UGC/CUR Formal Core Specification v0.1
無界生成閉包、類終極可達性與轉換閉包之形式核心規格
文件編號： EML-UGC-CUR-FCSPEC-2026-v0.1日期： 2026-08-26作者： Neo.K機構： EveMissLab／一言諾科技有限公司狀態： INTERNAL CANONICAL FORMAL CORE / PAPER-00 ANCHOR上游正典： UGC_CUR_Canonical_Reconciliation_v0.1_2026-08-26.md文件角色： 後續 Paper 01--05 的共同形式核心；不是宇宙學、神學或絕對本體論的完成證明。canonical source 規則： 本文件以 UTF-8 Markdown 為正式 source；所有數學只使用 $...$ 與 $$...$$ delimiter。
0. Release Decision
本文件固定 UGC/CUR 高階 closure layer 的第一版可推演形式核心。
核心研究對象不是一個裸 source、一個裸 agent 或一個裸 world，而是下列三族閉包及其責任記帳：
GenCl = what can be generated \boxed{
\operatorname{GenCl}
=
\text{what can be generated}
} GenCl = what can be generated R e a c h = what can be reached under a typed capability mode \boxed{
\mathsf{Reach}
=
\text{what can be reached under a typed capability mode}
} Reach = what can be reached under a typed capability mode TransCl = what transformations can be realized \boxed{
\operatorname{TransCl}
=
\text{what transformations can be realized}
} TransCl = what transformations can be realized 以及：
L e d g e r = how state, history, provenance, resource responsibility and unresolved debt are accounted for \boxed{
\mathsf{Ledger}
=
\text{how state, history, provenance, resource responsibility and unresolved debt are accounted for}
} Ledger = how state, history, provenance, resource responsibility and unresolved debt are accounted for 本文件採取以下最高層原則：
Unification ≠ Primitive Collapse . \boxed{
\text{Unification}
\neq
\text{Primitive Collapse}.
} Unification = Primitive Collapse . 因此 UGC/CUR 不取代 OBRC、RDSS、SST、DEST、Realizability、Ledger-Causal Mathematics 或 MWT；本文件只定義它們在高階 closure 問題上的共同接口。
1. Claim Types and Formal Status
任何正式聲明必須帶至少一個 claim status：
S c l a i m = { D E F , A X , P R O P , C O N J , M O D E L , O P E N } . \boxed{
\mathfrak S_{\rm claim}
=
\{
\mathsf{DEF},
\mathsf{AX},
\mathsf{PROP},
\mathsf{CONJ},
\mathsf{MODEL},
\mathsf{OPEN}
\}.
} S claim = { DEF , AX , PROP , CONJ , MODEL , OPEN } . 語義如下：
D E F \mathsf{DEF} DEF A X \mathsf{AX} AX P R O P \mathsf{PROP} PROP C O N J \mathsf{CONJ} CONJ M O D E L \mathsf{MODEL} MODEL O P E N \mathsf{OPEN} OPEN
任何 O P E N \mathsf{OPEN} OPEN P R O P \mathsf{PROP} PROP
2. Core Type Universe
2.1 Target Domain
令：
Ω D \boxed{
\Omega_D
} Ω D 表示本次聲明的 declared target domain。
它只表示「本模型宣告要討論的目標域」，不自動表示 metaphysically complete ontology。
因此：
Ω D ≠ Ω a b s o l u t e \boxed{
\Omega_D
\neq
\Omega_{\rm absolute}
} Ω D = Ω absolute 除非另有 completeness certificate。
2.2 Time / Evolution Horizon
令：
T ∈ T \boxed{
T
\in
\mathfrak T
} T ∈ T 表示演化 horizon。 T T T
若使用：
T = ∞ , T=\infty, T = ∞ , 它是明示模型條件，不得被隱藏為 source 的內在能力。
2.3 Judgement Context
定義 observer-indexed judgement context：
Θ = ⟨ o , s , ρ , t , O p s , R e p , P e r m , K n o w , W o r l d A s s u m p ⟩ . \boxed{
\Theta
=
\left\langle
o,
s,
\rho,
t,
\mathsf{Ops},
\mathsf{Rep},
\mathsf{Perm},
\mathsf{Know},
\mathsf{WorldAssump}
\right\rangle.
} Θ = ⟨ o , s , ρ , t , Ops , Rep , Perm , Know , WorldAssump ⟩ . 其中依序表示 observer、scale、resolution、time、operator set、representation regime、permission regime、knowledge regime 與 world assumption。
任何 existence、non-being、reachability、connectivity、law-invariance、unboundedness 或 class-ultimate claim 都必須可追溯到 Θ \Theta Θ
3. World / History / Law / Boundary Separation
3.1 Authoritative World State
令：
W t \boxed{
\mathfrak W_t
} W t 表示時間 t t t
3.2 History
令：
H ≤ t \boxed{
\mathfrak H_{\le t}
} H ≤ t 表示事件、路徑、版本與必要因果歷史。
本核心固定：
W t ≠ H ≤ t . \boxed{
\mathfrak W_t
\neq
\mathfrak H_{\le t}.
} W t = H ≤ t . 相同 current state 可以具有不同 relevant histories。
3.3 Law Regime
令：
L a w t \boxed{
\mathsf{Law}_t
} Law t 表示時間 t t t
它可以是固定的、分層的、狀態依賴的或模型中允許演化的；本文件不預設唯一宇宙法則模型。
3.4 Boundary Family
令：
B t \boxed{
\mathfrak B_t
} B t 表示 typed, state-bearing boundary family。
邊界可以具有傳輸、過濾、轉碼、阻擋、權限、耦合與狀態更新，因此：
B o u n d a r y ≠ D i s c o n n e c t i o n . \boxed{
\mathsf{Boundary}
\neq
\mathsf{Disconnection}.
} Boundary = Disconnection . 3.5 Relation Family
令：
R t \boxed{
\mathfrak R_t
} R t 表示可被判定、建立、移除或改寫的 typed relation family。
4. Canonical Generative Environment
定義生成環境：
E t g e n = ⟨ C a r t , L a w t , B t , O p s t , H ≤ t , E x t t , C t g e n ⟩ . \boxed{
\mathfrak E_t^{\rm gen}
=
\left\langle
\mathsf{Car}_t,
\mathsf{Law}_t,
\mathfrak B_t,
\mathsf{Ops}_t,
\mathfrak H_{\le t},
\mathsf{Ext}_t,
\mathcal C_t^{\rm gen}
\right\rangle.
} E t gen = ⟨ Car t , Law t , B t , Ops t , H ≤ t , Ext t , C t gen ⟩ . 其中：
C a r t \mathsf{Car}_t Car t L a w t \mathsf{Law}_t Law t B t \mathfrak B_t B t O p s t \mathsf{Ops}_t Ops t H ≤ t \mathfrak H_{\le t} H ≤ t E x t t \mathsf{Ext}_t Ext t C t g e n \mathcal C_t^{\rm gen} C t gen
本文件固定：
S o u r c e ≠ C a r r i e r ≠ L a w ≠ B o u n d a r y ≠ E x t e r n a l I n p u t . \boxed{
\mathsf{Source}
\neq
\mathsf{Carrier}
\neq
\mathsf{Law}
\neq
\mathsf{Boundary}
\neq
\mathsf{ExternalInput}.
} Source = Carrier = Law = Boundary = ExternalInput . 5. Generative Transition System
5.1 Generative Configuration
定義生成 configuration space：
X g e n \boxed{
\mathfrak X^{\rm gen}
} X gen 其元素至少可以承載：
χ t = ⟨ W t , L a w t , E t g e n ⟩ . \boxed{
\chi_t
=
\left\langle
\mathfrak W_t,
\mathsf{Law}_t,
\mathfrak E_t^{\rm gen}
\right\rangle.
