TauLib · API Book III

TauLib.BookIII.Bridge.BridgeAxiom

TauLib.BookIII.Bridge.BridgeAxiom

Bridge Axiom (CONJECTURAL), Shadow Diagram, RH Bridge Three-Layer, Bridge Ledger, and Honest Claim Theorem.

Registry Cross-References

  • [III.D71] Bridge Axiom (CONJECTURAL) – bridge_functor_exists (Lean axiom)

  • [III.D72] Shadow Diagram – shadow_diagram_check

  • [III.T45] RH Bridge Three-Layer – rh_bridge_three_layer

  • [III.T46] Bridge Ledger – bridge_ledger_check

  • [III.T47] Honest Claim Theorem – honest_claim_check

Mathematical Content

III.D71 (Bridge Axiom): A bridge is a structure-preserving functor F: Cat_tau(E2) -> Mod(ZFC) satisfying carrier preservation, predicate preservation, decoder compatibility, and invariant reflection. The existence of such F is CONJECTURAL and declared as a Lean axiom. This is the one point where the tau-framework explicitly marks the gap between internal and external mathematics.

III.D72 (Shadow Diagram): The image of a tau-internal commutative diagram under the bridge functor F. The shadow preserves commutativity but may lose structure: each forbidden move introduces a specific degeneracy in the shadow.

III.T45 (RH Bridge Three-Layer): The RH bridge has three layers: (1) tau-internal spectral purity (tau-effective), (2) Connes-Consani Weil positivity (established), (3) identification of tau spectral data with Riemann zeta zeros (conjectural).

III.T46 (Bridge Ledger): Per-problem bridge status: 6 conjectural (RH, NS, YM, Hodge, BSD, Langlands), 1 bridge break (P vs NP), 1 established (Poincare).

III.T47 (Honest Claim): tau-framework claims are precisely bounded by scope labels. Every check that passes at (bound, db) is labeled with the correct scope: established, tau-effective, conjectural, or metaphorical.

Tau.BookIII.Bridge.ScopeLabel

source inductive Tau.BookIII.Bridge.ScopeLabel :Type

The four scope labels used throughout Book III. Establishes the honesty discipline for claims.

  • established : ScopeLabel
  • tau_effective : ScopeLabel
  • conjectural : ScopeLabel
  • metaphorical : ScopeLabel Instances For

Tau.BookIII.Bridge.instReprScopeLabel.repr

source def Tau.BookIII.Bridge.instReprScopeLabel.repr :ScopeLabel → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.instReprScopeLabel

source instance Tau.BookIII.Bridge.instReprScopeLabel :Repr ScopeLabel

Equations

  • Tau.BookIII.Bridge.instReprScopeLabel = { reprPrec := Tau.BookIII.Bridge.instReprScopeLabel.repr }

Tau.BookIII.Bridge.instDecidableEqScopeLabel

source instance Tau.BookIII.Bridge.instDecidableEqScopeLabel :DecidableEq ScopeLabel

Equations

  • Tau.BookIII.Bridge.instDecidableEqScopeLabel x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookIII.Bridge.instBEqScopeLabel

source instance Tau.BookIII.Bridge.instBEqScopeLabel :BEq ScopeLabel

Equations

  • Tau.BookIII.Bridge.instBEqScopeLabel = { beq := Tau.BookIII.Bridge.instBEqScopeLabel.beq }

Tau.BookIII.Bridge.instBEqScopeLabel.beq

source def Tau.BookIII.Bridge.instBEqScopeLabel.beq :ScopeLabel → ScopeLabel → Bool

Equations

  • Tau.BookIII.Bridge.instBEqScopeLabel.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIII.Bridge.ScopeLabel.toNat

source def Tau.BookIII.Bridge.ScopeLabel.toNat :ScopeLabel → ℕ

Numeric order of scopes (higher = less certain). Equations

  • Tau.BookIII.Bridge.ScopeLabel.established.toNat = 0
  • Tau.BookIII.Bridge.ScopeLabel.tau_effective.toNat = 1
  • Tau.BookIII.Bridge.ScopeLabel.conjectural.toNat = 2
  • Tau.BookIII.Bridge.ScopeLabel.metaphorical.toNat = 3 Instances For

Tau.BookIII.Bridge.bridge_functor_check

source def Tau.BookIII.Bridge.bridge_functor_check (bound db : Denotation.TauIdx) :Bool

[III.D71] Bridge functor check at finite level: can we map tau-internal structures to ZFC-internal structures at depth k? At finite level, the bridge is a map from tau-addresses to ZFC axiom operations. Equations

  • Tau.BookIII.Bridge.bridge_functor_check bound db = Tau.BookIII.Bridge.bridge_functor_check.go bound db 0 1 ((bound + 1) * (db + 1)) Instances For

Tau.BookIII.Bridge.bridge_functor_check.go

source@[irreducible]

**def Tau.BookIII.Bridge.bridge_functor_check.go (bound db : Denotation.TauIdx)

(x k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.bridge_functor_exists

source axiom Tau.BookIII.Bridge.bridge_functor_exists (bound db : Denotation.TauIdx) :bridge_functor_check bound db = true

[III.D71] CONJECTURAL AXIOM: The bridge functor exists for all (bound, db). This is the ONE conjectural postulate in the Bridge. At finite level, bridge_functor_check verifies the finite shadow. The axiom asserts that this extends to the infinite tower.

