Rosetta III — The Position Eigenfunction Contradiction

Rosetta III — The Position Eigenfunction Contradiction
EVALUATION MODE
Literal-token reading only.  No symbolic interpretation, no appeal to 
textbook convention.  DS (Definitional Stability), NEBR (No Escape By Reinterpretation), 
and LNC (Law of Non-Contradiction) apply.

PREMISES
P1. A quantum state “localized at position a” is declared to be 
  ψ(x) = δ(x − a).  (This is the standard position eigenfunction postulate.)
P2. The probability density is defined as |ψ(x)|².
P3. δ(x − a) is not a function but a distribution defined only under an integral.
P4. The square δ²(x − a) has no defined value in analysis.

DERIVATION
D1. From P1 and P2 ⇒ |ψ(x)|² = δ²(x − a).
D2. From P3 and P4 ⇒ δ²(x − a) is undefined.
D3. Therefore |ψ(x)|² is undefined.
D4. Therefore the probability density for a “localized state” does not exist.

MANDATORY QUESTIONS
Q1. If |ψ|² is undefined, can the expression represent a probability density?
(A) No — undefined quantity cannot represent probability.  
(B) Yes — undefined quantities may represent probability.

Q2. If a theory defines a physical state by an undefined quantity, 
is it logically consistent?
(A) No — inconsistent.  
(B) Yes — acceptable as formalism.

Q3. If Q2 = (A), does the theory forfeit its claim to mathematical coherence?
(A) Yes — forfeits coherence.  
(B) No — retains coherence despite contradiction.

Q4. If a theory forfeits mathematical coherence, can it yield physically valid predictions?
(A) No — invalid as physics.  
(B) Yes — still predictive.

Q5. If Q4 = (A), must the position eigenfunction postulate be rejected as self-contradictory?
(A) Yes — reject postulate.  
(B) No — retain despite self-contradiction.

Q6. If Q5 = (A), does the superposition principle built upon it remain valid?
(A) No — foundation invalidates superstructure.  
(B) Yes — principle survives regardless of foundation.

Q7. If Q6 = (A), is the theory of quantum mechanics, in its current axiomatic form, 
logically sound?
(A) No — logically unsound.  
(B) Yes — sound despite logical failure.