G-SHARP: Cosmology, PDEs, and Surgical Imaging

• G-SHARP is a multidisciplinary topic involving an abrupt change in Newton’s constant in cosmology, a sharp pseudo-differential inequality on Lie groups, and a real-time surgical imaging pipeline.
• In cosmology, the G-step model addresses the Hubble tension by hypothesizing a sudden drop in G, though it faces stringent constraints from solar and geological data.
• In analysis and computer vision, G-SHARP underpins optimal lower bounds for subelliptic operators and accelerates high-fidelity, real-time 3D reconstruction in minimally invasive surgery.

G-SHARP refers to three distinct concepts in the research literature: (1) a cosmological model involving a sharp change in Newton’s constant $G$ to address the Hubble tension, (2) a subelliptic sharp Gårding inequality on compact Lie groups in pseudo-differential operator theory, and (3) a Gaussian Surgical Hardware Accelerated Real-time Pipeline for intra-operative 3D scene reconstruction. The following encyclopedic overview presents each interpretation in detail, emphasizing their mathematical, physical, or computational principles and their respective constraints or contributions.

1. G-SHARP in Cosmology: The $G$-Step Model and the Hubble Tension

The G-SHARP or "G-step model" (GSM) is a proposed solution to the Hubble tension, which postulates an abrupt decrease in Newton’s gravitational constant $G$ approximately 130 million years ago. The specific functional form is

$$G(t) = G_0 \times [1 + (\Delta G / G_0) H(t_s - t)],$$

where $H$ is the Heaviside step-function, $\Delta G < 0$ is the magnitude of the jump, $t_s$ is the step time, and $G_0$ is the present-day value. This abrupt modification in $G$ is theoretically motivated to make Type Ia supernovae (SNe Ia) in the distant Hubble flow intrinsically brighter (since their luminosities scale as a power of $G$), thus rendering them apparently further away at fixed redshift and potentially lowering the inferred Hubble rate at late times (Banik et al., 22 Nov 2024).

The mechanism can be summarized as follows:

• The Chandrasekhar mass sets the Ni-56 yield, so $M_{\mathrm{Ch}} \propto G^{-3/2}$ and $L_{\mathrm{peak}} \propto M_{\mathrm{Ch}}^\beta$ with $\beta \approx 1$–$1.2$.
• After empirical corrections, the SN luminosity scales as $L_\mathrm{SN} \propto G^\alpha$, $\alpha \approx 1.46$.
• A pre-step $G$ higher by $\Delta G / G_0 \sim 0.03$–$0.05$ would produce a $\sim 17\%$ increase in inferred SN luminosity, potentially reconciling the $H_0$ discrepancy.

However, GSM leads to several adverse physical and astrophysical consequences:

• Solar physics: $L_\odot \propto G^7$; a few percent drop in $G$ would precipitate a $\sim$30% decline in solar output, implying a planetary glaciation not seen in the geological record.
• Earth's orbital and rotational evolution: The orbital period $P_{\mathrm{orb}} \propto G^{-2}$; a sudden $3$–$5\%$ drop in $G$ would increase the year length by $\sim10\%$, inconsistent with cyclostratigraphic and paleontological records that constrain $\Delta G / G_0 \lesssim 0.003$.
• Solar age: The Sun would exhaust $\sim2/3$ of its hydrogen inventory, making its helioseismic age exceed that of the oldest meteorites; empirically, the two are in much tighter agreement, $|\Delta G / G_0| \lesssim 0.01$.
• Stellar chronology: Predicted stellar ages in the halo and globular clusters fall short by $2$–$3$\,Gyr, in conflict with observations.
• Cosmic chronometer and SN-CMB bounds: Measurements of $H(z)$ and the SN luminosity-redshift relation impose $|\Delta G / G_0| \lesssim 0.01$–$0.03$.

A comprehensive summary of these constraints is shown below:

Constraint	Maximum allowed $|\Delta G/G_0|$	Physical implication
Solar glaciation (Snowball Earth)	$\lesssim0.02$	Avoids global ice ages
Cyclostratigraphy (days/year)	$\lesssim0.003$	No abrupt year-length jumps
Helioseismic vs meteoritic age	$\lesssim0.01$	Solar age matches
Oldest stars (halo, GCs)	$\lesssim0.02$	No 3 Gyr age gap
Cosmic chronometers $H(z)$	$\lesssim0.01$	CCs agree with $\Lambda$CDM
Type Ia SN–CMB bound	$\lesssim0.03$	CMB+BAO consistent
GSM needed for $H_0$ tension	$0.03$–$0.05$	Resolves $H_0$ discrepancy

Even the weakest bound conclusively rules out the G-step amplitude needed to resolve the Hubble tension, with local Solar System and geological constraints decisively excluding any sharp, percent-level change in $G$ in the last $100$–$200$ Myr (Banik et al., 22 Nov 2024).