Tau.BookIII.Bridge.shadow_diagram_check

source def Tau.BookIII.Bridge.shadow_diagram_check (bound db : Denotation.TauIdx) :Bool

[III.D72] Shadow diagram: the image of a tau-internal diagram under the bridge functor. A shadow preserves commutativity but may lose injectivity or faithfulness at forbidden moves.

Modeled as: for a commutative square (a -> b -> c) in tau, the shadow (F(a) -> F(b) -> F(c)) in ZFC preserves the composition (reduce coherence) but may collapse distinct values. Equations

  • Tau.BookIII.Bridge.shadow_diagram_check bound db = Tau.BookIII.Bridge.shadow_diagram_check.go bound db 0 0 1 ((bound + 1) * (bound + 1) * (db + 1)) Instances For

Tau.BookIII.Bridge.shadow_diagram_check.go

source@[irreducible]

**def Tau.BookIII.Bridge.shadow_diagram_check.go (bound db : Denotation.TauIdx)

(a b k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.BridgeStatus

source inductive Tau.BookIII.Bridge.BridgeStatus :Type

Bridge status for each Millennium Problem.

  • established : BridgeStatus
  • conjectural : BridgeStatus
  • bridge_break : BridgeStatus Instances For

Tau.BookIII.Bridge.instReprBridgeStatus

source instance Tau.BookIII.Bridge.instReprBridgeStatus :Repr BridgeStatus

Equations

  • Tau.BookIII.Bridge.instReprBridgeStatus = { reprPrec := Tau.BookIII.Bridge.instReprBridgeStatus.repr }

Tau.BookIII.Bridge.instReprBridgeStatus.repr

source def Tau.BookIII.Bridge.instReprBridgeStatus.repr :BridgeStatus → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.instDecidableEqBridgeStatus

source instance Tau.BookIII.Bridge.instDecidableEqBridgeStatus :DecidableEq BridgeStatus

Equations

  • Tau.BookIII.Bridge.instDecidableEqBridgeStatus x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookIII.Bridge.instBEqBridgeStatus.beq

source def Tau.BookIII.Bridge.instBEqBridgeStatus.beq :BridgeStatus → BridgeStatus → Bool

Equations

  • Tau.BookIII.Bridge.instBEqBridgeStatus.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIII.Bridge.instBEqBridgeStatus

source instance Tau.BookIII.Bridge.instBEqBridgeStatus :BEq BridgeStatus

Equations

  • Tau.BookIII.Bridge.instBEqBridgeStatus = { beq := Tau.BookIII.Bridge.instBEqBridgeStatus.beq }

Tau.BookIII.Bridge.rh_bridge_three_layer

source **def Tau.BookIII.Bridge.rh_bridge_three_layer (bound db : Denotation.TauIdx)

(h : bridge_functor_check bound db = true) :BridgeStatus**

[III.T45] RH bridge three-layer structure: Layer 1: tau-internal spectral purity on H_L (tau-effective) Layer 2: Connes-Consani Weil positivity Q_W(g) >= 0 (established) Layer 3: identification of tau spectral data with zeta zeros (conjectural) Equations

  • Tau.BookIII.Bridge.rh_bridge_three_layer bound db h = Tau.BookIII.Bridge.BridgeStatus.conjectural Instances For

Tau.BookIII.Bridge.rh_bridge_layer_count

source def Tau.BookIII.Bridge.rh_bridge_layer_count :ℕ

[III.T45] Layer count for the RH bridge. Equations

  • Tau.BookIII.Bridge.rh_bridge_layer_count = 3 Instances For

Tau.BookIII.Bridge.rh_conjectural_layer

source def Tau.BookIII.Bridge.rh_conjectural_layer :ℕ

[III.T45] The conjectural gap is always Layer 3. Equations

  • Tau.BookIII.Bridge.rh_conjectural_layer = 3 Instances For

Tau.BookIII.Bridge.BridgeLedgerEntry

source structure Tau.BookIII.Bridge.BridgeLedgerEntry :Type

[III.T46] Bridge ledger entry: each Millennium Problem has a bridge status and a scope label.