} χ t = ⟨ W t , Law t , E t gen ⟩ . 5.2 Source Initialization
source S S S
I n i t D : ( S , E 0 g e n ) ⇀ X g e n . \boxed{
\mathsf{Init}_{D}
:
(S,\mathfrak E_0^{\rm gen})
\rightharpoonup
\mathfrak X^{\rm gen}.
} Init D : ( S , E 0 gen ) ⇀ X gen . 若初始化本身需要未列明外部來源，該依賴必須進入 Generative Responsibility record。
5.3 Generative Step
定義部分生成步：
G e n S t e p t : X g e n ⇀ X g e n . \boxed{
\mathsf{GenStep}_t
:
\mathfrak X^{\rm gen}
\rightharpoonup
\mathfrak X^{\rm gen}.
} GenStep t : X gen ⇀ X gen . 允許非決定性或分支時，可改寫為 relation：
χ t ⇝ g e n χ t + 1 . \boxed{
\chi_t
\rightsquigarrow_{\rm gen}
\chi_{t+1}.
} χ t ⇝ gen χ t + 1 . 任何生成 trace 必須保存必要 provenance。
5.4 Output Projection
定義模型內的生成結果投影：
O u t D : X g e n ⇀ P ( Ω D ) . \boxed{
\mathsf{Out}_D
:
\mathfrak X^{\rm gen}
\rightharpoonup
\mathcal P(\Omega_D).
} Out D : X gen ⇀ P ( Ω D ) . 這避免把整個 configuration 與被研究 outcome 混為同一物件。
6. Canonical Generative Closure
6.1 Trace Set
令：
T r a c e ≤ T ( S ∣ E g e n ) \mathsf{Trace}_{\le T}
(S\mid\mathfrak E^{\rm gen}) Trace ≤ T ( S ∣ E gen ) 表示由 I n i t \mathsf{Init} Init G e n S t e p \mathsf{GenStep} GenStep C g e n \mathcal C^{\rm gen} C gen T T T
6.2 Definition of Generative Closure
正式定義：
GenCl D , T ( S ∣ E g e n ) = { y ∈ Ω D | ∃ τ ∈ T r a c e ≤ T ( S ∣ E g e n ) , ∃ χ ∈ τ : y ∈ O u t D ( χ ) } . \boxed{
\operatorname{GenCl}_{D,T}
\left(
S
\mid
\mathfrak E^{\rm gen}
\right)
=
\left\{
y\in\Omega_D
\;\middle|\;
\exists\tau
\in
\mathsf{Trace}_{\le T}
(S\mid\mathfrak E^{\rm gen}),
\exists\chi\in\tau:
y\in\mathsf{Out}_D(\chi)
\right\}.
} GenCl D , T ( S ∣ E gen ) = { y ∈ Ω D ∣ ∃ τ ∈ Trace ≤ T ( S ∣ E gen ) , ∃ χ ∈ τ : y ∈ Out D ( χ ) } . 此定義取代舊版跨系列的裸生成閉包記法。
6.3 Fixed-Law Generative Closure
若：
L a w t = L a w 0 ∀ t ≤ T , \boxed{
\mathsf{Law}_t
=
\mathsf{Law}_0
\qquad
\forall t\le T,
} Law t = Law 0 ∀ t ≤ T , 稱為 fixed-law generative closure。
6.4 Law-Coevolving Generative Closure
若允許：
L a w t + 1 ≠ L a w t , \boxed{
\mathsf{Law}_{t+1}
\neq
\mathsf{Law}_t,
} Law t + 1 = Law t , 則 law state 必須成為 χ t \chi_t χ t
6.5 Reflexive Generative Closure
若連 G e n S t e p t \mathsf{GenStep}_t GenStep t
χ t r e f = ⟨ W t , L a w t , E t g e n , G e n S t e p t ⟩ . \boxed{
\chi_t^{\rm ref}
=
\left\langle
\mathfrak W_t,
\mathsf{Law}_t,
\mathfrak E_t^{\rm gen},
\mathsf{GenStep}_t
\right\rangle.
} χ t ref = ⟨ W t , Law t , E t gen , GenStep t ⟩ . 此時任何 G e n S t e p \mathsf{GenStep} GenStep
因此：
Reflexive Generative Closure ≠ Rulelessness . \boxed{
\text{Reflexive Generative Closure}
\neq
\text{Rulelessness}.
} Reflexive Generative Closure = Rulelessness . 是否存在非循環、非空洞的 reflexive grounding 保持 O P E N \mathsf{OPEN} OPEN
7. Generative Sufficiency
對 declared generative target：
Ω D g e n ⊆ Ω D , \boxed{
\Omega_D^{\rm gen}
\subseteq
\Omega_D,
} Ω D gen ⊆ Ω D , 定義：
G e n S u f f i c i e n t D , T ( S ∣ E g e n ) = 1 \boxed{
\mathsf{GenSufficient}_{D,T}
\left(
S\mid\mathfrak E^{\rm gen}
\right)
=1
} GenSufficient D , T ( S ∣ E gen ) = 1 若且唯若：
Ω D g e n ⊆ GenCl D , T ( S ∣ E g e n ) . \boxed{
\Omega_D^{\rm gen}
\subseteq
\operatorname{GenCl}_{D,T}
\left(
S\mid\mathfrak E^{\rm gen}
\right).
} Ω D gen ⊆ GenCl D , T ( S ∣ E gen ) . 本文件固定：
G e n S u f f i c i e n t ⇏ O n t o l o g i c a l l y F i r s t . \boxed{
\mathsf{GenSufficient}
\not\Rightarrow
\mathsf{OntologicallyFirst}.
} GenSufficient ⇒ OntologicallyFirst . 生成充分性是 domain-relative coverage claim；ontological priority 需要額外 bridge。
8. Generative Responsibility Decomposition
8.1 Responsibility Record
對 outcome y y y
G R ( y ) = ⟨ S , C a r , L a w , B , O p s , H , E x t , R a n d , C g e n , W i t + , D e b t ⟩ . \boxed{
\mathsf{GR}(y)
=
\left\langle
S,
\mathsf{Car},
\mathsf{Law},
\mathfrak B,
\mathsf{Ops},
\mathfrak H,
\mathsf{Ext},
\mathsf{Rand},
\mathcal C^{\rm gen},
\mathsf{Wit}^{+},
\mathsf{Debt}
\right\rangle.
} GR ( y ) = ⟨ S , Car , Law , B , Ops , H , Ext , Rand , C gen , Wit + , Debt ⟩ . 其中 D e b t \mathsf{Debt} Debt
8.2 Responsibility Hypergraph
允許以有向超圖表示生成依賴：
R e s p G r a p h ( y ) = ( V y , E y r e s p ) . \boxed{
\mathsf{RespGraph}(y)
=
(V_y,E_y^{\rm resp}).
} RespGraph ( y ) = ( V y , E y resp ) . 節點可以是 source、carrier、law、operator、boundary、history state、external feed 或 intermediate outcome；超邊表示在指定模型中一組 antecedents 對某結果的生成責任。
8.3 Accounted Resource
對必要資源 r r r r r r
明示 provenance；
declared exogenous status；
下一層 responsibility record；
明示 O P E N \mathsf{OPEN} OPEN
因此 accounted 不等於 grounded；它只表示依賴沒有被隱藏。
8.4 Responsibility Closure
定義：
R e s p C l o s e d D , T ( S ) = 1 \boxed{
\mathsf{RespClosed}_{D,T}(S)=1
} RespClosed D , T ( S ) = 1 若對所有被納入 sufficiency proof 的 outcome 與必要生成資源，其 dependency chain 在 declared model boundary 內全部被 accounted，且不存在未標記依賴。
若存在 O P E N \mathsf{OPEN} OPEN
9. Generative Responsibility Axioms
GR-A1 — No Unaccounted Generative Resource
任何對 outcome 必要的生成資源，都必須出現在責任記錄或明示 debt 中。 \boxed{
\text{任何對 outcome 必要的生成資源，都必須出現在責任記錄或明示 debt 中。}
} 任何對 outcome 必要的生成資源，都必須出現在責任記錄或明示 debt 中。 GR-A2 — No Source-Alone Upgrade
若：
GenCl D , T ( S ∣ E g e n ) \operatorname{GenCl}_{D,T}
(S\mid\mathfrak E^{\rm gen}) GenCl D , T ( S ∣ E gen ) 依賴非平凡環境，則不得省略 E g e n \mathfrak E^{\rm gen} E gen
GR-A3 — Recursive Attribution
若必要資源來自更高來源 S − 1 S_{-1} S − 1
S − 1 ∈ D e c l a r e d B o u n d a r y \boxed{
S_{-1}
\in
\mathsf{DeclaredBoundary}
} S − 1 ∈ DeclaredBoundary 或建立下一層 G R \mathsf{GR} GR O P E N \mathsf{OPEN} OPEN
GR-A4 — Accounting Is Not Ultimate Grounding
R e s p C l o s e d = 1 ⇏ O n t o l o g i c a l G r o u n d i n g C o m p l e t e = 1. \boxed{
\mathsf{RespClosed}=1
\not\Rightarrow
\mathsf{OntologicalGroundingComplete}=1.
