The Riemann Hypothesis: A Framework via Echo-Silence and Coprime-Diagonal Analysis
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Abstract
We present a new framework for investigating the Riemann Hypothesis through two companion papers that establish complementary approaches to the critical line.
Paper 1 (Echo-Silence): We prove an unconditional equivalence: RH holds if and only if a precisely defined mirror functional Mσ(y,T) vanishes uniformly on compact y-intervals for all σ ∈ [1/2+κ, 1-κ]. This reduces RH to verifying uniform echo-silence, providing a new analytic criterion.
Paper 2 (CDH Companion): Under a Type I/II hypothesis (Assumption A), we establish the required vanishing through two-sided Kuznetsov dispersion. The framework introduces angular dispersion in v = ½(log m - log n) and radial dispersion in u = ½(log m + log n) via Mellin shear, achieving a T-δ power saving uniformly in σ.
Key Results: (1) Unconditional: Echo-silence ⟺ RH equivalence via exponential polynomial theory; (2) Conditional: Two-sided dispersion delivers uniform vanishing under standard analytic hypotheses; (3) The bilinear constant c = 55/432 from classical exponent pairs; (4) Bandlimited weights enable Nikolskii upgrade from L² to L∞ control.
The papers also explore connections to physics (event horizons, quantum levels) and consciousness (recognition at boundaries), suggesting deeper mathematical-physical unities.
🔬 Key Contributions
🎯 Unconditional Equivalence
RH ⟺ echo-silence proven without assumptions
⚡ Two-Sided Dispersion
Angular + radial dispersion framework
📊 Explicit Constants
Bilinear constant c = 55/432 from classical exponents
🔢 Bandlimited Upgrade
Nikolskii: L² → L∞ for uniform control
✅ Mirror Functional
New analytic criterion for RH verification
🌐 Physics Bridge
Critical line as event horizon analogy
📚 How to Cite
Goodfellow, W. (2025). "Echo-Silence on the Critical Horizon and the Riemann Hypothesis." Preprint, January 2025.
CDH Companion Paper:
Goodfellow, W. (2025). "Vanishing Bounds for Mirror Functionals via Coprime-Diagonal Analysis." Preprint, January 2025.
Available at: https://shirania-branches.com/research/riemann-hypothesis/
🤝 Community Invitation
These papers present new approaches to the Riemann Hypothesis for community examination.
The Echo-Silence paper establishes an unconditional equivalence between RH and uniform vanishing of a mirror functional. The CDH Companion shows how two-sided Kuznetsov dispersion can deliver this vanishing under standard analytic hypotheses.
We invite rigorous peer review and discussion of these methods. The framework opens new avenues for investigating L-functions through mirror functionals and dispersion techniques, with intriguing connections to physics and consciousness studies.