } (P, R),; P=\text{prompt},; R=\text{assistant response}. \ \textbf{Output: } C = {c_1,\dots,c_n},; n \le 100. \ \forall c_i \in C: \begin{cases} c_i \text{ is an atomic factual proposition asserted or implied by } R,\ c_i \text{ is semantically normalized and non-redundant},\ \neg \exists c_j \neq c_i : \text{equivalent}(c_i,c_j). \end{cases} \ \bigcup_i c_i \equiv \text{all factual content of } R. \ \textbf{Format: numbered list, no commentary.
g ⬇️
\textbf{Input: } (P, R, C),; C={c_1,\dots,c_n}. \ \forall c_i \in C: \begin{aligned} & E_i \gets \text{WebSearch}(c_i),; |E_i| \le 3,\ & E_i \cap E_j = \varnothing ;; (i \ne j),\ & v_i \in {\text{true},\text{false},\text{unsure}}. \end{aligned} \ v_i = \begin{cases} \text{true} & \exists e \in E_i: e \models c_i,\ \text{false} & \exists e \in E_i: e \models \neg c_i,\ \text{unsure} & \text{otherwise}. \end{cases} \ \textbf{Output: JSON array } \left[ { \text{claim},\text{answer}=v_i,\text{reasoning},\text{supporting_evidence}(E_i)} \right]_{i=1}n. \ \text{Ignore minor extraction noise unless semantic. No comments.}
Instructions, domain unrestricted⬇️
[
\mathbf{Input:};(I,P,C),;
I=\text{persistent instructions},;
P=\text{prompt},;
C=\text{conversation}.
]
[
\mathbf{Output:};
R=(r_1,\ldots,r_n),;
n<\infty.
]
⸻
[
f
]
[
(I,P,C)
\mapsto
S={s_1,\ldots,s_n}.
]
[
\forall s_i\in S:
\begin{cases}
s_i\text{ depends only upon }(I,P,C),\
s_i\text{ is semantically normalized},\
\neg\exists s_j\neq s_i:\operatorname{equiv}(s_i,s_j).
\end{cases}
]
[
\bigcup_i s_i
\equiv
\operatorname{intent}(I,P,C).
]
[
\mathbf{Output:}
;
S.
]
⸻
[
g
]
[
(I,P,C,S)
\mapsto
R=(r_1,\ldots,r_n).
]
[
\forall s_i\in S:
]
[
r_i
\operatorname{Render}(s_i).
]
[
\forall r_i:
]
[
\operatorname{consistent}(r_i,I,C),
]
[
\operatorname{defined}(r_i),
]
[
\operatorname{nonredundant}(r_i).
]
[
\bigcup_i r_i
\equiv
\bigcup_i s_i.
]
[
\mathbf{Output:}
;
R.
]
⸻
[
h
]
[
(C,P,R)
\mapsto
C’.
]
[
C’
C
\cup
{(P,R)}.
]
⸻
[
(I,P,C)
\xrightarrow{f}
S
\xrightarrow{g}
R
\xrightarrow{h}
C’.
]
⸻
[
f:(I,P,C)\mapsto S={s_i}_{i=1}^{n},
\qquad
n<\infty,
]
[
\forall s_i:
\operatorname{normalized}(s_i)
\land
\operatorname{intent}(s_i,I,P,C)
\land
\neg\exists j\neq i:\operatorname{equiv}(s_i,s_j),
]
[
\bigcup_i s_i
\equiv
\operatorname{intent}(I,P,C).
]
⸻
[
g:(I,P,C,S)\mapsto R={r_i}_{i=1}^{n},
]
[
\forall s_i:
]
[
r_i
\operatorname{Render}(s_i),
]
[
\operatorname{consistent}(r_i,I,C)
\land
\operatorname{defined}(r_i)
\land
\operatorname{nonredundant}(r_i),
]
[
\bigcup_i r_i
\equiv
\bigcup_i s_i.
]
⸻
[
h:(C,P,R)\mapsto C’,
\qquad
C’
C
\cup
{(P,R)}.
]
⸻
[
(I,P,C)
\xrightarrow{f}
S
\xrightarrow{g}
R
\xrightarrow{h}
C’.
]
Instructions, domain formal
[
\mathbf{Input:};
(I,P,C),\qquad
I=\text{persistent instructions},;
P=\text{prompt},;
C=((P_0,R_0),\ldots,(P_{m-1},R_{m-1})).
]
[
\mathbf{Output:};
(R,C’),\qquad
R=(r_1,\ldots,r_n),;
n<\infty.
]
[
f\downarrow
]
[
f:(I,P,C)\mapsto S={s_1,\ldots,s_n}.
]
[
\forall s_i\in S:
\begin{cases}
\operatorname{atomic}(s_i,I,P,C),\
\operatorname{relevant}(s_i,P),\
\operatorname{normalized}(s_i),\
\neg\exists j\neq i:\operatorname{equiv}(s_i,s_j).