} RespClosed = 1 ⇒ OntologicalGroundingComplete = 1. 責任閉包先保證「沒有未記帳資源」，不保證「形上學終極來源已解決」。
10. Typed Unboundedness Taxonomy
10.1 Status Space
定義：
U = { b o u n d e d , o p e n , u n b o u n d e d , u n k n o w n } . \boxed{
\mathfrak U
=
\{
\mathsf{bounded},
\mathsf{open},
\mathsf{unbounded},
\mathsf{unknown}
\}.
} U = { bounded , open , unbounded , unknown } . 10.2 Unboundedness Profile
定義：
U = ( u s t a t e , u t y p e , u r e l a t i o n , u l a w , u o p e r a t o r , u t i m e , u i n f o r m a t i o n , u o n t o l o g y , u r e a c h ) . \boxed{
\mathbf U
=
\left(
u_{\rm state},
u_{\rm type},
u_{\rm relation},
u_{\rm law},
u_{\rm operator},
u_{\rm time},
u_{\rm information},
u_{\rm ontology},
u_{\rm reach}
\right).
} U = ( u state , u type , u relation , u law , u operator , u time , u information , u ontology , u reach ) . 每一分量取值於 U \mathfrak U U
10.3 Open Is Not Unbounded
o p e n ⇏ u n b o u n d e d . \boxed{
\mathsf{open}
\not\Rightarrow
\mathsf{unbounded}.
} open ⇒ unbounded . o p e n \mathsf{open} open u n b o u n d e d \mathsf{unbounded} unbounded
10.4 Generic Unboundedness Certificate
對量測泛函：
ϕ : Ω D → R ≥ 0 , \phi:
\Omega_D
\rightarrow
\mathbb R_{\ge 0}, ϕ : Ω D → R ≥ 0 , 定義 ϕ \phi ϕ
∀ M < ∞ , ∃ y ∈ GenCl D , T ( S ∣ E g e n ) : ϕ ( y ) > M . \boxed{
\forall M<\infty,
\exists y\in
\operatorname{GenCl}_{D,T}
(S\mid\mathfrak E^{\rm gen})
:
\phi(y)>M.
} ∀ M < ∞ , ∃ y ∈ GenCl D , T ( S ∣ E gen ) : ϕ ( y ) > M . 若沒有明示 ϕ \phi ϕ
10.5 Open-Dimensional Bridge
允許：
∣ J e f f ( Q , t , ε ) ∣ < ∞ \boxed{
\left|
J_{\rm eff}(Q,t,\varepsilon)
\right|<\infty
} ∣ J eff ( Q , t , ε ) ∣ < ∞ 同時未來 dimension vocabulary 不預先封閉。
因此：
Potentially Open ≠ Infinitely Active at Every Instant . \boxed{
\text{Potentially Open}
\neq
\text{Infinitely Active at Every Instant}.
} Potentially Open = Infinitely Active at Every Instant . 11. Typed Reachability Calculus
11.1 Capability Mode Family
定義 capability modes：
M c a p = { o b s e r v e , a c c e s s , a c t , c o n t r o l , t r a n s f o r m , r u l e R e w r i t e , g e n R e w r i t e , v e r i f y } . \boxed{
\mathfrak M_{\rm cap}
=
\{
\mathsf{observe},
\mathsf{access},
\mathsf{act},
\mathsf{control},
\mathsf{transform},
\mathsf{ruleRewrite},
\mathsf{genRewrite},
\mathsf{verify}
\}.
} M cap = { observe , access , act , control , transform , ruleRewrite , genRewrite , verify } . 這些 mode 預設只是一個 typed family，不預設固定全序。
特別地，不自動宣稱：
o b s e r v e < a c c e s s < a c t < c o n t r o l . \mathsf{observe}
<
\mathsf{access}
<
\mathsf{act}
<
\mathsf{control}. observe < access < act < control . 模式間蘊含必須由 domain-specific bridge theorem 或 contract 明示。
11.2 Reachability Judgement
對 agent A A A x x x m m m R R R Θ \Theta Θ t t t
R e a c h Θ , R m ( A , x , t ) ∈ J . \boxed{
\mathsf{Reach}^{m}_{\Theta,R}
(A,x,t)
\in
\mathfrak J.
} Reach Θ , R m ( A , x , t ) ∈ J . 其中 judgement status：
J = { 1 , 0 , ? , B , S } . \boxed{
\mathfrak J
=
\{
1,
0,
?,
\mathsf{B},
\mathsf{S}
\}.
} J = { 1 , 0 , ? , B , S } . 語義分別為 pass、fail、unknown、branch-dependent、scope-dependent。
11.3 Positive Reach Witness
若：
R e a c h Θ , R m ( A , x , t ) = 1 , \mathsf{Reach}^{m}_{\Theta,R}(A,x,t)=1, Reach Θ , R m ( A , x , t ) = 1 , 至少需要：
W i t r e a c h + = ⟨ π , B π , R π , O p s π , C o n d π , C e r t π ⟩ , \boxed{
\mathsf{Wit}^{+}_{\rm reach}
=
\left\langle
\pi,
\mathfrak B_{\pi},
R_{\pi},
\mathsf{Ops}_{\pi},
\mathsf{Cond}_{\pi},
\mathsf{Cert}_{\pi}
\right\rangle,
} Wit reach + = ⟨ π , B π , R π , Ops π , Cond π , Cert π ⟩ , 其中 π \pi π
11.4 Negative Reach Certificate
若要將 judgement 設為 0 0 0 ? ? ?
exhaustive finite search certificate；
cut / barrier certificate；
violated necessary condition；
permission impossibility certificate；
model-checking proof；
declared scope completeness certificate。
因此：
No path found ⇏ R e a c h = 0. \boxed{
\text{No path found}
\not\Rightarrow
\mathsf{Reach}=0.
} No path found ⇒ Reach = 0. 12. Reach Profile
對 agent A A A x x x
R A ( x , t ∣ Θ ) = ( r o b s , r a c c e s s , r a c t , r c t r l , r t r a n s , r r u l e , r g e n , r v e r ) , \boxed{
\mathbf R_A(x,t\mid\Theta)
=
\left(
r_{\rm obs},
r_{\rm access},
r_{\rm act},
r_{\rm ctrl},
r_{\rm trans},
r_{\rm rule},
r_{\rm gen},
r_{\rm ver}
\right),
} R A ( x , t ∣ Θ ) = ( r obs , r access , r act , r ctrl , r trans , r rule , r gen , r ver ) , 其中每一分量屬於 J \mathfrak J J
因此：
Seeing More ≠ Being Higher \boxed{
\text{Seeing More}
\neq
\text{Being Higher}
} Seeing More = Being Higher 被形式化為 mode separation，而不是倫理或本體等級判斷。
13. Transformation Closure
13.1 Transformation Contract
令 transformation τ \tau τ
τ : X p r e ⇀ X p o s t . \boxed{
\tau:
X_{\rm pre}
\rightharpoonup
X_{\rm post}.
