https://osf.io/y98bc/files/osfstorage/68f0afbacaed018c3cc3fd9b
Nested Uplifts Inevitability (INU) Assumption 3.3 and the Riemann Hypothesis: Engineering Relaxations, Conceptual Bridges, and What Current Evidence Allows
Abstract
This paper clarifies how INU Assumption 3.3 [1] (the “Δ → drift map”) interfaces with a geometric–dynamic reading of the Riemann Hypothesis (RH). From an engineering standpoint, we propose pragmatic relaxations of 3.3 that preserve stability and whitening while being easier to verify in experiments and data-driven systems. From a theoretical standpoint, we show how these relaxed conditions still support a closed-loop narrative—linking sequential-evidence whitening, curvature-balance in prime gaps, and a self-adjoint “collapse Laplacian”—which frames RH as a unique equilibrium of a coupled geometry–dynamics system. We finally comment on how far 3.3 can be softened in light of what is currently proved about ζ(s): zero-free regions, density theorems, a positive proportion of zeros on the critical line, and extensive numerical evidence. The upshot is twofold: (i) for engineering usage, it suffices to require local mean-reverting drift, sector-bounded monotonicity, or convex-potential (subgradient) structure with mild stochasticity; (ii) for RH-motivated inquiry, these same conditions remain compatible with a stability-based interpretation of the critical line as the unique whitening and energy-minimizing attractor.
1. Introduction
INU (Nested Uplifts Inevitability) models regime switching in open systems through sequential evidence, thresholds, and whitening criteria. A central postulate is Assumption 3.3, which posits a monotone “drift map” from a one-dimensional deviation Δ to an average corrective drift μ(Δ). In parallel, a geometric–dynamic perspective on RH (via Semantic Collapse Geometry, SCG) recasts prime-gap irregularities as curvature modes whose balanced configuration aligns with the zeta critical line. The two frameworks become mutually reinforcing when the whitening threshold in INU coincides with the curvature-balance locus in SCG; then RH emerges as the unique stable equilibrium of the closed loop.
This paper has three goals:
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Provide engineering-friendly relaxations of INU Assumption 3.3 that retain stability/whitening guarantees but are easier to validate experimentally.
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Use these relaxations to motivate deeper RH inquiries, preserving the conceptual bridge without demanding brittle assumptions.
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Discuss how far 3.3 can be weakened if we integrate what is known (and widely believed plausible) about ζ(s) and its zeros—while acknowledging that RH remains unproven.
2. INU Assumption 3.3 and Its Minimal Content
Assumption 3.3 (Δ → drift map): There exists a neighborhood 𝒩 of 0 such that Δ ∈ 𝒩 ⇒ μ(Δ) = h(Δ) with h strictly increasing; in particular, Δ > 0 ⇒ μ(Δ) > 0. Empirically, h can be estimated by local regression of u_t on proxies for g, β, γ.
Intuitively, Δ measures “how far” the system is from its target equilibrium (e.g., whitening threshold, spectral balance, or curvature neutrality). The drift μ(Δ) provides an average restoring force that drives Δ back to zero.
For dynamical clarity, one may write a coarse-grained evolution on a slow time τ:
dΔ/dτ = −h(Δ) + η(τ) (2.1)
where η(τ) is a zero-mean perturbation. If h is strictly increasing with h(0)=0 and h′(0)>0, then Δ=0 is locally asymptotically stable.
3. Engineering-Friendly Relaxations of Assumption 3.3
Strict global monotonicity is often stronger than necessary in real systems. Below are practical relaxations, each sufficient (with mild side conditions) to preserve stability and whitening for experiments and applications.
3.1 Sector-bounded local monotonicity
Require local bounds near 0:
k₁ ≤ h(Δ)/Δ ≤ k₂, for |Δ| ≤ δ, with k₁>0 (3.1)
This “sector condition” guarantees a minimum restoring gain without restricting global behavior. It is natural for identification via local regression and robust to mild nonlinearities.
3.2 Incremental passivity / mean-reversion inequality
A dissipativity-style condition around 0:
E[ μ(Δ) · Δ ] ≥ α · |Δ|², for |Δ| ≤ δ, with α>0 (3.2)
This statement tolerates stochasticity and model mismatch: on average, the drift consumes deviation energy and pushes toward Δ=0.
3.3 Subgradient structure (convex potential)
Assume μ is a (negative) subgradient of a convex potential V:
μ(Δ) ∈ −∂V(Δ), V convex with a unique minimum at Δ=0 (3.3)
Subgradient dynamics cover non-smooth effects (dead-zones, mild hysteresis). Convexity ensures global well-posedness for proximal or gradient-like schemes and supports Lyapunov proofs.