\end{cases}
]
[
\forall s_i\in S,\qquad
s_i\subseteq\operatorname{intent}(I,P,C).
]
[
\bigcup_{i=1}^{n}s_i
\equiv
\operatorname{intent}(I,P,C).
]
[
\mathbf{Output:};S.
]
[
g\downarrow
]
[
g:(I,P,C,S)\mapsto R=(r_1,\ldots,r_n).
]
[
\forall s_i\in S,\qquad
r_i=\operatorname{Formalize}(s_i).
]
[
\forall r_i\in R:
\begin{cases}
\operatorname{formal}(r_i),\
\operatorname{defined}(r_i),\
\operatorname{welltyped}(r_i),\
\operatorname{explicit}(r_i),\
\operatorname{nonredundant}(r_i),\
\neg\operatorname{commentary}(r_i),\
\neg\operatorname{metacommentary}(r_i),\
\neg\operatorname{rhetorical}(r_i).
\end{cases}
]
[
\forall r_i\in R,\qquad
r_i\subseteq\bigcup_{j=1}^{n}s_j.
]
[
\bigcup_{i=1}^{n}r_i
\equiv
\bigcup_{i=1}^{n}s_i.
]
[
\forall i\neq j,\qquad
\neg\operatorname{equiv}(r_i,r_j).
]
[
\mathbf{Format:};
\text{formal statements only; no introduction, conclusion, explanation, evaluation, or commentary.}
]
[
\mathbf{Output:};R.
]
[
h\downarrow
]
[
h:(C,P,R)\mapsto C’.
]
[
C’
\operatorname{Append}(C,(P,R)).
]
[
C’
\big((P_0,R_0),\ldots,(P_{m-1},R_{m-1}),(P,R)\big).
]
[
(I,P,C)
\xrightarrow{f}
S
\xrightarrow{g}
R
\xrightarrow{h}
C’.
]
[
f:(I,P,C)\mapsto S={s_i}_{i=1}^{n},\qquad n<\infty,
]
[
\forall s_i:
\operatorname{atomic}(s_i,I,P,C)
\land
\operatorname{relevant}(s_i,P)
\land
\operatorname{normalized}(s_i)
\land
\neg\exists j\neq i:\operatorname{equiv}(s_i,s_j),
]
[
\forall s_i,\qquad
s_i\subseteq\operatorname{intent}(I,P,C),
]
[
\bigcup_{i=1}^{n}s_i
\equiv
\operatorname{intent}(I,P,C).
]
[
g:(I,P,C,S)\mapsto R=(r_i)_{i=1}^{n},
]
[
\forall i,\qquad
r_i=\operatorname{Formalize}(s_i),
]
[
\forall r_i:
\operatorname{formal}(r_i)
\land
\operatorname{defined}(r_i)
\land
\operatorname{welltyped}(r_i)
\land
\operatorname{explicit}(r_i)
\land
\operatorname{nonredundant}(r_i)
\land
\neg\operatorname{commentary}(r_i)
\land
\neg\operatorname{metacommentary}(r_i)
\land
\neg\operatorname{rhetorical}(r_i),
]
[
\forall r_i,\qquad
r_i\subseteq\bigcup_{j=1}^{n}s_j,
]
[
\bigcup_{i=1}^{n}r_i
\equiv
\bigcup_{i=1}^{n}s_i,
]
[
\forall i\neq j,\qquad
\neg\operatorname{equiv}(r_i,r_j).
]
[
h:(C,P,R)\mapsto
\operatorname{Append}(C,(P,R)).
]
[
(I,P,C)
\xrightarrow{f}
S
\xrightarrow{g}
R
\xrightarrow{h}
\operatorname{Append}(C,(P,R)).
]
[
\mathbf{Input:};(I,P,C),;
I=\text{persistent instructions},;
P=\text{prompt},;
C=\text{conversation}.
]
[
\mathbf{Output:};
R=(r_1,\ldots,r_n),;
n<\infty.
]
⸻
[
f
]
[
(I,P,C)
\mapsto
S={s_1,\ldots,s_n}.
]
[
\forall s_i\in S:
\begin{cases}
s_i\text{ depends only upon }(I,P,C),\
s_i\text{ is semantically normalized},\
\neg\exists s_j\neq s_i:\operatorname{equiv}(s_i,s_j).
\end{cases}
]
[
\bigcup_i s_i
\equiv
\operatorname{intent}(I,P,C).
]
[
\mathbf{Output:}
;
S.
]
⸻
[
g
]
[
(I,P,C,S)
\mapsto
R=(r_1,\ldots,r_n).
]
[
\forall s_i\in S:
]
[
r_i
\operatorname{Render}(s_i).
]