2. Subelliptic Sharp Gårding Inequality: G-SHARP on Compact Lie Groups

G-SHARP also denotes the generalized sharp Gårding inequality for pseudo-differential operators on compact Lie groups. Let $G$ be compact, and let $\mathcal{R}$ be a positive Rockland (or sub-Laplacian) operator (homogeneous degree $\nu$). The global Hörmander symbol class $S^m_{\rho, \delta}(G)$ consists of symbols $a(x, \pi)$ satisfying

$$\| \Delta_\pi^\alpha \partial_x^\beta a(x, \pi) \|_{\text{op}} \leq C_{\alpha\beta} \langle \pi \rangle^{m - \rho |\alpha| + \delta |\beta|}$$

for $0 \leq \delta < \rho \leq 1$. Pseudo-differential quantization is

$$(Au)(x) = \sum_{\pi \in \widehat{G}} d_\pi \operatorname{Tr}\left[ \pi(x) a(x, \pi) \widehat u(\pi) \right].$$

Assuming $a(x, \pi)$ is Hermitian, nonnegative, and other technical conditions, the G-SHARP theorem asserts: $$\operatorname{Re} \langle Au, u \rangle_{L^2(G)} \geq - C \|u\|_{H^s_\mathcal{R}(G)}^2$$ with Sobolev norm (relative to $\mathcal{R}$),

$$\| u \|_{H^s_\mathcal{R}} = \| (1 + \mathcal{R})^{s/(2\nu)} u \|_{L^2(G)}, \quad s = m - (\rho - \delta).$$

Key features:

• The result covers the full range $0 \leq \delta < \rho \leq 1$, including elliptic ($\rho=1$, $\nu=2$) and subelliptic ($\nu>2$) contexts.
• The "sharpness" of the lower bound is optimal: attempts to reduce the loss (lower $s$) fail on explicit counter-examples, especially on tori.
• Technical reach extends to global quantizations, subelliptic sums of squares, and the use of Rockland operators on stratified groups (Cardona et al., 2021).

The proof strategy employs a decomposition $A = P + Q$, constructing a positive operator $P$ matched to $a(x, \pi)$ and using the remainder $Q$'s mapping properties, together with a frequency-dependent weight to localize the nonnegative contributions. This provides a unified framework for sharp lower bounds across analytic, geometric, and representation-theoretic subelliptic settings.

3. G-SHARP: Gaussian Surgical Hardware Accelerated Real-time Pipeline

G-SHARP also refers to a real-time, commercially compatible surgical scene reconstruction framework targeting minimally invasive procedures (Nath et al., 2 Dec 2025). It leverages a GSplat-based differentiable Gaussian rasterization pipeline deployed on edge hardware (e.g., NVIDIA IGX Orin, Thor) via the Holoscan SDK to enable high-fidelity 3D modeling of deformable tissue at video rates in the operating room.

Pipeline architecture:

• Offline Training: Multiframe point-cloud initialization combines depth, color, and tool/tissue masks from the EndoNeRF "pulling" benchmark, yielding a dense Gaussian point cloud (32K–64K Gaussians). A two-stage optimization first fits mean, covariance, and opacity, then refines view-dependent color via spherical harmonics ($l \leq 3$) and learns temporal deformation.
• Real-time Rendering: Camera poses stream in at video rates. The pipeline loads the pretrained model, applies per-Gaussian deformation via a learned MLP operating on HexPlane features (XY, XZ, YZ, XT, YT, ZT, resolution 64$^3$×100), and renders via differentiable GSplat composite kernels, achieving $>60$ FPS at 640×512 resolution.