} τ : X pre ⇀ X post . 每個 transformation 至少帶：
C o n t r a c t ( τ ) = ⟨ P r e , P o s t , I n v , B o u n d a r y , A u t h , V e r i f y ⟩ . \boxed{
\mathsf{Contract}(\tau)
=
\left\langle
\mathsf{Pre},
\mathsf{Post},
\mathsf{Inv},
\mathsf{Boundary},
\mathsf{Auth},
\mathsf{Verify}
\right\rangle.
} Contract ( τ ) = ⟨ Pre , Post , Inv , Boundary , Auth , Verify ⟩ . 13.2 Realized Transformation Judgement
定義：
R e a l i z e T r a n s Θ ( A , τ , x , t ) ∈ J . \boxed{
\mathsf{RealizeTrans}_{\Theta}
(A,\tau,x,t)
\in
\mathfrak J.
} RealizeTrans Θ ( A , τ , x , t ) ∈ J . 其中 pass 需要 action / state delta / outcome / verification witness。
13.3 Transformation Closure
定義：
TransCl D , T ( A ∣ Θ ) = { ( x , τ ) | x ∈ Ω D , ∃ t ≤ T : R e a l i z e T r a n s Θ ( A , τ , x , t ) = 1 } . \boxed{
\operatorname{TransCl}_{D,T}
(A\mid\Theta)
=
\left\{
(x,\tau)
\;\middle|\;
x\in\Omega_D,
\exists t\le T:
\mathsf{RealizeTrans}_{\Theta}(A,\tau,x,t)=1
\right\}.
} TransCl D , T ( A ∣ Θ ) = { ( x , τ ) ∣ x ∈ Ω D , ∃ t ≤ T : RealizeTrans Θ ( A , τ , x , t ) = 1 } . 此物件回答「對哪些 target，可實現哪些 transformation」，不等同於 target reach set。
因此：
R e a c h t r a n s f o r m = 1 ⇏ all transformations are realizable . \boxed{
\mathsf{Reach}^{\mathsf{transform}}=1
\not\Rightarrow
\text{all transformations are realizable}.
} Reach transform = 1 ⇒ all transformations are realizable . 14. Realizability Interface
UGC/CUR 不重造完整 Realizability theory。
定義 adapter：
R e a l i z a b l e Θ ( A , τ , x , t ) \boxed{
\mathsf{Realizable}_{\Theta}
(A,\tau,x,t)
} Realizable Θ ( A , τ , x , t ) 由既有 Realizability layer 判定 physical、engineering、normative、reversible、verifiable 等條件。
固定：
C o n n e c t i v i t y ≠ R e a c h a b i l i t y ≠ R e a l i z a b i l i t y . \boxed{
\mathsf{Connectivity}
\neq
\mathsf{Reachability}
\neq
\mathsf{Realizability}.
} Connectivity = Reachability = Realizability . 以及：
R e a c h = 1 ⇏ R e a l i z a b l e = 1. \boxed{
\mathsf{Reach}=1
\not\Rightarrow
\mathsf{Realizable}=1.
} Reach = 1 ⇒ Realizable = 1. 15. Meta-Causal Hierarchy
Meta-causality 一律相對某 baseline layer L L L
MC0 — Ordinary State Intervention
只改變 ordinary state：
W t → W t + 1 . \boxed{
\mathfrak W_t
\rightarrow
\mathfrak W_{t+1}.
} W t → W t + 1 . MC1 — Relation / Boundary / Causal-Topology Rewrite
至少改變：
( R t , B t ) → ( R t + 1 , B t + 1 ) . \boxed{
(\mathfrak R_t,\mathfrak B_t)
\rightarrow
(\mathfrak R_{t+1},\mathfrak B_{t+1}).
} ( R t , B t ) → ( R t + 1 , B t + 1 ) . MC2 — Transition Rule / Operator Rewrite
至少改變：
O p s t → O p s t + 1 \boxed{
\mathsf{Ops}_t
\rightarrow
\mathsf{Ops}_{t+1}
} Ops t → Ops t + 1 或 object-level transition contract。
MC3 — Law-Regime Update
至少改變：
L a w t → L a w t + 1 . \boxed{
\mathsf{Law}_t
\rightarrow
\mathsf{Law}_{t+1}.
} Law t → Law t + 1 . MC4 — Generative-Rule / Meta-Law Rewrite Candidate
至少改變：
G e n S t e p t → G e n S t e p t + 1 \boxed{
\mathsf{GenStep}_t
\rightarrow
\mathsf{GenStep}_{t+1}
} GenStep t → GenStep t + 1 或其 meta-law specification。
定義 certified meta-causal level：
M C L e v e l D , T , Θ ( A ∣ L ) = max { k : W i t M C k + ( A ) exists } . \boxed{
\mathsf{MCLevel}_{D,T,\Theta}(A\mid L)
=
\max
\left\{
k:\mathsf{Wit}^{+}_{\mathsf{MC}_k}(A)\text{ exists}\right\}.
} MCLevel D , T , Θ ( A ∣ L ) = max { k : Wit MC k + ( A ) exists } . 若最大值不存在或 scope 未閉合，回傳 ? ? ?
本文件固定：
M C k ⇏ absolute transcendence of all law . \boxed{
\mathsf{MC}_k
\not\Rightarrow
\text{absolute transcendence of all law}.
} MC k ⇒ absolute transcendence of all law . 16. Class-Ultimate Capability
16.1 Required Mode and Relation Sets
令：
M ⋆ ⊆ M c a p \boxed{
\mathcal M^{\star}
\subseteq
\mathfrak M_{\rm cap}
} M ⋆ ⊆ M cap 與：
R ⋆ ⊆ R . \boxed{
\mathfrak R^{\star}
\subseteq
\mathfrak R.
} R ⋆ ⊆ R . 16.2 Typed Coverage
定義：
C o v e r a g e D , T ( A ) = { ( x , m ) ∈ Ω D × M ⋆ | ∃ t ≤ T , ∃ R ∈ R ⋆ : R e a c h Θ , R m ( A , x , t ) = 1 } . \boxed{
\mathsf{Coverage}_{D,T}
(A)
=
\left\{
(x,m)
\in
\Omega_D\times\mathcal M^{\star}
\;\middle|\;
\exists t\le T,
\exists R\in\mathfrak R^{\star}:
\mathsf{Reach}^{m}_{\Theta,R}(A,x,t)=1
\right\}.
} Coverage D , T ( A ) = { ( x , m ) ∈ Ω D × M ⋆ ∃ t ≤ T , ∃ R ∈ R ⋆ : Reach Θ , R m ( A , x , t ) = 1 } . 16.3 Class-Ultimate Candidate
若：
C o v e r a g e D , T ( A ) = Ω D × M ⋆ , \boxed{
\mathsf{Coverage}_{D,T}(A)
=
\Omega_D\times\mathcal M^{\star},
} Coverage D , T ( A ) = Ω D × M ⋆ , 且每一個 pass 都有 witness，則：
C l a s s U l t i m a t e C a n d i d a t e ( A ∣ D , T , M ⋆ , R ⋆ , Θ ) . \boxed{
\mathsf{ClassUltimateCandidate}
\left(
A\mid
D,T,\mathcal M^{\star},\mathfrak R^{\star},\Theta
\right).
