The Necessity of the Information Axiom and the Octonions C; Arithmetic Rigidity Theorem: The Uniqueness of the Scale Factor λ = e2π from Self-Referential Closure Conditions and Modular Forms

This paper demonstrates that within a self-referential recursive system, the
scale factor λ is locked to the unique mathematical constant e2π. This conclu
sion stems from the simultaneous solution of the self-referential closure condition
and the positivity constraint of the spectral gap. By equivalently transforming
the self-referential closure condition into a scaling relation of the heat kernel trace,
we establish recurrence equations for the coefficients of the asymptotic expansion
of the heat kernel at the infrared fixed point, revealing the central role of λ as a
scaling factor. Introducing the Jones subfactor theory imposes spectral constraints
on the recursive algebraic sequence. Analysis shows that the discrete spectrum
case cannot satisfy the index lower bound under the condition λ > 1 and is there
fore excluded; the continuous spectrum case allows λ to take any value not less
than 4, compressing the possible range of λ to [4,∞). Using the Rankin–Selberg
asymptotic formula for the mean square of the Ramanujan τ function, the exponen
tial factor e4π appearing therein matches the recurrence relation λ2 = e4π for the
coefficient of the squared curvature term, thereby uniquely determining λ = e2π.
This value lies within the continuous spectrum interval and is compatible with the
Jones subfactor theory. The uniqueness proof shows that any scale factor satisfying
the self-referential closure condition must satisfy this equation, hence the solution
is unique. This paper reveals a deep connection between the scaling behavior of
self-referential systems, the arithmetic depth of modular forms, and the continuous
spectrum structure of Jones subfactor theory, providing an arithmetic foundation
for understanding discrete scale invariance in quantum gravity