Scalar Memory Blindness

Abstract

Scalar Euler behavior forgets every hidden-memory coordinate.

Theorem 1.1 (Every finite scalar behavior is blind to memory).

$$\forall r\in \mathrm{Fin}\left(3\right),s\in \mathbb{C},v\in Nat.Primes\to \left(\mathrm{Fin}\left(2\right)\to \mathbb{C}\right),\forall z\in \mathbb{C},zprime\in \mathbb{C},z=zprime\Rightarrow \left(\left(\forall w\in \mathrm{List}\left(Nat.Primes\right),m\in \mathrm{Fin}\left(2\right)\to \mathbb{C},mprime\in \mathrm{Fin}\left(2\right)\to \mathbb{C},\mathrm{snd}\left(\mathrm{runWord}\left(\mathrm{scalarMemoryUpdate}\left(r,s,v\right),w,(m,z)\right)\right)=\mathrm{snd}\left(\mathrm{runWord}\left(\mathrm{scalarMemoryUpdate}\left(r,s,v\right),w,(mprime,zprime)\right)\right)\right)\wedge \left(\forall m\in \mathrm{Fin}\left(2\right)\to \mathbb{C},mprime\in \mathrm{Fin}\left(2\right)\to \mathbb{C},\mathrm{completionProjection}\left(\mathrm{scalarMemoryUpdate}\left(r,s,v\right),snd,(m,z)\right)=\mathrm{completionProjection}\left(\mathrm{scalarMemoryUpdate}\left(r,s,v\right),snd,(mprime,z)\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencyHolonomy/ScalarMemoryBlindness.scalar_memory_blindness (✓ std3). ∎

Source. Repository-derived.

Commentary.

The layer ranges over the source-fixed three residual local factors. The spectral parameter and the entire prime-indexed channel family remain public parameters.

Each prime step applies the imported Fibonacci substitution to the two-dimensional memory, adds the local-factor channel forcing, and multiplies the scalar by that same local factor.

The scalar coordinate therefore evolves without reading memory. Induction over every finite prime word gives equal scalar readouts, and the canonical controlled-behavior quotient identifies the full memory fiber over each scalar.

References

• Truth anchor: D5/S3/Observer/AgencyHolonomy/ScalarMemoryBlindness.scalar_memory_blindness
• Dependency: D5/S1/Scale/FibonacciEigen
• Dependency: D5/S3/ObserverMemory/Prediction/ControlledBehaviorUniversality