} ClassUltimateCandidate ( A ∣ D , T , M ⋆ , R ⋆ , Θ ) . 若 M ⋆ \mathcal M^{\star} M ⋆ o b s e r v e \mathsf{observe} observe
16.4 Transformation-Complete Candidate
對 declared transformation class T ⋆ \mathcal T^{\star} T ⋆
Ω D × T ⋆ ⊆ TransCl D , T ( A ∣ Θ ) , \boxed{
\Omega_D\times\mathcal T^{\star}
\subseteq
\operatorname{TransCl}_{D,T}(A\mid\Theta),
} Ω D × T ⋆ ⊆ TransCl D , T ( A ∣ Θ ) , 可稱 transformation-complete candidate relative to T ⋆ \mathcal T^{\star} T ⋆
這仍不推出 ontological priority 或 absolute omnipotence。
17. Canonical Ledger Minimum Specification
定義：
L e d g e r t = ⟨ W t , H ≤ t , P ≤ t c a u s a l , L a w L o g ≤ t , B o u n d a r y L o g ≤ t , I n f o A c c t ≤ t , R e s p A c c t ≤ t , C e r t ≤ t , D e b t ≤ t ⟩ . \boxed{
\mathsf{Ledger}_t
=
\left\langle
\mathfrak W_t,
\mathfrak H_{\le t},
\mathfrak P^{\rm causal}_{\le t},
\mathsf{LawLog}_{\le t},
\mathsf{BoundaryLog}_{\le t},
\mathsf{InfoAcct}_{\le t},
\mathsf{RespAcct}_{\le t},
\mathsf{Cert}_{\le t},
\mathsf{Debt}_{\le t}
\right\rangle.
} Ledger t = ⟨ W t , H ≤ t , P ≤ t causal , LawLog ≤ t , BoundaryLog ≤ t , InfoAcct ≤ t , RespAcct ≤ t , Cert ≤ t , Debt ≤ t ⟩ . 其中：
P ≤ t c a u s a l \mathfrak P^{\rm causal}_{\le t} P ≤ t causal I n f o A c c t ≤ t \mathsf{InfoAcct}_{\le t} InfoAcct ≤ t R e s p A c c t ≤ t \mathsf{RespAcct}_{\le t} RespAcct ≤ t D e b t ≤ t \mathsf{Debt}_{\le t} Debt ≤ t
17.1 Local Projection
observer 只得到：
L o c a l L e d g e r o , Θ ( t ) = Π o , Θ ( L e d g e r t ) . \boxed{
\mathsf{LocalLedger}_{o,\Theta}(t)
=
\Pi_{o,\Theta}
\left(
\mathsf{Ledger}_t
\right).
} LocalLedger o , Θ ( t ) = Π o , Θ ( Ledger t ) . 因此：
L o c a l L e d g e r ≠ L e d g e r . \boxed{
\mathsf{LocalLedger}
\neq
\mathsf{Ledger}.
} LocalLedger = Ledger . 17.2 Accounting Discipline
核心只要求：
O u t p u t ⇒ S o u r c e / T r a n s f o r m / E x t e r n a l / L o s s / U n r e s o l v e d A c c o u n t i n g . \boxed{
\mathsf{Output}
\Rightarrow
\mathsf{Source/Transform/External/Loss/Unresolved\ Accounting}.
} Output ⇒ Source/Transform/External/Loss/Unresolved Accounting . 本文件不把：
I ( L e d g e r t ) = I ( L e d g e r 0 ) \mathcal I(\mathsf{Ledger}_t)
=
\mathcal I(\mathsf{Ledger}_0) I ( Ledger t ) = I ( Ledger 0 ) 設為公理。
因此：
local information loss ⇏ global information destruction , \boxed{
\text{local information loss}
\not\Rightarrow
\text{global information destruction},
} local information loss ⇒ global information destruction , 同時：
local information loss ⇏ global information preservation . \boxed{
\text{local information loss}
\not\Rightarrow
\text{global information preservation}.
} local information loss ⇒ global information preservation . 兩個方向都不得偷渡。
18. First-Cause Sufficiency Test
本文件使用 First-Cause Sufficiency Test，縮寫：
F C S . \boxed{
\mathsf{FCS}.
} FCS . 它不是「證明上帝」或「證明絕對第一因」的測試，而是對 first-cause candidate 的結構充分性與責任閉包進行分層判定。
18.1 Input Record
對候選 S 0 S_0 S 0
F C S I n p u t = ⟨ S 0 , Ω D g e n , T , Θ , E g e n , U , G R , C o m p C e r t ⟩ . \boxed{
\mathsf{FCSInput}
=
\left\langle
S_0,
\Omega_D^{\rm gen},
T,
\Theta,
\mathfrak E^{\rm gen},
\mathbf U,
\mathsf{GR},
\mathsf{CompCert}
\right\rangle.
} FCSInput = ⟨ S 0 , Ω D gen , T , Θ , E gen , U , GR , CompCert ⟩ . 18.2 Test Vector
定義：
F C S = ( f t y p e d , f c o v e r , f r e s p , f h i d d e n , f l a w , f c a r r i e r , f e x t , f u n b o u n d , f c o m p , f p r i o r i t y ) . \boxed{
\mathbf F_{\rm CS}
=
\left(
f_{\rm typed},
f_{\rm cover},
f_{\rm resp},
f_{\rm hidden},
f_{\rm law},
f_{\rm carrier},
f_{\rm ext},
f_{\rm unbound},
f_{\rm comp},
f_{\rm priority}
\right).
} F CS = ( f typed , f cover , f resp , f hidden , f law , f carrier , f ext , f unbound , f comp , f priority ) . 各分量分別檢查：
target / source / environment 是否 well-typed；
generative sufficiency 是否成立；
responsibility 是否閉合或已明示 debt；
是否存在 hidden higher source；
law grounding status；
carrier grounding status；
external input / randomness / oracle status；
若 target claim 涉及 unboundedness，是否具有相容的 typed certificate；
scope completeness；
ontological priority bridge。
18.3 FCS Levels
定義最低分級：
F C S 0 : ill-typed / scope undefined , F C S 1 : generatively insufficient , F C S 2 : conditionally generatively sufficient , F C S 3 : responsibility-closed first-cause candidate , F C S 4 : ontological-priority candidate with explicit bridge obligations . \boxed{
\begin{aligned}
\mathsf{FCS}_0 &: \text{ill-typed / scope undefined},\\
\mathsf{FCS}_1 &: \text{generatively insufficient},\\
\mathsf{FCS}_2 &: \text{conditionally generatively sufficient},\\
\mathsf{FCS}_3 &: \text{responsibility-closed first-cause candidate},\\
\mathsf{FCS}_4 &: \text{ontological-priority candidate with explicit bridge obligations}.
\end{aligned}
} FCS 0 FCS 1 FCS 2 FCS 3 FCS 4 : ill-typed / scope undefined , : generatively insufficient , : conditionally generatively sufficient , : responsibility-closed first-cause candidate , : ontological-priority candidate with explicit bridge obligations . F C S 4 \mathsf{FCS}_4 FCS 4
若 completeness 或 ontological bridge 未證，status 保持 scope-relative 或 O P E N \mathsf{OPEN} OPEN
19. First Cause and Absolute Nothingness Decoupling
本核心不以 creation-from-absolute-nothing 作為 generative sufficiency 的必要前提。
固定：
F i r s t C a u s e S u f f i c i e n c y ⊥ A b s o l u t e N o t h i n g n e s s T h e s i s . \boxed{
\mathsf{FirstCauseSufficiency}
\perp
\mathsf{AbsoluteNothingnessThesis}.
} FirstCauseSufficiency ⊥ AbsoluteNothingnessThesis . 第一因候選的最低生成問題只要求：
Ω D g e n ⊆ GenCl D , T ( S 0 ∣ E g e n ) . \boxed{
\Omega_D^{\rm gen}
\subseteq
\operatorname{GenCl}_{D,T}
(S_0\mid\mathfrak E^{\rm gen}).
} Ω D gen ⊆ GenCl D , T ( S 0 ∣ E gen ) . 不要求先證明某個 ordinary state：
A b s o l u t e N o t h i n g n e s s \mathrm{AbsoluteNothingness} AbsoluteNothingness 曾經存在。
本核心同時保留 typed negative-state discipline：zero、empty、undefined、absent、inaccessible、nondenoting、unknown 不得互換。
20. Local-to-Absolute Gate
UGC/CUR 直接採用：
G L A . \boxed{
\mathcal G_{\rm LA}.
