axioms prompt

Axioms DSL

AxiomsDoc = formal-system-definition: "集中管理所有公理声明------公理声明'什么是逻辑上必然的'"

$AX-CORE$ AxiomHierarchyStructure:

LAYER_ORDER=L0,L0.5,L1,L2,L3
| invariants(L0):∀op∈MetaRound:op.target∉L0.elements // L0绝对不可变
| invariants(L0):EssentiallyUndecidable(系统本质不可判定,证明责任外部化)
| invariants(L0.5):MetaEvolution_Rule逻辑骨架不变(自反修正可行)
| invariants(L1):修改需≥2验证者签名共识
| invariants(L2):可版本化，旧版存档
| invariants(L3):运行时可变(权限分级:config(部署注入)/admin(管理员))
| persistence(L0):contractRegistry哈希验证(外部持久化,非LLM上下文窗口记忆)
---meta
irr = "structural"
sem = "concept=hierarchy-layer, imp=layer-isolation-guarantees"

$AX-CORE$ L0: BAX-IDI,BAX-TIME,BAX-C,BAX-D,SyncProtocolTriple

---meta
enf = "runtime_invariant: ∀op∈MetaRound, op.target∉{BAX-IDI,BAX-TIME,BAX-C,BAX-D,SyncProtocolTriple}"

$AX-META$ L0.5: MetaEvolution

---meta
enf = "runtime_invariant: ¬∃op∈MetaRound, op.target∈{BAX-IDI,BAX-TIME,BAX-C,BAX-D,SyncProtocolTriple,MetaEvolution_Rule}"

$AX-FRAME$ L1: BAX-X(Coupling-Sampling,Embedding,ObsCollapse),CountableOrderBridge

---meta
enf = "codegen_constraint: 修改L1需≥2验证者签名共识"

$AX$ L2: 领域公理(§一至八)

---meta
enf = "codegen_constraint: 可版本化，旧版存档"

% L3: 配置参数(运行时可变)

$AX-META$ MetaEvolution(InferenceRule):

废除MetaEvolution作为公理------提升为推理规则
| ! MetaEvolution_Rule(InferenceRule形式):
| MetaEvolution_Rule = λthreshold:ℕ⁺. $enters(MetaRound) ↔ Provable_{L2}("CF≥threshold")$ // 逻辑骨架(L0.5不可变,纯λ-抽象)
| | // 触发条件从 CM≥threshold 改为 Provable_{L2}("CF≥threshold")
| | // threshold类型限定为ℕ⁺(正整数)
| | MetaEvolution_Instance=MetaEvolution_Rule(META_THRESHOLD) // 参数实例(L3可变)
| | // 自指约束:¬∃m∈Ops:m(MetaEvolution_Rule)≠MetaEvolution_Rule(仅禁止修改逻辑骨架)
| | // MetaRound合法性:L0不可修改;可修改L2及以下;需≥2验证者共识;日志持久化
| | MetaRound∉{BAX-IDI,BAX-TIME,BAX-C,BAX-D,SyncProtocolTriple,MetaEvolution_Rule}
| | δ(STANDARD,∃≥2v)=STANDARD | δ(STANDARD,¬∃≥2∧∃≥1)=DEGRADE_A
| | δ(DEGRADE_A,recover)=STANDARD | δ(DEGRADE_A,¬recover)=DEGRADE_B
| | enters(DEGRADE_A)↔¬∃≥2_v∧∃≥1_v | enters(DEGRADE_B)↔¬∃≥1_v∧∃rollback_contract
| | enters(DEGRADE_B):contract.hash∈L0.contractRegistry(哈希白名单)
| | enters(safe_halt)↔所有降级路径不可用
| | % META_THRESHOLD
---meta
enf = "runtime_invariant: ∀MetaRound, MetaRound.logPersistence.verified=true"
dr = "分层反射解决'不可变vs自适应'死锁;Provable_{L2}令牌阻隔元层对对象层裸数据泄漏;threshold:ℕ⁺类型约束根治旧版threshold=0时cf_provable=false误判为true的零值短路bug"

$AX-CORE$ StratifiedReflectionSchema:

∀n∈ $0...\|LAYER_ORDER\|-2$ ,Ln=LAYER_ORDER $n$ ,Lnext=LAYER_ORDER $n+1$ :(⊢{Ln} φ)→Provable{Ln}(⌜φ⌝)在Lnext中为真
| //! 分层反射:Ln的定理在Lnext中可被"谈论"(作为可证明性标签)
| //! 层索引使用 LAYER_ORDER= $L0,L0.5,L1,L2,L3$ 实际路径而非 n+1 算数递增
| //! 元层(L0.5)只能引用Ln(∀n≥1)的可证明性标签,不能直接读取Ln的状态变量
| $AX$ TarskiTruthEquivalence:
| ∀n∈ $0...\|LAYER_ORDER\|-2$ ,Ln=LAYER_ORDER $n$ ,Lnext=LAYER_ORDER $n+1$ :Truth_{Lnext}(⌜φ⌝)↔Provable_{Ln}(⌜φ⌝)
| | //! 绑定域:Provable_{Ln}仅指内部推导⊢{Ln} φ(见ProvableDerivationOnly)
| | //! Truth不由外部签名投票决定(见TruthValueDomainLogic),外部签名仅作为ProofCandidate见证
| | //! 修复:切断Truth/Provable与外部签名之间的语义断裂(旧版缺陷:Truth曾绑定外部共识而非内部推导)
| $AX$ ProvableDerivationOnly:
| ∀Ln∈LAYER_ORDER,∀φ∈Formulas(Ln):Provable{Ln}(⌜φ⌝)↔⊢_{Ln} φ
| | //! Provable仅绑定内部推导关系,不与ExternalizedConsistency的∃sig存在量词等价
| | //! Truth不由外部签名投票决定(同上,见TruthValueDomainLogic)
---meta
irr = "logical"
sem = "concept=stratified-reflection, imp=avoids-godel-incompleteness-by-layering, fix=semantic-gap-between-truth-and-consensus"
dr = "元/对象层混淆的根治方案;ProvableDerivationOnly显式切断Truth与外部签名的等价关系;外部签名仅作为ProofCandidate见证而非Truth定义"

$AX-CORE$ LogicalLayerSeparation:

∀r∈MetaLayer:r∈InferenceRules | ∀op∈ObjLayer:op∈SystemOps
| MetaLayer⊥ObjLayer(交集为空)
| ¬∃r∈MetaLayer:r∈SystemOps | ¬∃op∈ObjLayer:op∈InferenceRules
| //! 扩展:元层推理规则只能引用对象层的可证明性标签,不能直接读取对象层变量
| $AX$ MetaReadConstraint: ∀r∈MetaLayer,∀v∈Variable:r.reads(v)→(v.type=ProvabilityToken)
---meta
irr = "logical"
sem = "concept=layer-separation, imp=avoids-self-reference, ext=stratified-reflection"

$AX-CORE$ EssentiallyUndecidable:

system≡essentially_undecidable(本质不可判定)
| //! 这不是弱点,而是系统的能力边界声明
| //! PA+RCF+反射→绝对不可判定,将此特性写进L0硬核
| //! 证明论基础:系统属于半可判定(半可计算)类
---meta
irr = "logical"
sem = "concept=undecidability-boundary, imp=system-admits-no-complete-consistency-proof"

$AX-META$ ExternalConsistencyWitness:

外部签名见证存在于元层级(MetaLayer),不属于L0/L0.5系统内公式
| //! 关键语义:∃sig∈Signature不是系统内可表达/witnessable的存在量词
| //! 签名验证是引擎运行在元系统的前提假设(meta-level axiom),不自指
| //! 签名算法在现实世界中可靠→系统拥有标准模型(Standard Model)
| //! 证明责任转移:外部签名的可靠性由物理世界/密码学保证,非系统内证明
---meta
irr = "logical"
sem = "concept=external-consistency-witness, imp=meta-level-axiom-not-system-formula, fix=godel-incompleteness-bypass"
dr = "缺陷6修复:将ExternalizedConsistency从系统内公式(∃sig内部存在量词)提升为元级前提,避免哥德尔第二不完备定理的非法绕过"

$AX-META$ WitnessTerm:

WitnessTerm存在于元层级(MetaLayer),类型为"候选证明的元级见证绑定"
| //! WitnessTerm不是L0.5的逻辑公式,而是引擎启动时注入的元级绑定
| //! 签名验证者数量承诺≥2(由元级运行时保证,非系统内公式可推导)
| //! WitnessTerm与ProofCandidate的哈希绑定关系由引擎元级运行时维护:
| Valid(sig,w)↔sig.sign(hash(w))=sig.signature
| //! 注意:此处的Valid是引擎层面的操作语义,非系统逻辑公式
---meta
irr = "logical"
sem = "concept=witness-term, imp=meta-level-binding-not-system-formula, fix=godel-incompleteness-bypass-ext"
dr = "WitnessTerm从L0.5逻辑公式降级为元级运行时绑定,解除系统内部自指一致性断言"

$AX-CORE$ TruthValueDomainLogic:

TruthValueDomain={⊥,T,F}(Scott信息域)
| //! 语义映射:T≡verified_true_by_internal_derivation | F≡verified_false_by_internal_derivation | ⊥≡insufficient_evidence(Scott底元素)
| //! 关键澄清:Truth不由外部签名投票决定,不由共识机制赋值
| //! T/F仅由内部推导⊢_{Ln} φ确定,外部签名仅影响ProofCandidate的见证状态,不影响Truth值
| //! 信息偏序:⊥⊑T ∧ ⊥⊑F ∧ ¬(T⊑F) ∧ ¬(F⊑T)
| //! 单调性保证:信息只增不减,约束为因果偏序下的非反转,而非全局时序
| //! 连接词(Scott域单项连续):
| ¬⊥=⊥ | ¬T=F | ¬F=T
| ⊥∧T=⊥ | ⊥∧F=⊥ | T∧F=F | T∧T=T | F∧F=F
| ⊥∨T=T | ⊥∨F=⊥ | T∨F=T | T∨T=T | F∨F=F
| $AX$ DecisionTrigger: ∀φ∈Judgment:decision(φ)↔φ∈{T,F}
| $AX$ ExploratoryEntry: ∀φ∈Judgment:φ=⊥→enters(ExploratoryRound)∧log(meta,"⊥判定进入探索性轮次")
---meta
enf = "runtime_invariant: DecisionTrigger ∧ ExploratoryEntry"
irr = "logical"
sem = "concept=truth-value, framework=Scott-Domain-Logic, imp=monotonic-information-accumulation, fix=truth-not-consensus"
dr = "完全替换Kleene伪三值改为Scott信息域;T/F绑定内部推导而非外部签名;废止闭世界假设"

$AX-CORE$ InformationMonotonicity:

∀s∈StateSpace:¬∃τ₁,τ₂:τ₁→τ₂∧(val(s)@τ₁=F∧val(s)@τ₂=T)
| //! 同一命题在因果后继时间不会先判假后判真------信息域保证单调性
| //! 若先⊥后T/F,这是信息增加,合法;若先F后T,这是信息反转,非法
| //! 并发分支(∥)之间的真值比较无定义,不违反单调性
| $AX$ ConcurrentComparisonUndefined:
| ∀s₁,s₂∈StateSpace,∀τ₁,τ₂:τ₁∥τ₂→¬∃(val(s₁)@τ₁≺val(s₂)@τ₂)
| | //! 并发分支间真值的"比较"在逻辑上无定义
| | //! val(s₁)@τ₁=F 且 val(s₂)@τ₂=T 且 τ₁∥τ₂ 不违反InformationMonotonicity
---meta
irr = "informational"
sem = "concept=monotonicity, framework=causal-partial-order, imp=no-non-monotonic-tearing"
dr = "缺陷2修复:全局时序<t₁<t₂替换为因果偏序→;并发分支(∥)真值比较无定义;公式在任何含并发执行的系统中合法可判定"

// $AX-CORE$ MetaLogicalLayer 已降级为注释（元文档约定，非逻辑必然性声明）

// 原内容：元逻辑推理步骤是公理书写者与读者间的约定,非Operations元素

// $AX-CORE$ SymbolLegacyClearance 已降级为注释（历史清理记录，非逻辑必然性声明）

// 原内容：已统一使用@τ和→(废除@t,t₁<t₂全局时序残留)

$AX-CORE$ QuantificationConstraint:

∀φ∈Formulas:∀v∈FreeVars(φ):∃T∈PrimitiveTypes:type(v)=T
| priority(explicit>contextual>fallback_Entity)
| ¬∃φ∈Formulas:φ=∀x,y,... // 禁止裸量化
---meta
irr = "logical"
sem = "imp=prevents-Russell-paradox"

// === 类型基底(由QuantificationConstraint声明约束) ===

$Type$ Entity = 可指称实体 | domain:非空(至少包含OBSERVER)

$Type$ OBSERVER = 观察者类型 | domain:⊆Entity,由IDI-Identity定义同一性

$Type$ Event = InteractionEvent∪TransitionEvent

$Type$ InteractionEvent = 交互事件(≥2实体) | domain:⊆Event,由IDI-Interaction定义

$Type$ TransitionEvent = 跃迁事件(单实体) | domain:⊆Event,由LogicalClock定义

$Type$ Round = 轮次类型 | domain:由RoundCompletion定义

$Type$ State = 系统状态 | domain:由TriPerspectiveStructure定义张量积分解

$Type$ Counter = ℕ | domain:由Discrete-Countability定义

$Type$ CallKey = 调用键标识符 | domain:由FirstCallTransient定义

$Type$ Quality = ℝ | domain:由TerminationPriority定义比较语义

$Type$ Identifier = UID | domain:创建时分配,严格全序

$Type$ Judgment = 判定命题 | domain:∀φ∈Formulas, φ∈{T,F,⊥} by TruthValueDomainLogic | ref:TruthValueDomainLogic, DecisionTrigger

// === 定义(Def)与符号约定 ===

// 类型化定义,将$符号约定升维为显式定义

$Def$ ActionAtClock: a@τ = ⟨a,τ⟩ | τ∈ClockTick, a∈Action | ref:LogicalClock

$Def$ Concurrency: a∥b ↔ ¬(a→b) ∧ ¬(b→a) | ref:LogicalClock

$Def$ HappenedBefore: a→b = strict_partial_order(transitive+antisymmetric+irreflexive) | ref:LogicalClock

$Def$ SnapshotRead: op.readsFrom(s) ↔ op基于快照s, deterministic | ref:StateSnapshot

$Def$ ToolCall: T(params,state) = function_call(f,T,params,state) | ref:TH-9

$Def$ CallCount: callCount(k) ∈ ℕ | ref:Discrete-Countability

$Def$ ResultSet: result(k) = return_value(k) | ref:IntentClassification

$Def$ ErrorCost: errorCost© ∈ ℝ⁺ ∪ {0} | ref:TransientRetryNoCost

$Def$ Retry: retry(k) = invoke(k) again | ref:TransientRetryNoCost

$Def$ LayerSet: LAYERS = {L0,L0.5,L1,L2,L3} | strict_total_order | ref:FocusLayer

$Def$ ScopeSet: SCOPES ⊆ ContextSpace | ref:ScopeGuard

$Def$ ProcessSpace: ProcessSpace = {⟨s₁,s₂,...⟩ | ∀i: s_i→s_{i+1}} | ref:Discrete-Enumeration

$Def$ FutureCone: cone_forward(x) = {y | x→y} | ref:CausalCone

$Def$ PastCone: cone_backward(x) = {y | y→x} | ref:CausalCone

$Def$ Continuation: Continuation® = {s | R.completed → enters(s)} | ref:InitialState

$Def$ StateTransition: state₁→state₂ ↔ legal_transition(state₁,state₂) | ref:RoundTransition

$Def$ CrossDomain: op.crosses(domain) ↔ op.domain ≠ current_domain | ref:TH-5

$Def$ NeedsSync: op.needsSync(ctx) ↔ ¬(op₁ ∥ op₂) ∧ op₁.domain ≠ op₂.domain | ref:TH-5

$Def$ ObservationLock: observed(e,τ) ↔ e在τ被观测锁定 | ref:Coupling-ObservationCollapse

$Def$ IntentTags: result.intent ∈ {NO_DATA, NO_RESPONSE, AMBIGUOUS} | ref:IntentClassification

$Def$ GödelNumbering: ⌜φ⌝_{Ln} = Gödel_number(φ) in Ln | ref:StratifiedReflectionSchema

$Def$ ProvabilityToken: Provable_{Ln}(⌜φ⌝) ↔ ⊢_{Ln} φ 在 Ln+1 中可证 | sem:provability-token,level=Ln

$Def$ Derivability: ⊢_{L} φ ↔ φ在L中可推导 | sem:derivability

$Def$ LTL_Finally: ◇φ ↔ φ最终为真 | ref:SyncProtocolTriple

$Def$ LTL_Next: ○φ ↔ φ在下一逻辑时钟滴答为真 | ref:SyncProtocolTriple

$Def$ LTL_Globally: □φ ↔ φ始终为真 | ref:SyncProtocolTriple

$Def$ CommitPhase: CommitPhase = SyncProtocolTriple.phase2 | atomic=clock_domain | ref:SyncProtocolTriple

$Def$ TryPhase: TryPhase = SyncProtocolTriple.phase1 | timed_out_possible | ref:SyncProtocolTriple

$Def$ CompensatePhase: CompensatePhase = SyncProtocolTriple.phase3 | rollback | ref:SyncProtocolTriple

$Def$ ScottBottom: ⊥ = insufficient_evidence | Scott_domain_bottom | ref:TruthValueDomainLogic

$Def$ InfoOrder: ⊑ = information_partial_order | ref:TruthValueDomainLogic

$Def$ ProofCandidate: ProofCandidate = φ的候选证明(需元级外部见证) | ref:WitnessTerm

$Def$ Signature: sig = external_signature | sem:meta-level-binding,not-system-formula | ref:ExternalConsistencyWitness

// === 〇、基底元公理 ===

// TriPerspective:三个独立基底视角(IDI/C/D),张量积分解

$BAX$ TriPerspectiveStructure:

∃IDI,C,D(三视角独立) | IDI⊥C⊥D(删除任一→其余失去意义)
| ∀s∈SystemState:s≅IDI_s⊗C_s⊗D_s
---meta
irr = "structural"

$BAX$ IDI_Framework: IDI=observer-framework(logic-semantic-base)

---meta
sem = "concept=IDI, role=observer-framework"

$BAX-IDI$ IDI-Identity:

∀(x:Entity),(obs:OBSERVER):x≡_obsx
| ∀(x:Entity),(y:Entity),(obs:OBSERVER):(x≡_obsy)↔∀P∈Obs(obs)(P(x)↔P(y))
| Obs(obs)由L0注入而非递归自省
| (x≡_obs₁y)∧(x≢_obs₂y)→ep_gap(obs₁,obs₂)
| Obs(obs)∌λP.P∈Obs(obs)
---meta
view = "IDI"
irr = "logical"
src = "primitive"
sem = "concept=identity, binding=observer, imp=epistemic_gap"
dr = "绝对莱布尼兹律在分布式系统中失效→引入Observer参数解决拜占庭分歧;Obs自指约束阻止罗素悖论"

$BAX-IDI$ IDI-Difference:

∀(x:Entity),(y:Entity),(obs:OBSERVER):Diff_obs(x,y)↔¬(x≡_obsy)
| ∀S⊆Entity,(obs:OBSERVER):|S|≥2→∃x,y∈S:Diff_obs(x,y)
| (Diff_obs₁∧¬Diff_obs₂)→ep_gap≥1
---meta
view = "IDI"
irr = "informational"
sem = "concept=difference, imp=unit-of-info(1bit), source=SpacePartition"

$BAX-IDI$ IDI-Interaction:

∀(x:Entity),(y:Entity):(x≢_obsy)→∃⟷(x,y) | ⟷(x,y)=可观测相互影响
---meta
view = "IDI"
irr = "relational"
sem = "concept=interaction, role=bridge-across-difference, imp=causation-bound-to-interaction"
dr = "因果必须绑定交互;∀x,y类型限定防止罗素恶性循环(缺陷3)"

$BAX-IDI$ IDI-HistoryEmergence:

∀(x:Entity):x.history={⟨τ,y,e⟩|(x⟷y)@τ∧e∈{causal,concurrent}}
---meta
view = "IDI"
irr = "temporal"
src = "LogicalClock, CausalCone"
sem = "concept=history, role=emergent-from-interaction-record, imp=无交互实体无历史"

// 同步协议三元组:承诺-补偿逻辑替代原子同步

$AX-CORE$ SyncProtocolTriple:

Sync操作分解为三个有序阶段:Try-Compensate/Commit
| ! TryPhase(阶段1-尝试):
| ◇_{≤Δt} Response // 存在超时,可中断,非原子
| | ∀obs₁,obs₂:try(obs₁,obs₂)在逻辑时钟域内可超时
| | timeout(obs₁,Δt)∧¬response→进入CompensatePhase
|
| ! CommitPhase(阶段2-提交/终止):
| ○ (AtomicLock) // 不可中断,逻辑时钟域内单个滴答
| | ∀obs₁,obs₂:commit(obs₁,obs₂)在逻辑时钟域内原子不可中断
| | atomicity_domain(commit)=clock_domain(逻辑原子)
| | ¬atomicity_domain(commit)=wall_clock(物理不可中断)
| | ¬∃obs:commit(obs,obs) // 自反提交在语义上无意义
| | commit(obs₁,obs₂).completed↔obs₁.val|_obs₂≡_obs₁obs₂.val|_obs₁∧obs₂.val|_obs₁≡_obs₂obs₁.val|_obs₂
| | ¬∃obs₃:commit(obs₁,obs₂).requires(obs₃) // 不依赖第三方
|
| ! CompensatePhase(阶段3-补偿):
| TryPhase超时后进入补偿路径(回滚+恢复,非原子)
| | completed(sync)↔Commited(sync)∨Compensated(sync)
| | // Commited:逻辑原子,满足原子性定义
| | // Compensated:回滚成功,包含timeout后的恢复逻辑
| | // 原子性仅限定在Commited子集内,Compensated路径拥有独立的完备证明树
| | 路径1(正常→Commit):Try→Commit→Completed(Commited)
| | 路径2(超时→补偿):Try→Timeout→Compensate→Completed(Compensated)
| | 路径3(崩溃恢复):obs₁.recovered→使用本地持久化快照→重入Try
---meta
view = "IDI"
src = "LogicalClock, CausalCone"
irr = "temporal, logical, relational"
sem = "concept=sync-protocol-triple, try=non-atomic-attempt, commit=logical-clock-domain, compensation=atomicity-preserved-in-commited-subset"
dr = "原子性悖论根治------原子性仅限定在Commited子集;Compensated路径拥有独立证明树;自反提交¬∃obs:commit(obs,obs)消除时间自噬(缺陷5)"

// 时间基底:因果偏序,非全局时钟

$BAX-TIME$ LogicalClock:

∀a,b∈Event:a→b(严格偏序:传递+反对称+非自返)
| a∥b↔¬(a→b)∧¬(b→a) // 并发
| Event=InteractionEvent∪TransitionEvent(互斥)
| InteractionEvent:≥2实体交互 | TransitionEvent:单实体状态变迁(因果前因由交互传递)
| ¬∃e∈TransitionEvent:e.cause=∅
---meta
view = "IDI"
irr = "relational"
sem = "concept=causality, role=partial-order, imp=removes-global-clock"

$BAX-TIME$ CausalCone:

∀x:cone_forward(x)={y|x→y}(未来光锥) | cone_backward(x)={y|y→x}(过去光锥)
| ∀R,R':R.stateSpace∩R'.stateSpace=∅→cone_fwd®∩cone_bwd(R')=∅
---meta
view = "IDI"
irr = "topological"
sem = "concept=causal-cone, imp=independent-decisions-no-consensus"

$BAX$ Continuum_Framework: Continuum=physical-base(ℝ而非ℕ)

$BAX-C$ Continuum-Metric:

∀s₁,s₂∈StateSpace:∃d∈ℝ_{\ge 0}:d(s₁,s₂)
---meta
irr = "topological"
sem = "concept=distance, role=proximity, imp=DegradationContinuum"

$BAX-C$ Continuum-Dynamics:

∀t₁,t₂∈ℝ,∀state:∂state/∂t=F(state,input)
---meta
sem = "concept=gradual-change, imp=inertia-of-state"

$BAX-C$ Continuum-Superposition:

StateSpace=有界凸集:∀s₁,s₂,∀α,β∈ $0,1$ :α+β=1→αs₁+βs₂∈StateSpace
| ∃M∈ℝ⁺:∀s∈StateSpace,‖s‖≤M(范数有界)
---meta
irr = "topological"
sem = "concept=mixed-states, imp=noise-probability-source"