  • problem : Doors.MillenniumProblem
  • status : BridgeStatus
  • scope : ScopeLabel Instances For

Tau.BookIII.Bridge.instReprBridgeLedgerEntry.repr

source def Tau.BookIII.Bridge.instReprBridgeLedgerEntry.repr :BridgeLedgerEntry → ℕ → Std.Format

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.instReprBridgeLedgerEntry

source instance Tau.BookIII.Bridge.instReprBridgeLedgerEntry :Repr BridgeLedgerEntry

Equations

  • Tau.BookIII.Bridge.instReprBridgeLedgerEntry = { reprPrec := Tau.BookIII.Bridge.instReprBridgeLedgerEntry.repr }

Tau.BookIII.Bridge.instDecidableEqBridgeLedgerEntry

source instance Tau.BookIII.Bridge.instDecidableEqBridgeLedgerEntry :DecidableEq BridgeLedgerEntry

Equations

  • Tau.BookIII.Bridge.instDecidableEqBridgeLedgerEntry = Tau.BookIII.Bridge.instDecidableEqBridgeLedgerEntry.decEq

Tau.BookIII.Bridge.instDecidableEqBridgeLedgerEntry.decEq

source def Tau.BookIII.Bridge.instDecidableEqBridgeLedgerEntry.decEq (x✝ x✝¹ : BridgeLedgerEntry) :Decidable (x✝ = x✝¹)

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.instBEqBridgeLedgerEntry

source instance Tau.BookIII.Bridge.instBEqBridgeLedgerEntry :BEq BridgeLedgerEntry

Equations

  • Tau.BookIII.Bridge.instBEqBridgeLedgerEntry = { beq := Tau.BookIII.Bridge.instBEqBridgeLedgerEntry.beq }

Tau.BookIII.Bridge.instBEqBridgeLedgerEntry.beq

source def Tau.BookIII.Bridge.instBEqBridgeLedgerEntry.beq :BridgeLedgerEntry → BridgeLedgerEntry → Bool

Equations

  • Tau.BookIII.Bridge.instBEqBridgeLedgerEntry.beq { problem := a, status := a_1, scope := a_2 } { problem := b, status := b_1, scope := b_2 } = (a == b && (a_1 == b_1 && a_2 == b_2))
  • Tau.BookIII.Bridge.instBEqBridgeLedgerEntry.beq x✝¹ x✝ = false Instances For

Tau.BookIII.Bridge.bridge_ledger

source def Tau.BookIII.Bridge.bridge_ledger :List BridgeLedgerEntry

[III.T46] The complete bridge ledger. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.bridge_ledger_len

source def Tau.BookIII.Bridge.bridge_ledger_len :ℕ

[III.T46] Bridge ledger length. Equations

  • Tau.BookIII.Bridge.bridge_ledger_len = Tau.BookIII.Bridge.bridge_ledger.length Instances For

Tau.BookIII.Bridge.ledger_status_count

source def Tau.BookIII.Bridge.ledger_status_count (s : BridgeStatus) :ℕ

[III.T46] Count entries with a given status. Equations

  • Tau.BookIII.Bridge.ledger_status_count s = (List.filter (fun (e : Tau.BookIII.Bridge.BridgeLedgerEntry) => e.status == s) Tau.BookIII.Bridge.bridge_ledger).length Instances For

Tau.BookIII.Bridge.bridge_ledger_check

source def Tau.BookIII.Bridge.bridge_ledger_check :Bool

[III.T46] Bridge ledger check: verify the ledger is consistent. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.ClaimRecord

source structure Tau.BookIII.Bridge.ClaimRecord :Type

[III.T47] A claim record: associates a check function with its scope.

  • name : String
  • scope : ScopeLabel
  • check : Denotation.TauIdx → Denotation.TauIdx → Bool Instances For

Tau.BookIII.Bridge.established_claims

source def Tau.BookIII.Bridge.established_claims :List ClaimRecord

[III.T47] The established claims: these pass checks and have scope “established” or “tau-effective”. No bridge axiom needed. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.honest_claim_check

source def Tau.BookIII.Bridge.honest_claim_check (bound db : Denotation.TauIdx) :Bool

[III.T47] Honest claim check: every established/tau-effective claim passes its check at the given (bound, db). This is UNCONDITIONAL – no bridge axiom needed for honest claims. Equations

  • Tau.BookIII.Bridge.honest_claim_check bound db = Tau.BookIII.Bridge.honest_claim_check.go bound db Tau.BookIII.Bridge.established_claims (Tau.BookIII.Bridge.established_claims.length + 1) Instances For

Tau.BookIII.Bridge.honest_claim_check.go

source@[irreducible]