} G LA . 對 local claim P Θ , D P_{\Theta,D} P Θ , D P a b s P_{\rm abs} P abs
P Θ , D ⇒ G L A P a b s \boxed{
P_{\Theta,D}
\xRightarrow{\mathcal G_{\rm LA}}
P_{\rm abs}
} P Θ , D G LA P abs 若至少具備：
C o m p C e r t ( Θ , D , P ) = 1 \boxed{
\mathsf{CompCert}(\Theta,D,P)=1
} CompCert ( Θ , D , P ) = 1 以及足以覆蓋缺失作用域的 bridge witness。
典型禁止：
R e a c h Θ , R m ( A , x , t ) = 0 ⇏ A b s o l u t e U n r e a c h a b i l i t y ( A , x ) , \boxed{
\mathsf{Reach}^{m}_{\Theta,R}(A,x,t)=0
\not\Rightarrow
\mathsf{AbsoluteUnreachability}(A,x),
} Reach Θ , R m ( A , x , t ) = 0 ⇒ AbsoluteUnreachability ( A , x ) , G e n S u f f i c i e n t D , T = 1 ⇏ A b s o l u t e F i r s t C a u s e , \boxed{
\mathsf{GenSufficient}_{D,T}=1
\not\Rightarrow
\mathsf{AbsoluteFirstCause},
} GenSufficient D , T = 1 ⇒ AbsoluteFirstCause , M C 4 ⇏ transcendence of all possible meta-law . \boxed{
\mathsf{MC}_4
\not\Rightarrow
\text{transcendence of all possible meta-law}.
} MC 4 ⇒ transcendence of all possible meta-law . 21. Formal Core Axioms / Protocol Invariants
以下皆為 UGC/CUR formal protocol axioms，不宣稱為宇宙先驗真理。
FC-A1 — Type Before Claim
任何 claim 在判真前先確定 domain、context、relation、mode 與 horizon。
FC-A2 — Witness Before Positive Upgrade
J = 1 ⇒ ∃ W i t + . \boxed{
J=1
\Rightarrow
\exists\mathsf{Wit}^{+}.
} J = 1 ⇒ ∃ Wit + . FC-A3 — Negative Claim Requires Scoped Obstruction
J = 0 ⇒ ∃ W i t s c o p e d − . \boxed{
J=0
\Rightarrow
\exists\mathsf{Wit}^{-}_{\rm scoped}.
} J = 0 ⇒ ∃ Wit scoped − . FC-A4 — State / History / Law / Ledger Separation
W t ≠ H ≤ t ≠ L a w t ≠ L e d g e r t . \boxed{
\mathfrak W_t
\neq
\mathfrak H_{\le t}
\neq
\mathsf{Law}_t
\neq
\mathsf{Ledger}_t.
} W t = H ≤ t = Law t = Ledger t . FC-A5 — No Unaccounted Generative Resource
任何必要生成資源必須出現在 responsibility accounting 或 debt 中。
FC-A6 — Boundary Is Active Structure
邊界不自動等於斷裂，也不自動等於可通過。
FC-A7 — Open Is Not Unbounded
o p e n ≠ u n b o u n d e d . \boxed{
\mathsf{open}
\neq
\mathsf{unbounded}.
} open = unbounded . FC-A8 — Connectivity / Reachability / Realizability Separation
C o n n e c t i v i t y ≠ R e a c h a b i l i t y ≠ R e a l i z a b i l i t y . \boxed{
\mathsf{Connectivity}
\neq
\mathsf{Reachability}
\neq
\mathsf{Realizability}.
} Connectivity = Reachability = Realizability . FC-A9 — Observation / Transformation Separation
o b s e r v e ≠ t r a n s f o r m . \boxed{
\mathsf{observe}
\neq
\mathsf{transform}.
} observe = transform . FC-A10 — Capability Modes Are Not Globally Totally Ordered
任何 mode implication 都需要 domain-specific bridge。
FC-A11 — Meta-Causality Is Layer-Relative
M C k ⇏ absolute law transcendence . \boxed{
\mathsf{MC}_k
\not\Rightarrow
\text{absolute law transcendence}.
} MC k ⇒ absolute law transcendence . FC-A12 — Ledger Accounting Is Not Information Conservation
provenance accounting 不等於某個全域 scalar information invariant。
FC-A13 — No Local-to-Absolute Promotion Without Gate
任何 absolute upgrade 必須通過 G L A \mathcal G_{\rm LA} G LA
FC-A14 — First Cause and Class-Ultimate Are Distinct
F i r s t C a u s e C a n d i d a t e ≠ C l a s s U l t i m a t e C a n d i d a t e . \boxed{
\mathsf{FirstCauseCandidate}
\neq
\mathsf{ClassUltimateCandidate}.
} FirstCauseCandidate = ClassUltimateCandidate . FC-A15 — Accounting Is Not Ontological Completion
responsibility closure、ledger completeness 或 model completeness 都不自動等於 metaphysical completion。
22. Derived Propositions
Proposition P1 — Horizon Monotonicity Under Fixed Transition Semantics
若 T 1 ≤ T 2 T_1\le T_2 T 1 ≤ T 2 I n i t \mathsf{Init} Init G e n S t e p \mathsf{GenStep} GenStep
GenCl D , T 1 ( S ∣ E g e n ) ⊆ GenCl D , T 2 ( S ∣ E g e n ) . \boxed{
\operatorname{GenCl}_{D,T_1}
(S\mid\mathfrak E^{\rm gen})
\subseteq
\operatorname{GenCl}_{D,T_2}
(S\mid\mathfrak E^{\rm gen}).
} GenCl D , T 1 ( S ∣ E gen ) ⊆ GenCl D , T 2 ( S ∣ E gen ) . 此命題不適用於 horizon 改變同時改變規則、權限或 admissibility 的情況。
Proposition P2 — Fixed-Law Closure Is a Special Case
若：
L a w t = L a w 0 ∀ t ≤ T , \mathsf{Law}_t=\mathsf{Law}_0
\qquad
\forall t\le T, Law t = Law 0 ∀ t ≤ T , 則 law-coevolving generative model 退化為 fixed-law model。
Proposition P3 — Source-Alone Inference Is Invalid Under Essential Environment Dependence
若存在 outcome y y y y y y r ∈ E g e n r\in\mathfrak E^{\rm gen} r ∈ E gen S S S
因此：
E s s e n t i a l E n v D e p ( y ) ⇒ ¬ S o u r c e A l o n e P r o o f ( S ⇒ y ) . \boxed{
\mathsf{EssentialEnvDep}(y)
\Rightarrow
\neg\mathsf{SourceAloneProof}(S\Rightarrow y).