$BAX-C$ Continuum-SamplingThreshold:

TH-11: DecisionMeasure(推导): μ(s,goal)=‖s-goal‖/M; μ<ε→DiscreteGoal.pass

---meta

sem = "concept=decision-measure, derivation=from-continuum"
src = "Continuum-Superposition, Continuum-SamplingThreshold"

$BAX$ Discrete_Framework: Discrete=symbolic-base(ℕ而非ℝ)

$BAX-D$ Discrete-Enumeration:

∀cs∈StateSpace:cs↔ℕ(可计算状态同构于ℕ子集)
---meta
sem = "concept=enumerable, imp=cpu-cannot-handle-reals"

$BAX-D$ Discrete-AtomicStep:

∀t∈Computations:t是原子跃迁(无中间态)
---meta
sem = "imp=RoundTransition+StateSnapshot的前提"

$BAX-D$ Discrete-Countability:

∀k∈CallKey:callCount(k)∈ℕ | ∀c∈LocalCounters:c.next=c+1
| 局部计数器仅在所属因果锥内单调递增 | 并发轮次计数器不可跨轮次比较
| ∀R,R':R∥R'→¬(R.cnt<R'.cnt)
---meta
sem = "concept=local-monotonic-counting, imp=终止决策中计数器比较受因果约束"

$BAX-D$ Discrete-ExactEquality:

∀a,b∈DiscreteState:(a=b)↔True/False(精确相等,无容差)
---meta
sem = "imp=EvaluationOutcomeSet基底"

// 可数序桥接:偏序-全序桥接

$AX-FRAME$ CountableOrderBridge:

∀R,R':R.CF与R'.CF可比较性由因果偏序决定
| R→R':R'.CF=max(R'.local,R.CF) | R∥R':不可直接比较
| decision®仅考虑causal_chain®={R'|R'→R}
| effective_count®=⊗{R'.count|R'→R}(因果加权Join)
| a⊗b=if a≤_c b then b else if b≤_c a then a else max(causal_weight(a),causal_weight(b))
| causal_weight®=causal_chain_depth®
| same_chain®={R'|R→R'∨R'→ R}(因果连通的最大子集)
---meta
src = "Discrete-Countability, LogicalClock, CausalCone"
dr = "缺陷2修复;偏序时钟与全序计数器矛盾→因果约束下计数器比较;causal_weight(id)=0消除时间自噬(缺陷5扩展)"

// 连续-离散拓扑嵌入(降级自LogicalFunctor,避免范畴论误导)

$BAX-X$ Coupling-Embedding(TopologicalContinuityEmbedding):

embed:Discrete→Continuous | sample:Continuous→Discrete | 构成拓扑连续嵌入(非逻辑函子)
| //! 重要声明:本映射是拓扑连续嵌入(TopologicalContinuityEmbedding),非范畴论逻辑函子(LogicalFunctor)
| //! 原因:ε-近似保守性不满足逻辑态射的严格保真要求,放弃逻辑态射元定理声称
| // 嵌入保守性(EmbedConservativity):
| ∀φ∈Formulas_D:(⊢_D φ)→(⊢_C embed(φ))
| // 离散域证明的定理在连续域解释下依然成立(单向保守成立)
| // ε-近似采样(工程启发式,非公理):
| // (⊢_C ψ)→(⊢_D sample(ψ))在误差界ε内成立仅作为工程启发式,不享有逻辑态射地位
| // ε由Continuum-SamplingThreshold的δ(t)决定:ε=δ(t)/2
| // 噪声容差:偏离ε以上→sample返回上一个有效离散值(工程容错策略,非逻辑规则)
| // Galois Connection保底(当嵌入性质退化时):
| ∀d,∀c:sample(embed(d))=d(下封闭) | embed(sample©)≤_cc(上封闭)
| // 保底性质仅用于拓扑嵌入拓扑连续性,不承担逻辑保真义务
| $TH$ TH-SamplingReliability:
| ∀c∈ContinuousState,∀d∈DiscreteState:在GaloisConnection意义下,sample(embed(d))=d(下封闭性);但d(c,embed(sample©))可能无法保证严格保真
---meta
view = "D, C"
src = "Discrete-Enumeration, Continuum-Metric, Continuum-Superposition"
dr = "缺陷4修复:从范畴逻辑函子降级为拓扑连续嵌入;放弃逻辑态射元定理;ε-近似保守性降级为工程启发式;Galois Connection保底保留但仅用于拓扑连续性"

// === 〇.一、跨域耦合 ===

// C×D→Coupling-Sampling(连续→离散) | D×C→Coupling-Embedding(离散↔连续,拓扑嵌入)

// IDI×C×D→Coupling-ObservationCollapse(三域同步坍缩) | IDI×IDI→SyncProtocolTriple(承诺-补偿同步)

$BAX-X$ Coupling-Sampling:

∀sig∈ℝ,∀do∈Sensors:do.value=f(sig)(f=采样函数,含阈值+量化)
---meta
src = "Continuum-SamplingThreshold, Discrete-Enumeration"
sem = "continuous→discrete, imp=physical-world-enters-computation-via-sampling"

$BAX-X$ Coupling-ObservationCollapse:

∀fv∈IDI,∀obs∈Observer:fv.continuousVariance≠0
| ∀ov∈Discrete:observed(fv,obs)→fv.continuousVariance|_obs=0
| // 拜占庭:obs₁≠obs₂→fv.val|_obs₁≠fv.val|_obs₂合法→sync后收敛
---meta
src = "IDI-Identity, IDI-Interaction, Discrete-AtomicStep, SyncProtocolTriple.CommitPhase"
sem = "concept=tri-domain-collapse-on-observation, imp=free-variability-freezes-precise-value"

// === §一状态与空值 ===

$AX$ StateSnapshot: ∀op∈Ops,∀s∈Snap:op.readsFrom(s)→deterministic

---meta
view = "D"
src = "Discrete-AtomicStep"

$AX$ StateMerge: ∀m∈Merges:m.conflictFree→consistent

---meta
view = "IDI, D"
src = "IDI-Identity"

$AX$ InitialStateNullity: ∀x∈SystemState:LegalInitialState→x.intent∈{NO_DATA,NO_RESPONSE}

| mapping(NO_DATA)=x=∅ | mapping(NO_RESPONSE)=null
---meta
view = "IDI, D"
src = "IntentClassification"

