Golden Image Recovery Obstruction

Abstract

The projective golden boundary image forgets every observer rapidity.

Theorem 1.1 (A fixed projective boundary image cannot reconstruct the observer).

$$\mathrm{let}futurePoint:\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right)=\mathrm{class}\left(\mathrm{pair}\left(\phi ,1\right)\right),\mathrm{let}pastPoint:\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right)=\mathrm{class}\left(\mathrm{pair}\left({\phi }^{\prime },1\right)\right),\mathrm{let}ObserverOrbit:Type=\mathrm{Subtype}\left(\mathrm{range}\left((eta:\mathrm{Real}\left(\right)\mapsto \mathrm{pair}\left(h\left(eta\right),\mathrm{tangent}\left(eta\right)\right))\right)\right),\mathrm{let}Pi:ObserverOrbit\to \mathrm{Prod}\left(\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right),\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right)\right)=(state:ObserverOrbit\mapsto \mathrm{pair}\left(\mathrm{class}\left(\mathrm{fst}\left(state\right)+\mathrm{snd}\left(state\right)\right),\mathrm{class}\left(\mathrm{snd}\left(state\right)-\mathrm{fst}\left(state\right)\right)\right)),\mathrm{let}image:\mathrm{Real}\left(\right)\to \mathrm{Prod}\left(\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right),\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right)\right)=(eta:\mathrm{Real}\left(\right)\mapsto Pi\left(\mathrm{pair}\left(h\left(eta\right),\mathrm{tangent}\left(eta\right)\right)\right)),\left(\forall eta\in \mathrm{Real}\left(\right),image\left(eta\right)=\mathrm{pair}\left(futurePoint,pastPoint\right)\right)\wedge \left(\left(\forall eta1\in \mathrm{Real}\left(\right),eta2\in \mathrm{Real}\left(\right),eta1\ne eta2\Rightarrow \left(image\left(eta1\right)=image\left(eta2\right)\wedge \mathrm{pair}\left(h\left(eta1\right),\mathrm{tangent}\left(eta1\right)\right)\ne \mathrm{pair}\left(h\left(eta2\right),\mathrm{tangent}\left(eta2\right)\right)\right)\right)\wedge \left(\left(\neg \mathrm{Injective}\left(Pi\right)\right)\wedge \left(\left(\forall recoverRapidity\in \mathrm{Prod}\left(\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right),\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right)\right)\to \mathrm{Real}\left(\right),\neg \left(\forall eta\in \mathrm{Real}\left(\right),recoverRapidity\left(image\left(eta\right)\right)=eta\right)\right)\wedge \left(\forall recoverObserver\in \mathrm{Prod}\left(\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right),\mathrm{Projectivization}\left(\mathrm{Real}\left(\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right)\right)\to \mathrm{Prod}\left(\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right),\mathrm{Prod}\left(\mathrm{Real}\left(\right),\mathrm{Real}\left(\right)\right)\right),\neg \left(\forall eta\in \mathrm{Real}\left(\right),recoverObserver\left(image\left(eta\right)\right)=\mathrm{pair}\left(h\left(eta\right),\mathrm{tangent}\left(eta\right)\right)\right)\right)\right)\right)\right).$$

Proof. Machine-checked in Lean as D5/S3/Observer/HyperbolicTransport/GoldenImageRecoveryObstruction.golden_image_recovery_obstruction (✓ std3). ∎

Source. Repository-derived.

Commentary.

The observer event and its genuine tangent are the canonical objects constructed by the preceding null-direction theorem. The boundary projection is defined on their concrete orbit and sends each state’s sum and difference to Mathlib’s projective quotient.

Positive rapidity-dependent amplitudes disappear in projective space, so every rapidity has the same ordered pair of golden null points. The event-tangent states themselves remain distinct.

The theorem states the resulting non-injectivity directly and rules out both a rapidity decoder and a concrete observer-state decoder from the complete boundary image.

References

• Truth anchor: D5/S3/Observer/HyperbolicTransport/GoldenImageRecoveryObstruction.golden_image_recovery_obstruction
• Dependency: D5/S3/Observer/HyperbolicTransport/ObserverEventNullDirections