3.4 Locally odd symmetry with small bias
If the deviation cost is roughly symmetric left/right,
h(−Δ) ≈ −h(Δ) near 0, possibly with a small bias term (3.4)
This captures practical asymmetries while preserving the net mean-reversion structure.
3.5 Stochastic mean reversion with bounded diffusion
Let the drift be corrupted by bounded diffusion σ(Δ):
μ(Δ) = h(Δ) + σ(Δ)·ξ_τ, E[ξ_τ]=0, |σ(Δ)| ≤ σ_max (3.5)
If (3.1) or (3.2) holds in expectation, stability in probability (and whitening) can be established for small enough σ_max.
3.6 Dead-zone and saturation (everyday physics)
A common, intuitive actuator model:
μ(Δ) = −k(|Δ|) · sat(Δ/δ₀) (3.6)
Here sat(·) is a bounded, odd saturation; k(|Δ|) ≥ k_min > 0 for |Δ| ≥ δ₀ and may taper near 0 (dead-zone). This captures the spring + damping + dry friction metaphor engineers find natural. Local mean-reversion persists if the effective gain beyond the dead-zone is positive.
4. Mapping 3.3 to the RH Stability Picture
To link INU to RH, we need a meaningful Δ that tracks deviation from the RH-aligned equilibrium. Three interchangeable “coordinates” are useful:
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Critical-line offset
Δ_line = Re(s) − 1/2.
Δ_line = 0 corresponds to being on the critical line. -
Whitening gap
Δ_white = R★ − R_log, where R_log measures whiteness on the natural semilog axis and R★ is the threshold.
Δ_white = 0 means residuals are whitened precisely at the threshold. -
Curvature mismatch (SCG)
Δ_curv summarizes phase/energy misalignment for the discrete curvature field extracted from prime gaps (e.g., mismatch of dominant curvature modes with the balanced spectrum).
All three aim at the same fixed point—the RH-aligned equilibrium—but are measured in different modalities.
A unified evolution (continuous-time caricature):
dΔ/dτ = −h(Δ) + η(τ) (4.1)
Lyapunov-type functional that couples energy and whitening:
𝓛(Δ) = V(Δ) + γ · (R★ − R_log(Δ))_+ (4.2)
If μ(Δ) ∈ −∂V(Δ) and whitening holds only at Δ=0 (R_log ≥ R★ ⇔ Δ=0), then 𝓛 is nonincreasing along trajectories and minimized only at the RH-aligned equilibrium.
5. Deeper Paths Toward RH Inspired by 3.3
Assumption 3.3 suggests local restoring structure around the equilibrium. In the SCG picture, that equilibrium corresponds to curvature balance and a spectral alignment with a self-adjoint “collapse Laplacian” Δ_c constructed from prime-gap curvature. The following steps are natural:
5.1 SCG operator construction
Define a weighted discrete Laplacian on curvature sequences κ = (κ_n):
(Δ_c x)n = w{n−1}(x_n − x_{n−1}) − w_n(x_{n+1} − x_n) (5.1)
with positive, slowly varying weights w_n derived from gap statistics. Under mild conditions this operator is self-adjoint on ℓ², ensuring a real spectrum. Its resonant frequencies then play the role of zero ordinates t = Im(ρ).
5.2 Spectral equivalence target
Normalize Δ_c so that its resonant set {t_k} matches the ζ zero ordinates {Im(ρ_k)}:
Spec_target: {t_k} = {Im(ρ_k)} (5.2)
This is a Hilbert–Pólya flavored objective, but now geometrically generated from prime-gap curvature.
5.3 Closed-loop stability and whitening
If the residual processes (e.g., Chebyshev error) whiten only when phases are driven by {t_k}, then the INU whitening threshold selects the same equilibrium as SCG’s energy minimum. The relaxed 3.3 conditions in Section 3 suffice to guarantee local attraction without needing global strict monotonicity.
5.4 Practical research program
• Empirically estimate κ and w_n, construct finite-section Δ_c, and track spectral convergence.
• Measure whitening metrics R_log on residuals and test alignment with Δ_c modes.
• Fit relaxed drift laws (e.g., sector bounds, subgradient forms) directly on Δ-proxies to validate 3.3-like behavior.