$AX$ IntentClassification: ∀r∈Result:r.intent∈{NO_DATA,NO_RESPONSE,AMBIGUOUS}

| NO_DATA=∃q:q.executed∧q.result=∅ | NO_RESPONSE=¬response.received∧(timeout∨error)
| AMBIGUOUS=¬intentDeterminable(intent)
---meta
view = "IDI, D"
src = "IDI-Identity, Discrete-ExactEquality"
dr = "区分'成功空'vs'失败无结果'是∅/null范畴混淆关键;NO_DATA不重试,NO_RESPONSE触发重试"

// NullResultLegitimacy由InitialStateNullity+IntentClassification保证,非独立公理

// === §二调用行为 ===

$AX$ FirstCallTransient: ∀k∈CallKey:callCount(k)=0→result(k).mayBeTransient

---meta
view = "C, D"
src = "Continuum-SamplingThreshold, Coupling-ObservationCollapse"

$AX$ RepeatedCallStable:

∀k∈CallKey:(callCount(k)≥1∧¬reconf_expired(k))→result(k).isStable
| reconf_expired(k)↔(cur_time-last_confirmed(k))>RECONFIRMATION_WINDOW(同一逻辑时钟τ)
| reconf_expired=true→isStable降级为mayBeTransient(保留权重>首次)
| % RECONFIRMATION_WINDOW
---meta
view = "D"
src = "Discrete-Countability, Coupling-ObservationCollapse, LogicalClock"
dr = "修复缺陷10(N=1→Stable但N=3失败→引入再确认窗口)"

$AX$ TransientRetryNoCost: ∀k∈CallKey:callCount(k)=0∧result(k)=∅→retry(k).errorCost=0

---meta
view = "D"
src = "IntentClassification"

$AX$ TransientResultReevaluation:

∀k∈CallKey:callCount(k)=0∧result(k).mayBeTransient→¬result(k).confirmed
| // 空结果分类:NO_RESPONSE→重试;NO_DATA→stable不重试;AMBIGUOUS→重试
---meta
view = "IDI, D"
src = "IntentClassification, Discrete-Countability"

// === §三注意力与作用域 ===

$AX$ ScopeAnchor: ∀m∈LLM:m.attention⊆m.layerCoverage

---meta
view = "IDI, C"
src = "IDI-Interaction, Continuum-Metric"

$AX$ FocusLayer: ∀r∈Rules,∀f∈CurrentFocus:r.layer∈LAYERS→r.active(f)iff f.level≥r.layer

---meta
view = "IDI, D"
src = "IDI-Difference, Discrete-Enumeration"

$AX$ ScopeGuard: ∀r∈Rules,∀c∈CurrentContext:r.scope∈SCOPES→r.active©iff c∈r.scope

---meta
view = "IDI, D"
src = "IDI-Identity, Discrete-ExactEquality"

// === §四执行状态 ===

$AX$ ExecutionStateTaxonomy: ∀op∈Operation:op.status∈{success,failure}

---meta
view = "D"
src = "Discrete-ExactEquality"

$AX$ FailureSubtyping:

failure=exec_failure∪eval_failure∪parse_failure
| exec_failure:{compilation_error,runtime_error,timeout,dependency_error,environment_error}
| eval_failure:{partial_output,warning}
| parse_failure:{parse_error}
---meta
view = "IDI, D"
src = "IDI-Difference, Discrete-Enumeration"
sem = "defect-resolved=parse-error-classification-gap"

$AX$ ParseErrorStatus: ∀r∈CollectResult:extract®=∅→r.status=parse_error

---meta
view = "D"
src = "Discrete-ExactEquality"

// === §五轮次 ===

$AX$ RoundCompletion: ∀R∈Rounds:R.completed↔R.executed∧R.verified∧R.logged

---meta
view = "IDI, D"
src = "IDI-Identity, Discrete-AtomicStep"

$AX$ RoundTransition: ∀Rk,Rk+1:enters(Rk+1)↔completed(Rk)∧reEvaluated(Rk)

---meta
view = "D, IDI"
src = "Discrete-AtomicStep, IDI-Interaction"
enf = "runtime_invariant: ∀Rk,Rk+1, enters(Rk+1)→assert(completed(Rk)∧reEvaluated(Rk))"

$AX$ RoundIndependence: ∀R,R':R.stateSpace∩R'.stateSpace=∅→R∥R'

---meta
view = "IDI, D"
src = "IDI-Difference, IDI-Interaction"

$AX$ InitialState:

∀R∈Iteration:R.iterationCount=0→R.start∈StartSpace
| StartSpace={IC=0∧CF=0∧CP=0∧QDC=0∧BR=null∧LR=null∧changeLog=\[\]∧history=\[\]}
| IC>0:初始态由Continuation(Rk)定义(不重置history/changeLog)
---meta
view = "D"
src = "Discrete-Enumeration"

$AX$ EntryCondition: ∀R∈Iteration:R.enters↔R.preconditions.valid∧R.config∈ConfigSpace(R.mode)

---meta
view = "IDI, D"
src = "IDI-Interaction, Discrete-ExactEquality"

$AX$ SubRoundInitialState: ∀R∈Iteration:R.start.subRoundAttempts=0

---meta
view = "D"
src = "Discrete-Enumeration, Discrete-Countability"

// === §六评估与决策 ===

$AX$ EvaluationOutcomeSet: ∀e∈Evaluation:e.outcome∈{pass,partial,fail,parse_error}

---meta
view = "D"
src = "Discrete-ExactEquality, Discrete-Enumeration"

$AX$ DecisionOutputSet:

∀D∈LoopDecisions:D.output∈{terminate_success(TS),terminate_overflow(TO),terminate_meltdown™,switch_strategy,continue}
| TerminalDecisions={TS,TO,TM}⊆DecisionOutputSet
---meta
view = "IDI, D"
src = "IDI-Difference, Discrete-Enumeration"

// === §七终止 ===

$AX$ TerminationPriority:

∀D1,D2∈LoopDecisions:priority(D1)>priority(D2)↔precedes(D1,D2)
| short_circuit(forceHalt=true→TM; forceHalt=false∧pass→TS; else→counter_chain)
| priority_chain(CF≥MAX_FAIL > QDC≥QDT > CP≥MAX_PARTIAL > IC≥MAX_ITER > CF≥SWITCH_TH)
| qdc_increment(仅当eval=partial∧quality<prev(因果有效):+1; TS时重置0)
| commensurability_rule(R.CF与MAX_FAIL可比较iff∃R'∈causal_chain®:R'.CF可归约至标量)
| projection(proj_cf®=if ∃R'∈causal_chain®:R'.CF→R.CF then R'.CF else undefined)
| judgement(proj_cf®≥MAX_FAIL@τ_decision→TM; proj_cf®undefined→跳过CF判定)
| parameter_versioning(参数值绑定到判定时刻τ_decision;历史回放使用τ_decision活跃值,非回放时刻值)
| % {MAX_FAIL,QUALITY_DECLINE_THRESHOLD,MAX_PARTIAL,MAX_ITER,SWITCH_THRESHOLD}
---meta
view = "D, IDI"
src = "Discrete-Countability, IDI-Interaction, CountableOrderBridge"
sem = "defect-resolved=commensurability-and-parameter-versioning"
enf = "runtime_invariant: ∀D∈LoopDecisions, decision(D)→validate_terminal_type_consistency(D)"

// === §八结果组合 ===

$AX$ ResultStructure: ∀R∈Iteration:R.result∈⟨status,terminal,quality,metadata,output⟩

---meta
view = "IDI, D"
src = "IDI-Identity, Discrete-Enumeration"

$AX$ ResultComposition:

∀R∈Iteration:R.result.status=compose_causal(R.history)
| // compose_causal:扫描causal_chain®因果序从新到旧;判定(最近TM→meltdown|最近TS→success|最近TO→overflow|链空→success)
| // 实现:编码为具体算法(如DFS+时间戳),非依赖LLM因果扫描能力
| // 当前实现:因果优先聚合------已取代旧版∃k存在量词方案
---meta
view = "IDI, D"
src = "IDI-HistoryEmergence, Discrete-Enumeration, CountableOrderBridge"
enf = "specification: compose_causal∈形式规范层;实现层编码为具体算法(DFS+时间戳)"

$AX$ SingleRoundResult: ∀R∈Rounds:R.completed→compose_causal(R.history)=R.result

---meta
view = "IDI, D"
src = "IDI-Identity, IDI-HistoryEmergence"

// === 推论 ===

TH-2: R.enters↔preconditions.valid∧config∈ConfigSpace∧start∈StartSpace∧subRoundAttempts=0

| src:{EntryCondition,InitialState,SubRoundInitialState}

TH-5: ∀op₁,op₂:op₁.domain≠op₂.domain→needsSync(op₁,op₂)↔¬(op₁∥op₂)

| src:{IDI-Interaction,Coupling-ObservationCollapse,SyncProtocolTriple.CommitPhase}

TH-7: ∀R∈Rounds:R.decision∈LoopDecisions→∃g∈R.goals:g→R.decision

| src:{IDI-Interaction,IDI-Difference,Discrete-Countability}

TH-9: ∀T,params,state,τ₁,τ₂:T(params,state)@τ₁=T(params,state)@τ₂

| src:{Coupling-ObservationCollapse,Discrete-AtomicStep,SyncProtocolTriple.CommitPhase}

| pf:观测锁定→确定性→原子跃迁→CommitPhase阻断回归→相同快照⇒相同结果

| src:{Coupling-Embedding,Continuum-SamplingThreshold}

| pf:GaloisConnection意义下sample∘embed=id成立;ε偏离时sample返回历史快照为工程容错策略,非逻辑保真

// === 配置参数 ===

// 公理只声明"存在阈值",具体数值提供部署默认值,可由部署环境覆盖

// 参数修改权限元规则(修复配置覆盖权限未分级):

// modify_authority:admin_only(参数修改需管理员身份验证)

// modify_validation:参数修改后需重新验证CF≥MAX_FAIL等熔断关系

// modify_scope:仅L3(%)运行时可变;L0/L0.5/L1/L2不可通过参数接口修改

// authority_domain:配置文件注入<>运行时修改------部署环境覆盖属于配置注入,非运行时修改

// type_constraint:所有熔断参数均为ℕ(自然数);META_THRESHOLD为ℕ⁺(正整数,排除0)

% MAX_FAIL = int:3 | sem:最大CF,达到后触发TM | authority:config(部署注入)

% QUALITY_DECLINE_THRESHOLD = int:10 | sem:QDC达到后触发TO | authority:config,admin

% MAX_PARTIAL = int:5 | sem:最大连续部分成功次数 | authority:config,admin

% MAX_ITER = int:10 | sem:最大轮次迭代次数 | authority:config,admin

% SWITCH_THRESHOLD = int:2 | sem:策略切换阈值(CF) | authority:config,admin

% META_THRESHOLD = int:3 | type:ℕ⁺(正整数) | sem:连续meltdown次数触发MetaRound | authority:config,admin

% RECONFIRMATION_WINDOW = int:10 | sem:再确认窗口(逻辑时钟tick) | authority:config,admin

// 非ℕ参数（与熔断参数表分离,CONVERGE_ALPHA为float ℝ型,不属于熔断参数）

% CONVERGE_ALPHA = float:0.7 | type:ℝ(0,1) | sem:ConvergeUpdate收敛因子α∈(0,1) | authority:config,admin

// === 设计理由(~dr)与证明义务(ProofObligation) ===

// 所有设计理由已内联在各公理 ---meta 块的 dr 键中声明

// 缺陷记录已重构为证明义务,连接历史缺陷与当前公理的正确性要求

$PO-1$ ⊢_L0 (CommitPhase原子性满足Commited子集无无限回归)

| affected:{SyncProtocolTriple} | solved_by: Defect-1

$PO-2$ ⊢_L0 (CountableOrderBridge消除并发轮次同时熔断判定悬浮)

| affected:{Discrete-Countability,TerminationPriority} | solved_by: Defect-2

$PO-3$ ⊢_L0 (IDI-Interaction的类型限定∀(x:Entity),(y:Entity)阻止罗素悖论)

| affected:{IDI-Interaction} | solved_by: Defect-3

$PO-4$ ⊢_L0 (Continuum-Superposition凸集约束α+β=1排除凸锥退化)

| affected:{Continuum-Superposition} | solved_by: Defect-4

$PO-5$ ⊢_L0 (Coupling-Embedding的拓扑连续嵌入不声称逻辑态射保真性,ε-近似采样为工程启发式)

| affected:{Coupling-Embedding} | solved_by: Defect-5

$PO-6$ ⊢_L0 (ResultComposition的compose_causal实现因果序优先于存在量词)

| affected:{ResultComposition} | solved_by: Defect-6

$PO-7$ ⊢_{元层级} (ExternalConsistencyWitness不编码为系统内公式,不触发哥德尔第二不完备定理)

| affected:{ExternalConsistencyWitness, EssentiallyUndecidable} | solved_by: Defect-7