Mathematical details:

• Each Gaussian $i$ is parameterized as

$$\mu_i \in \mathbb{R}^3,\quad \Sigma_i = R(q_i)\,\mathrm{diag}(s_i^2)\,R(q_i)^\top,\quad \alpha_i \in [0,1]$$

with view-dependent color $c_i(\omega)$ expanded in SH $(l=0,1,2,3)$. The 3D density contribution is

$$g_i(x) = \alpha_i \exp\!\left( -\frac12 (x - \mu_i)^\top \Sigma_i^{-1} (x - \mu_i) \right).$$

Rendering aggregates the contributions along camera rays.

• The composite loss in fine-stage optimization consists of RGB, depth, SSIM, and total-variation penalties (the latter targeted to unobserved tissue behind tools), as well as deformation smoothness and grid-based TV on the HexPlanes.

Occlusion and deformation modeling:

• Tool occlusions are managed by binary masks in initialization/loss terms; TV is only imposed in never-directly-observed ("invisible") regions.
• Deformation is modeled per-Gaussian using HexPlanes features and is MLP-predicted at each frame.

Performance:

• Training: $\sim2$ min/scene on a single A100 GPU.
• Inference: $>60$ FPS at 640×512 on edge hardware.
• PSNR: 37.98 dB on EndoNeRF "pulling" benchmark (full-scene).
• Ablations demonstrate that multi-frame Gaussians, invisible-mask TV, and temporal smoothness penalties are critical for visual fidelity and stable operation.

Edge deployment:

• Holoscan SDK orchestrates inference modules (pose streaming, Gaussian checkpoint loading, per-frame deformation/rasterization, visualization, and output). Efficient memory and compute trade-offs are provided by reducing SH degree, Gaussian count, or disabling deformation for ultra-low latency needs.

Experimental and practical insights:

• Offline precomputed tool and depth masks, multi-view initialization, and tool-region Gaussian reservation improve occlusion and tissue coverage.
• Real OR tests confirm stable throughput ($60\pm2$ FPS, $\sim16$ ms latency), robust to thermal or load fluctuations.

Current research directions include reducing end-to-end latency to the sub-10 ms regime, optical-see-through AR integration, haptic-force–augmented modeling, and incorporation of language-based OR scene understanding. The open-source, Apache-2.0 GSplat base ensures commercial extensibility (Nath et al., 2 Dec 2025).

4. Comparative Table: Three G-SHARP Paradigms

Interpretation	Core Concept	Primary Domain
Cosmological G-step	Abrupt $G$ change for Hubble tension	Physical cosmology
Sharp Gårding (Lie Gps.)	Lower bounds for pseudo-differential operators	Harmonic analysis/PDE
Surgical Pipeline	Real-time 3D Gaussian scene reconstruction	Surgical computer vision

Each G-SHARP paradigm is fundamentally distinct in mathematical, physical, and computational scope, yet each is built around the notion of a "sharp" transition, bound, or real-time edge in their target application.

5. Implications, Limitations, and Future Directions

In cosmology, the G-SHARP ($G$-step) hypothesis is strongly constrained or entirely ruled out by multi-disciplinary data, including heliophysics, stratigraphy, and cosmic chronometers, with the lowest upper bound on allowed $|\Delta G/G_0|$ determined by continuous records of Earth's orbital period and solar output. Extensions or variants with time-dependent $G$ lacking an abrupt feature could be less constrained, though no viable solution to the Hubble tension is presently supported within this framework (Banik et al., 22 Nov 2024).

In harmonic analysis and PDE theory, the subelliptic G-SHARP theorem extends the reach of Gårding-type inequalities to a broad class of compact groups and geometric subelliptic settings, with exponents and symbol classes now shown to be optimal across a wide functional-analytic landscape (Cardona et al., 2021). Ongoing research is exploring analogous bounds on other Lie-type structures and for more general non-commuting symbol classes.

In intra-operative vision, the G-SHARP pipeline operationalizes differentiable Gaussian splatting for real-time, high-fidelity reconstruction, robust to occlusion and soft-tissue deformation, and deployable with commercial constraints (Nath et al., 2 Dec 2025). Future work focuses on further reduction in latency, multi-modal and AR fusion, and dynamic modeling incorporating allied sensor data.

• "Challenges to a sharp change in $G$ as a solution to the Hubble tension" (Banik et al., 22 Nov 2024)
• "Subelliptic sharp Gårding inequality on compact Lie groups" (Cardona et al., 2021)
• "G-SHARP: Gaussian Surgical Hardware Accelerated Real-time Pipeline" (Nath et al., 2 Dec 2025)