6. How Far Can 3.3 Be Relaxed Given Current ζ-Evidence?
RH is unproven, but a substantial body of results and computations exists: classical zero-free regions to the right of the critical line; zero-density theorems; proofs that infinitely many zeros lie on the critical line; and extensive numerical verification of billions of zeros on the line, along with pair-correlation phenomena predicted by random matrix theory. While these do not prove RH, they support the view that the critical line is a distinguished stability locus.
Against that backdrop, what relaxations of 3.3 are justified?
6.1 Local (not global) requirements
Given the partial results and strong numerics, it is reasonable to assume only local mean-reversion near Δ=0:
E[ μ(Δ) · Δ ] ≥ α · |Δ|² for |Δ| ≤ δ, with α>0 (6.1)
This is strictly weaker than global strict monotonicity and aligns with “the critical line is locally stabilizing.”
6.2 Sector conditions and subgradient structure
Assuming a sector-bounded slope near 0 or a convex potential V with μ ∈ −∂V provides robust, estimation-friendly alternatives:
k₁ ≤ h(Δ)/Δ ≤ k₂ for |Δ| ≤ δ (6.2)
μ(Δ) ∈ −∂V(Δ), V convex, unique minimizer at 0 (6.3)
These are compatible with existing evidence and avoid over-commitment to a specific parametric law.
6.3 Stochastic perturbations and ISS-type robustness
Allow bounded stochasticity and require stability in probability or in mean:
dΔ/dτ = −h(Δ) + σ(Δ)·ξ_τ, |σ(Δ)| ≤ σ_max (6.4)
With (6.1)–(6.3), one can obtain input-to-state stability (ISS) bounds and whitening at equilibrium despite noise, reflecting the empirically noisy character of arithmetic error processes.
6.4 Dead-zones and saturation are acceptable
Because none of the current evidence pinpoints a global linear law, it is reasonable to tolerate mild dead-zones, saturation, or asymmetries, provided the net local restoring effect holds. This mirrors what control engineers implement routinely without undermining convergence.
Bottom line: In light of what is proved and strongly supported numerically, Assumption 3.3 can be relaxed to local mean reversion plus mild regularity (sector bounds or convex subgradient) with stochastic robustness. This maintains the conceptual bridge to RH while being realistic for engineering and data analysis.
7. Conclusions
INU’s Assumption 3.3 encodes a simple idea: small deviations should elicit an average restoring drift. For engineering purposes, global strict monotonicity is unnecessary; local sector-bounded monotonicity, incremental passivity, or convex-subgradient structure with bounded stochasticity are more than adequate and easier to verify. For theory, these same relaxations still support the geometric–dynamic interpretation of the critical line as the unique stability/whitening equilibrium, in harmony with the SCG operator picture.
Given present knowledge about ζ(s)—zero-free regions, density results, a positive proportion of zeros on the line, and compelling numerics—3.3 can be softened to local mean-reversion with modest regularity without losing its role in the stability narrative for RH. This dual perspective preserves practical feasibility for experiments while keeping open a structured research path: construct the curvature-driven operator, test spectral alignment, and couple it to whitening metrics under relaxed, empirically grounded drift laws.
Practical Checklist (for experiments)
• Define Δ as one (or a combination) of: critical-line offset, whitening gap, curvature mismatch.
• Fit μ(Δ) using monotone or convex-regularized regression to obtain sector bounds near 0.
• Check whitening: verify R_log ≥ R★ only at Δ≈0.
• Build Δ_c from curvature weights and compare its leading spectral modes with zero ordinates.
• Stress-test under dead-zone and saturation models; confirm local mean-reversion persists.
If these checks pass, your system already satisfies a relaxed 3.3 that is faithful to INU’s intent and fully compatible with the stability-oriented reading of the Riemann Hypothesis.
Reference
[1] Nested Uplifts Inevitability: A Sequential-Evidence and Small-Gain Theory of Regime Switching in Open Dissipative Systems
https://osf.io/ne89a/files/osfstorage/68effd340c8fad784bc40616
[2] Semantic Collapse Geometry and Nested Uplifts Inevitability: A Geometric–Dynamic Path Toward the Riemann Hypothesis
https://osf.io/y98bc/files/osfstorage/68f03034e9e93b23f27f2f3b
[3] Semantic Collapse Geometry: A Unified Topological Model Linking Gödelian Logic, Attractor Dynamics, and Prime Number Gaps
https://osf.io/7jzpq
© 2025 Danny Yeung. All rights reserved. 版权所有 不得转载
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This book is the product of a collaboration between the author and OpenAI's GPT-5, Wolfram's GPTs language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.
This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.
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