Completion Thread Fiber

Abstract

A constant completed readout has a nontrivial thread fiber, while adjoining the blow-up origin restores injectivity and proves that no completed-value decoder can reconstruct every thread.

Theorem 1.1 (Completion Value Constant).

$$\begin{gathered} \forall o_1:GoldenThreadObserver,o_2:GoldenThreadObserver, \\ (completionValue{o}_1=completionValue{o}_2). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completion_value_constant (✓ std3). ∎

Source. Repository-derived.

Commentary.

Every pair of threads lies in the same zeroth-order completion fiber.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.2 (Completion Value Not Injective).

$$(\neg Function.InjectivecompletionValue).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completion_value_not_injective (✓ std3). ∎

Source. Repository-derived.

Commentary.

Zeroth-order completion is not injective.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.3 (Blowup Value Injective).

$$(Function.InjectiveblowupValue).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.blowup_value_injective (✓ std3). ∎

Source. Repository-derived.

Commentary.

The first blow-up readout is injective on this normalized thread family.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.4 (Completed Jet Readout Injective).

$$(Function.InjectivecompletedJetReadout).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completed_jet_readout_injective (✓ std3). ∎

Source. Repository-derived.

Commentary.

Adjoining the first jet to the completion value restores injectivity.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.5 (No Completion Value Decoder).

$$(\neg \exists decode:\mathbb{R}\to \mathbb{R},\forall observer:GoldenThreadObserver,decode(completionValueobserver)=observer.origin).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.no_completion_value_decoder (✓ std3). ∎

Source. Repository-derived.

Commentary.

No function of the completed value alone can recover every origin coefficient.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.6 (No Completion Thread Reconstructor).

$$(\neg \exists reconstruct:\mathbb{R}\to GoldenThreadObserver,\forall observer:GoldenThreadObserver,reconstruct(completionValueobserver)=observer).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.no_completion_thread_reconstructor (✓ std3). ∎

Source. Repository-derived.

Commentary.

Any putative reconstruction of the full normalized observer from the completed value would induce a forbidden origin decoder.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.7 (Completion Fiber Contains All Origins).

$$\begin{gathered} \forall c:\mathbb{R}, \\ (completionValue⟨c⟩=Real.goldenRatio). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completion_fiber_contains_all_origins (✓ std3). ∎

Source. Repository-derived.

Commentary.

The common completion fiber is infinite, witnessed by the embedding of all real origin coefficients.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

References

• Truth anchor: D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.blowup_value_injective
• Truth anchor: D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completed_jet_readout_injective
• Truth anchor: D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completion_fiber_contains_all_origins
• Truth anchor: D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completion_value_constant
• Truth anchor: D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.completion_value_not_injective
• Truth anchor: D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.no_completion_thread_reconstructor
• Truth anchor: D5/S3/CompletionDynamics/DynamicReal/CompletionThreadFiber.no_completion_value_decoder
• Dependency: D5/S3/CompletionDynamics/GoldenMobius/GoldenThreadBlowup