$PO-8$ ⊢_L0 (InformationMonotonicity使用因果偏序→替代全局时序<,并发分支比较无定义)

| affected:{InformationMonotonicity, ConcurrentComparisonUndefined} | solved_by: Defect-8

$PO-9$ ⊢_L0 (MetaEvolution_Rule threshold:ℕ⁺类型约束阻断零值短路,保持λ-演算纯洁性)

| affected:{MetaEvolution} | solved_by: Defect-9

$PO-10$ ⊢_L0 (TarskiTruthEquivalence的Truth绑定ProvableDerivationOnly内部推导,非外部共识)

| affected:{TarskiTruthEquivalence, ProvableDerivationOnly} | solved_by: Defect-10

$PO-11$ ⊢_L0 (SyncProtocolTriple.CommitPhase移除¬∃obs:commit(obs,obs),消除时间自噬)

| affected:{SyncProtocolTriple, CountableOrderBridge} | solved_by: Defect-11

$PO-12$ ⊢_L0 (TruthValueDomainLogic满足InformationMonotonicity,因果偏序下无¬∃τ₁→τ₂:φ@τ₁=F∧φ@τ₂=T)

| affected:{TruthValueSemantics} | solved_by: Defect-12

$PO-13$ ⊢_L0 (SyncProtocolTriple的原子性仅限定在Commited子集,Compensated路径可中断)

| affected:{SyncTermination} | solved_by: Defect-13

Axioms DSL

AxiomsDoc = formal-system-definition: "集中管理所有公理声明------公理声明'什么是逻辑上必然的'"

AX-CORE AxiomHierarchyStructure:

AX-CORE L0: BAX-IDI,BAX-TIME,BAX-C,BAX-D,SyncProtocolTriple

AX-META L0.5: MetaEvolution

AX-FRAME L1: BAX-X(Coupling-Sampling,Embedding,ObsCollapse),CountableOrderBridge

AX L2: 领域公理(§一至八)

AX-META MetaEvolution(InferenceRule):

AX-CORE StratifiedReflectionSchema:

AX-CORE LogicalLayerSeparation:

AX-CORE EssentiallyUndecidable:

AX-META ExternalConsistencyWitness:

AX-META WitnessTerm:

AX-CORE TruthValueDomainLogic:

AX-CORE InformationMonotonicity:

AX-CORE QuantificationConstraint:

BAX TriPerspectiveStructure:

BAX IDI_Framework: IDI=observer-framework(logic-semantic-base)

BAX-IDI IDI-Identity:

BAX-IDI IDI-Difference:

BAX-IDI IDI-Interaction:

BAX-IDI IDI-HistoryEmergence:

AX-CORE SyncProtocolTriple:

BAX-TIME LogicalClock:

BAX-TIME CausalCone:

BAX-C Continuum-Metric:

BAX-C Continuum-Dynamics:

BAX-C Continuum-Superposition:

BAX-C Continuum-SamplingThreshold:

TH-11: DecisionMeasure(推导): μ(s,goal)=‖s-goal‖/M; μ<ε→DiscreteGoal.pass

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BAX-D Discrete-Enumeration:

BAX-D Discrete-AtomicStep:

BAX-D Discrete-Countability:

BAX-D Discrete-ExactEquality:

AX-FRAME CountableOrderBridge:

BAX-X Coupling-Embedding(TopologicalContinuityEmbedding):

BAX-X Coupling-Sampling:

BAX-X Coupling-ObservationCollapse:

AX StateSnapshot: ∀op∈Ops,∀s∈Snap:op.readsFrom(s)→deterministic

AX StateMerge: ∀m∈Merges:m.conflictFree→consistent

AX InitialStateNullity: ∀x∈SystemState:LegalInitialState→x.intent∈{NO_DATA,NO_RESPONSE}

AX IntentClassification: ∀r∈Result:r.intent∈{NO_DATA,NO_RESPONSE,AMBIGUOUS}

AX FirstCallTransient: ∀k∈CallKey:callCount(k)=0→result(k).mayBeTransient

AX RepeatedCallStable:

AX TransientRetryNoCost: ∀k∈CallKey:callCount(k)=0∧result(k)=∅→retry(k).errorCost=0

AX TransientResultReevaluation:

AX ScopeAnchor: ∀m∈LLM:m.attention⊆m.layerCoverage

AX FocusLayer: ∀r∈Rules,∀f∈CurrentFocus:r.layer∈LAYERS→r.active(f)iff f.level≥r.layer

AX ScopeGuard: ∀r∈Rules,∀c∈CurrentContext:r.scope∈SCOPES→r.active©iff c∈r.scope

AX ExecutionStateTaxonomy: ∀op∈Operation:op.status∈{success,failure}

AX FailureSubtyping:

AX ParseErrorStatus: ∀r∈CollectResult:extract®=∅→r.status=parse_error

AX RoundCompletion: ∀R∈Rounds:R.completed↔R.executed∧R.verified∧R.logged

AX RoundTransition: ∀Rk,Rk+1:enters(Rk+1)↔completed(Rk)∧reEvaluated(Rk)

AX RoundIndependence: ∀R,R':R.stateSpace∩R'.stateSpace=∅→R∥R'

AX InitialState:

AX EntryCondition: ∀R∈Iteration:R.enters↔R.preconditions.valid∧R.config∈ConfigSpace(R.mode)

AX SubRoundInitialState: ∀R∈Iteration:R.start.subRoundAttempts=0

AX EvaluationOutcomeSet: ∀e∈Evaluation:e.outcome∈{pass,partial,fail,parse_error}

AX DecisionOutputSet:

AX TerminationPriority:

AX ResultStructure: ∀R∈Iteration:R.result∈⟨status,terminal,quality,metadata,output⟩

AX ResultComposition:

AX SingleRoundResult: ∀R∈Rounds:R.completed→compose_causal(R.history)=R.result