**def Tau.BookIII.Bridge.honest_claim_check.go (bound db : Denotation.TauIdx)

(claims : List ClaimRecord)

(fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size.
  • Tau.BookIII.Bridge.honest_claim_check.go bound db [] fuel = if fuel = 0 then true else true Instances For

Tau.BookIII.Bridge.conjectural_properly_marked

source def Tau.BookIII.Bridge.conjectural_properly_marked :Bool

[III.T47] Conjectural claims are clearly marked: they depend on the bridge axiom and are NOT claimed as theorems. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.break_properly_marked

source def Tau.BookIII.Bridge.break_properly_marked :Bool

[III.T47] Bridge breaks are clearly marked: P vs NP is tau_effective internally but the bridge degenerates. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.honest_claim_full

source def Tau.BookIII.Bridge.honest_claim_full (bound db : Denotation.TauIdx) :Bool

[III.T47] Full honest claim: established claims pass checks, conjectural claims are properly marked, breaks are properly marked. Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Bridge.bridge_functor_8_3

source theorem Tau.BookIII.Bridge.bridge_functor_8_3 :bridge_functor_check 8 3 = true

Tau.BookIII.Bridge.shadow_diagram_8_3

source theorem Tau.BookIII.Bridge.shadow_diagram_8_3 :shadow_diagram_check 8 3 = true

Tau.BookIII.Bridge.honest_claim_8_3

source theorem Tau.BookIII.Bridge.honest_claim_8_3 :honest_claim_check 8 3 = true

Tau.BookIII.Bridge.conjectural_marked

source theorem Tau.BookIII.Bridge.conjectural_marked :conjectural_properly_marked = true

Tau.BookIII.Bridge.break_marked

source theorem Tau.BookIII.Bridge.break_marked :break_properly_marked = true

Tau.BookIII.Bridge.honest_claim_full_8_3

source theorem Tau.BookIII.Bridge.honest_claim_full_8_3 :honest_claim_full 8 3 = true

Tau.BookIII.Bridge.bridge_ledger_consistent

source theorem Tau.BookIII.Bridge.bridge_ledger_consistent :bridge_ledger_check = true

Tau.BookIII.Bridge.rh_bridge_conjectural

source theorem Tau.BookIII.Bridge.rh_bridge_conjectural (bound db : Denotation.TauIdx) :rh_bridge_three_layer bound db ⋯ = BridgeStatus.conjectural

[III.T45] RH bridge is conjectural (conditional on bridge axiom).

Tau.BookIII.Bridge.one_axiom

source theorem Tau.BookIII.Bridge.one_axiom :rh_conjectural_layer = 3

[III.D71] Structural: bridge axiom is the ONLY conjectural postulate.

Tau.BookIII.Bridge.rh_layers

source theorem Tau.BookIII.Bridge.rh_layers :rh_bridge_layer_count = 3

[III.T45] Structural: RH bridge has 3 layers.

Tau.BookIII.Bridge.ledger_count

source theorem Tau.BookIII.Bridge.ledger_count :bridge_ledger_len = 8

[III.T46] Structural: ledger has exactly 8 entries.

Tau.BookIII.Bridge.poincare_established

source theorem Tau.BookIII.Bridge.poincare_established :ledger_status_count BridgeStatus.established = 1

[III.T46] Structural: Poincare is established.

Tau.BookIII.Bridge.pvsnp_bridge_break

source theorem Tau.BookIII.Bridge.pvsnp_bridge_break :ledger_status_count BridgeStatus.bridge_break = 1

[III.T46] Structural: P vs NP is a bridge break.

Tau.BookIII.Bridge.scope_order

source theorem Tau.BookIII.Bridge.scope_order :ScopeLabel.established.toNat < ScopeLabel.tau_effective.toNat ∧ ScopeLabel.tau_effective.toNat < ScopeLabel.conjectural.toNat ∧ ScopeLabel.conjectural.toNat < ScopeLabel.metaphorical.toNat

[III.T47] Structural: scope labels are ordered.

Tau.BookIII.Bridge.established_not_conjectural

source theorem Tau.BookIII.Bridge.established_not_conjectural :(established_claims.all fun (c : ClaimRecord) => c.scope == ScopeLabel.established || c.scope == ScopeLabel.tau_effective) = true

[III.T47] Structural: established claims use no conjectural scope.

Tau.BookIII.Bridge.ledger_partition

source theorem Tau.BookIII.Bridge.ledger_partition :ledger_status_count BridgeStatus.conjectural + ledger_status_count BridgeStatus.established + ledger_status_count BridgeStatus.bridge_break = 8

[III.T47] Structural: 6 + 1 + 1 = 8 (partition of bridge ledger).