explorations/

Heuristic, out-of-scope explorations — NOT part of Oscillating Brane Theory V8.2.

Nothing in this folder is included in the PDF, the validation pipeline, the seven sacred theory files, or any academic claim of V8.2. These are speculative seeds for possible future work (a hypothetical V9.0 on holographic quantum gravity), kept deliberately separate to protect the epistemological integrity of V8.2 — which is a macroscopic phenomenological cosmology paper and must remain one.

decoherence_riemann.py

A visualization of the Riemann explicit formula: the truncated sum over the non-trivial zeros of the Riemann zeta function reconstructs the Chebyshev psi(x) staircase (the counting function of prime powers). As more zeros are added, the continuous oscillatory sum converges toward the discrete step function. It illustrates the general principle “a sum of continuous modes can build discrete structure.”

Status: heuristic analogy, not a derivation.

The conjectured holographic thread (V9.0 research direction)

There is a non-trivial — but unproven — chain of connections worth recording:

• Berry-Keating / Hilbert-Polya: the Riemann zeros are conjectured to be the spectrum of a Hermitian operator; the classical Hamiltonian H = xp is the proposed candidate.
• H = xp is the dilatation generator of Conformal Quantum Mechanics (CQM, de Alfaro-Fubini-Furlan 1976).
• CQM is the boundary dual (CFT_1) of AdS_2 geometry.
• The near-horizon throat of an extremal / near-extremal black hole is AdS_2 x S^2 (textbook: near-horizon extremal Reissner-Nordstrom).

If the sub-critical PBH capillaries of OBT possess near-horizon AdS_2 x S^2 throats, their resonance spectrum might connect to the Riemann zeros. This is a research direction, not a result: it would require a full AdS_2/CFT_1 holographic derivation that does not exist yet.

Peer-review caveats (honest accounting)

• The residual ~9% overshoot near each step is the Gibbs phenomenon — a standard artifact of truncated mode sums (non-uniform convergence). Its amplitude is constant; only its width narrows (as ~1/N). It is not a perfect right-angle staircase and should not be over-interpreted.
• A genuine kernel of physics does exist: truncating the mode sum at N modes imposes a bandwidth, hence a finite spatial resolution (transition width ~1/N). That bandwidth-resolution tradeoff is the Fourier uncertainty relation, which becomes Heisenberg under p = hbar k. But this is encoded in the transition width, not in the Gibbs overshoot — they are distinct effects. Identifications with Compton wavelength / Zitterbewegung are evocative but not literal.

Run

pip install numpy matplotlib mpmath
python decoherence_riemann.py