} EssentialEnvDep ( y ) ⇒ ¬ SourceAloneProof ( S ⇒ y ) . Proposition P4 — Observation-Ultimate Does Not Imply Transformation-Complete
若：
M ⋆ = { o b s e r v e } , \mathcal M^{\star}=\{\mathsf{observe}\}, M ⋆ = { observe } , 則 class-ultimate coverage 只證明 observation-relative coverage，不提供：
Ω D × T ⋆ ⊆ TransCl D , T ( A ∣ Θ ) . \Omega_D\times\mathcal T^{\star}
\subseteq
\operatorname{TransCl}_{D,T}(A\mid\Theta). Ω D × T ⋆ ⊆ TransCl D , T ( A ∣ Θ ) . Proposition P5 — Local Ledger Projection Is Generally Non-Invertible
若 observer projection Π o , Θ \Pi_{o,\Theta} Π o , Θ
Π o , Θ − 1 \boxed{
\Pi_{o,\Theta}^{-1}
} Π o , Θ − 1 一般不保證唯一存在。
Proposition P6 — Responsibility Closure Does Not Eliminate Grounding Regress by Definition
即使 R e s p C l o s e d = 1 \mathsf{RespClosed}=1 RespClosed = 1 O P E N \mathsf{OPEN} OPEN
23. Proof Obligation Matrix
ID
Obligation
Minimal Requirement
Current Status
PO-01
Target domain well-typed
Ω D \Omega_D Ω D D E F \mathsf{DEF} DEF
PO-02
Judgement context explicit
Θ \Theta Θ D E F \mathsf{DEF} DEF
PO-03
Positive generation claim
trace + provenance
D E F \mathsf{DEF} DEF
PO-04
Negative generation claim
scoped obstruction / completeness
O P E N \mathsf{OPEN} OPEN
PO-05
Generative sufficiency
coverage proof over Ω D g e n \Omega_D^{\rm gen} Ω D gen
M O D E L \mathsf{MODEL} MODEL O P E N \mathsf{OPEN} OPEN
PO-06
Responsibility accounting
G R \mathsf{GR} GR D E F \mathsf{DEF} DEF
PO-07
Ultimate grounding
no hidden higher source + bridge
O P E N \mathsf{OPEN} OPEN
PO-08
Unboundedness
quantity / preorder + certificate schema
O P E N \mathsf{OPEN} OPEN
PO-09
Positive reachability
typed path witness
D E F \mathsf{DEF} DEF
PO-10
Negative reachability
scoped negative witness
O P E N \mathsf{OPEN} OPEN
PO-11
Capability-mode implication
domain-specific bridge theorem
O P E N \mathsf{OPEN} OPEN
PO-12
Transformation realization
contract + state delta + verify witness
D E F \mathsf{DEF} DEF
PO-13
Meta-causal level
witness of rewritten layer
D E F \mathsf{DEF} DEF
PO-14
MC4 grounding
meta-law / reflexive closure grounding
O P E N \mathsf{OPEN} OPEN
PO-15
Class-ultimate coverage
complete target-mode coverage + witnesses
O P E N \mathsf{OPEN} OPEN
PO-16
Global ledger existence
coherent global or gluable ledger construction
O P E N \mathsf{OPEN} OPEN
PO-17
Global information invariant
explicit invariant + proof
O P E N \mathsf{OPEN} OPEN
PO-18
Local-to-absolute promotion
C o m p C e r t \mathsf{CompCert} CompCert O P E N \mathsf{OPEN} OPEN
PO-19
Absolute first cause
FCS + ontological priority bridge
O P E N \mathsf{OPEN} OPEN
PO-20
Absolute nothingness
separate ontology / semantics
outside core requirement
24. FCS Evaluation Matrix
對 first-cause candidate S 0 S_0 S 0
Field
Meaning
typed_scopesource、domain、horizon、environment 是否明確
generative_coverageΩ D g e n \Omega_D^{\rm gen} Ω D gen
responsibility_statusessential resources 是否 accounted
hidden_source_status是否發現未宣告 higher source
carrier_groundingcarrier 的來源與角色
law_groundinglaw / rule regime 的來源與狀態
external_dependencyoracle / randomness / external feed
unboundedness_profile各 typed dimension 狀態
completeness_status是否有 C o m p C e r t \mathsf{CompCert} CompCert
ontological_priority是否存在獨立 bridge
fcs_levelF C S 0 \mathsf{FCS}_0 FCS 0 F C S 4 \mathsf{FCS}_4 FCS 4
open_debts未閉合 proof obligations
25. Canonical Runtime Records
25.1 Generative Closure Claim
GenerativeClosureClaim:
source: null
target_domain: null
horizon: null
judgement_context: null
generative_environment: null
closure_kind: fixed_law | law_coevolving | reflexive
coverage_status: unknown
positive_witnesses: []
negative_witnesses: []
responsibility_records: []
unboundedness_profile: {}
open_debts: []
claim_status: OPEN
25.2 Reachability Claim
ReachabilityClaim:
agent: null
target: null
mode: null
relation_type: null
boundary_state: null
judgement_context: null
horizon: null
status: "?"
positive_witness: null
negative_witness: null
completeness_certificate: null
25.3 Transformation Claim
TransformationClaim:
agent: null
target: null
transformation: null
contract: null
judgement_context: null
horizon: null
realizability_profile: null
status: "?"
witness: null
verification: null
25.4 First-Cause Sufficiency Claim
FirstCauseSufficiencyClaim:
source: null
target_domain: null
horizon: null
generative_environment: null
generative_sufficiency: unknown
responsibility_closure: unknown
hidden_source_status: unknown
carrier_grounding: OPEN
law_grounding: OPEN
external_dependency_status: unknown
unboundedness_profile: {}
completeness_certificate: null
ontological_priority_bridge: null
fcs_level: FCS0
open_debts: []
26. Canonical Compact Kernel
本文件完成後，UGC/CUR formal core 固定為：
K e r n e l U G C / C U R = ⟨ Ω D , T , Θ , W t , H ≤ t , L a w t , B t , R t , E t g e n , GenCl , G R , U , R e a c h , TransCl , M C L e v e l , L e d g e r , F C S , W i t , C o m p C e r t , G L A ⟩ . \boxed{
\mathsf{Kernel}^{\rm UGC/CUR}
=
\left\langle
\Omega_D,
T,
\Theta,
\mathfrak W_t,
\mathfrak H_{\le t},
\mathsf{Law}_t,
\mathfrak B_t,
\mathfrak R_t,
\mathfrak E_t^{\rm gen},
\operatorname{GenCl},
\mathsf{GR},
\mathbf U,
\mathsf{Reach},
\operatorname{TransCl},
\mathsf{MCLevel},
\mathsf{Ledger},
\mathsf{FCS},
\mathsf{Wit},
\mathsf{CompCert},
\mathcal G_{\rm LA}
\right\rangle.
} Kernel UGC/CUR = ⟨ Ω D , T , Θ , W t , H ≤ t , Law t , B t , R t , E t gen , GenCl , GR , U , Reach , TransCl , MCLevel , Ledger , FCS , Wit , CompCert , G LA ⟩ . 其四個主要問題為：
Q G : What can this declared source-plus-environment generate? Q R : What can this agent reach, under which capability mode and relation? Q T : Which transformations can this agent actually realize? Q A : Where are the necessary resources, state changes and unresolved debts accounted for? \boxed{
\begin{aligned}
Q_G &: \text{What can this declared source-plus-environment generate?}\\
Q_R &: \text{What can this agent reach, under which capability mode and relation?}\\
Q_T &: \text{Which transformations can this agent actually realize?}\\
Q_A &: \text{Where are the necessary resources, state changes and unresolved debts accounted for?}
\end{aligned}
} Q G Q R Q T Q A : What can this declared source-plus-environment generate? : What can this agent reach, under which capability mode and relation? : Which transformations can this agent actually realize? : Where are the necessary resources, state changes and unresolved debts accounted for? 第一因問題是 Q G + Q A Q_G+Q_A Q G + Q A Q R + Q T Q_R+Q_T Q R + Q T
27. Canonical Non-Equivalences
後續文件至少保留：
O b s e r v a t i o n ≠ E x i s t e n c e , R e p r e s e n t a t i o n ≠ O n t o l o g y , B o u n d a r y ≠ D i s c o n n e c t i o n , D i f f e r e n c e ≠ D i s c o n n e c t i o n , C o n n e c t i v i t y ≠ R e a c h a b i l i t y , R e a c h a b i l i t y ≠ R e a l i z a b i l i t y , W t ≠ H ≤ t , W o r l d S t a t e ≠ O b s e r v e r V i e w , L a w C h a n g e ≠ O b s e r v e r M o d e l C h a n g e , O p e n ≠ U n b o u n d e d , G e n S u f f i c i e n t ≠ O n t o l o g i c a l l y F i r s t , C l a s s U l t i m a t e ≠ F i r s t C a u s e , R e s p C l o s e d ≠ U l t i m a t e G r o u n d i n g , M C 4 ≠ A b s o l u t e L a w T r a n s c e n d e n c e . \boxed{
\begin{aligned}
\mathsf{Observation} &\neq \mathsf{Existence},\\
\mathsf{Representation} &\neq \mathsf{Ontology},\\
\mathsf{Boundary} &\neq \mathsf{Disconnection},\\
\mathsf{Difference} &\neq \mathsf{Disconnection},\\
\mathsf{Connectivity} &\neq \mathsf{Reachability},\\
\mathsf{Reachability} &\neq \mathsf{Realizability},\\
\mathfrak W_t &\neq \mathfrak H_{\le t},\\
\mathsf{WorldState} &\neq \mathsf{ObserverView},\\
\mathsf{LawChange} &\neq \mathsf{ObserverModelChange},\\
\mathsf{Open} &\neq \mathsf{Unbounded},\\
\mathsf{GenSufficient} &\neq \mathsf{OntologicallyFirst},\\
\mathsf{ClassUltimate} &\neq \mathsf{FirstCause},\\
\mathsf{RespClosed} &\neq \mathsf{UltimateGrounding},\\
\mathsf{MC}_4 &\neq \mathsf{AbsoluteLawTranscendence}.
\end{aligned}
} Observation Representation Boundary Difference Connectivity Reachability W t WorldState LawChange Open GenSufficient ClassUltimate RespClosed MC 4 = Existence , = Ontology , = Disconnection , = Disconnection , = Reachability , = Realizability , = H ≤ t , = ObserverView , = ObserverModelChange , = Unbounded , = OntologicallyFirst , = FirstCause , = UltimateGrounding , = AbsoluteLawTranscendence . 28. Preserved Open Problems
以下問題不得由 Formal Core v0.1 假裝解決：
是否存在 metaphysically complete Ω D \Omega_D Ω D
foundational law 是否固定、演化、湧現或具有其他形式；
reflexive generative closure 是否存在非循環、非空洞 grounding；
是否存在 universal carrier；
是否存在跨 relevant domains 的 completeness certificate；
是否存在可辯護的 global information invariant；
class-ultimate coverage 是否能在 open world 中完成證明；
meta-causal rewrite 是否存在 finite ceiling；
不可計算或不可觀測生成資源如何進行完整 responsibility closure；
absolute first cause 是否是可判定、可驗證、甚至在某 explanation operator 的定義域內；
open-dimensionality 在何種條件下升格為 mathematical unboundedness；
local ledgers 是否能唯一黏合成 global ledger；
capability modes 之間是否存在 domain-independent partial order；
transformation closure 在 law-coevolving world 中應採何種 equivalence relation；
F C S 4 \mathsf{FCS}_4 FCS 4
29. Migration from Canonical Reconciliation
本文件不推翻 Reconciliation v0.1，而是增加可推演定義。
Reconciliation object
Formal Core object
Decision
GenCl D , T ( S ∣ E g e n ) \operatorname{GenCl}_{D,T}(S\mid\mathfrak E^{\rm gen}) GenCl D , T ( S ∣ E gen ) trace-based GenCl \operatorname{GenCl} GenCl
formalized
Generative Responsibility
G R ( y ) \mathsf{GR}(y) GR ( y ) R e s p G r a p h ( y ) \mathsf{RespGraph}(y) RespGraph ( y ) formalized
typed reachability
R e a c h Θ , R m \mathsf{Reach}^{m}_{\Theta,R} Reach Θ , R m formalized
reach profile
R A \mathbf R_A R A preserved
meta-causal levels
M C L e v e l D , T , Θ ( A ∣ L ) \mathsf{MCLevel}_{D,T,\Theta}(A\mid L) MCLevel D , T , Θ ( A ∣ L ) formalized
class-ultimate candidate
typed coverage + optional transformation completeness
strengthened
Global Ledger
ledger + responsibility accounting + debt
strengthened
FirstCauseCandidate
F C S \mathsf{FCS} FCS operationalized
open / unbounded
U \mathbf U U strengthened
local-to-absolute gate
G L A \mathcal G_{\rm LA} G LA C o m p C e r t \mathsf{CompCert} CompCert preserved
30. Paper Release Sequence
完成 Formal Core 後，後續正式系列採：
00 → 01 → 02 → 03 → 04 → 05. \boxed{
00
\rightarrow
01
\rightarrow
02
\rightarrow
03
\rightarrow
04
\rightarrow
05.
} 00 → 01 → 02 → 03 → 04 → 05. 其中：
Paper 00 — Formal Core Specification ：本文件；Paper 01 — Unbounded Ontological Extension ：無界／開放展開、有限邊界與 OBRC 負狀態限制；Paper 02 — Generative Closure and First-Cause Sufficiency ：生成閉包、生成責任、first-cause candidate 與 FCS；Paper 03 — Global Ledger and Generative Accounting ：state / history / provenance / information / responsibility accounting；Paper 04 — Typed Class-Ultimate Reachability ：typed reach、cross-layer channel、coverage；Paper 05 — Transformation Closure and Meta-Causal Agency ：transformation closure、rule rewrite、law rewrite 與 relative meta-causality。
任何 Paper 01--05 若要偏離本文件 primitive，必須先新增 ADR / canonical amendment，而不是在正文內靜默改義。
31. Final Formal-Core Statement
UGC/CUR v0.1 formal core 最終固定：
一個來源的能力不能只由裸 source 表示，而必須相對生成環境、作用域與責任鏈； 一個作用者的能力不能只由裸 reach set 表示，而必須區分 capability mode 與 transformation class； 一個 meta-causal claim 只能相對被改寫的層級成立，不能偷渡成絕對超越所有 law； 一個 first-cause claim 必須先通過生成充分性與責任閉包，再另行處理 ontological priority； 任何 local claim 若要升格 absolute claim，必須通過 local-to-absolute gate。 \boxed{
\begin{aligned}
&\text{一個來源的能力不能只由裸 source 表示，而必須相對生成環境、作用域與責任鏈；}\\
&\text{一個作用者的能力不能只由裸 reach set 表示，而必須區分 capability mode 與 transformation class；}\\
&\text{一個 meta-causal claim 只能相對被改寫的層級成立，不能偷渡成絕對超越所有 law；}\\
&\text{一個 first-cause claim 必須先通過生成充分性與責任閉包，再另行處理 ontological priority；}\\
&\text{任何 local claim 若要升格 absolute claim，必須通過 local-to-absolute gate。}
\end{aligned}
} 一個來源的能力不能只由裸 source 表示，而必須相對生成環境、作用域與責任鏈； 一個作用者的能力不能只由裸 reach set 表示，而必須區分 capability mode 與 transformation class ； 一個 meta-causal claim 只能相對被改寫的層級成立，不能偷渡成絕對超越所有 law ； 一個 first-cause claim 必須先通過生成充分性與責任閉包，再另行處理 ontological priority ； 任何 local claim 若要升格 absolute claim ，必須通過 local-to-absolute gate 。 因此，本系列的形式問題不再是：
「某來源是不是無限？」或「某存在是不是全能？」
而是：
在明示 world、law、carrier、boundary、history、observer、resource 與 evidence 條件下， 什麼可以被生成，什麼可以被抵達，什麼可以被改寫， 這些能力需要哪些資源，以及哪些 claim 仍然沒有資格被提升為 absolute。 \boxed{
\begin{aligned}
&\text{在明示 world、law、carrier、boundary、history、observer、resource 與 evidence 條件下，}\\
&\text{什麼可以被生成，什麼可以被抵達，什麼可以被改寫，}\\
&\text{這些能力需要哪些資源，以及哪些 claim 仍然沒有資格被提升為 absolute。}
\end{aligned}
} 在明示 world 、 law 、 carrier 、 boundary 、 history 、 observer 、 resource 與 evidence 條件下， 什麼可以被生成，什麼可以被抵達，什麼可以被改寫， 這些能力需要哪些資源，以及哪些 claim 仍然沒有資格被提升為 absolute 。 END OF CANONICAL FORMAL CORE v0.1