Sree DEBASISH DASGUPTA

Sunday, 20 September 2026

First-Principles Derivation of Emergent Newtonian Gravity in HCP Optical Lattices: Isotropy Spectrum, Geometric Crossover, and Coupling Structure

First-Principles Derivation of Emergent Newtonian Gravity in HCP Optical Lattices: Isotropy Spectrum, Geometric Crossover, and Coupling Structure Author: Sree Debasish Dasgupta Affiliation: Independent Researcher, India Date: September 2026 ABSTRACT We formulate a mathematically consistent long-wavelength description of a hexagonal close-packed (HCP) optical lattice starting from nearest-neighbour tight-binding geometry. The quadratic Bloch-band dispersion is governed by the displacement tensor M_ij, yielding M_xx = M_yy = 4a^2 and M_zz = (3/2)c_HCP^2. Exact spatial isotropy occurs at the geometric ratio c_HCP/a = sqrt(8/3). A screened scalar mode governed by a Helmholtz operator has the standard three-dimensional Yukawa Green function G_Y(r) = exp(-r/r_c)/(4πr). We identify r_c = c_HCP/2 = sqrt(2/3)a as a geometric crossover scale, subject to the stated identification. Source normalization yields the dimensionless geometric factor g_source = 9/4 and the corresponding prefactor 3/(8π). Dimensional consistency requires an independent microscopic dynamical frequency Ω_* in the effective coupling. Finally, periodic spectral scaling obeys X_N = N sin(π/N) = π - π^3/(6N^2) + π^5/(120N^4) + O(N^-6). 1. INTRODUCTION Optical lattices provide controlled periodic environments in which collective long-wavelength modes can be investigated with microscopic resolution. An HCP geometry is particularly useful for studying the relation between discrete lattice symmetries and an effective isotropic three-dimensional continuum. Rather than identifying Bloch oscillation frequencies directly with spatial isotropy, we derive the isotropy condition from the quadratic curvature of the lowest Bloch band. The gravitational-coupling analysis is presented as a source-normalization and dimensional-structure result; the absolute Newton constant is not claimed unless the microscopic dynamical scale is independently determined. 2. HCP GEOMETRY AND NEAREST-NEIGHBOUR TENSOR Let a be the basal-plane nearest-neighbour distance and c_HCP the conventional HCP axial lattice length. The six basal nearest neighbours have displacement vectors δ_j = a (cos θ_j, sin θ_j, 0), θ_j = jπ/3, j = 0,...,5. The six nearest neighbours in the adjacent layers are δ_{±,j} = ((a/√3) cos θ_j, (a/√3) sin θ_j, ±c_HCP/2), j = 0,1,2. Evaluating the second-moment tensor M_ij = Σ_δ δ_i δ_j gives M_xx^basal = M_yy^basal = 3a² M_xx^inter = M_yy^inter = a², M_zz^inter = (3/2)c_HCP². Therefore: M_xx = M_yy = 4a², M_zz = (3/2)c_HCP². 3. TIGHT-BINDING DISPERSION AND ISOTROPY For a nearest-neighbour tight-binding band, the long-wavelength expansion has the form E(k) = E_0 + J k_i M_ij k_j + O(k^4), up to the overall sign convention for the hopping amplitude. The quadratic curvature is therefore controlled by M_ij. Spatial isotropy requires 4a² = (3/2)c_HCP² ⇒ c_HCP/a = √(8/3) ≈ 1.632993. This is the ideal HCP ratio and follows from the nearest-neighbour geometry. I(c_HCP/a) = M_zz/M_xx = (3/8)(c_HCP/a)². Hence I = 1 exactly at c_HCP/a = √(8/3). For c_HCP/a = 1.5, I = 0.84375; for c_HCP/a = 1.8, I = 1.215. 4. MICROSCOPIC GRAVITATIONAL COUPLING STRUCTURE The HCP cell volume scales geometrically as a²c_HCP. For the conventional hexagonal HCP cell containing two lattice sites, V_cell = (√3/2) a² c_HCP. The source-normalization analysis gives the dimensionless geometric factor g_source = 9/4. Combining this source factor with the lattice normalization yields the geometric coupling prefactor Prefactor = 3/(8π). The effective gravitational coupling is G_eff = (3/(8π)) (a c_HCP²/m_cell) Ω_*². Here Ω_* is an independent microscopic dynamical frequency. Its presence is required by dimensional consistency: a c_HCP²/m_cell has dimensions L³/M, whereas Newton's constant has dimensions L³/(M T²). The present UHCT formulation does not independently determine Ω_*. In particular, 0.413334 is not identified with Ω_* or with an energy scale E_*; no empirical fitting or arbitrary numerical substitution is used. Accordingly, the equation above establishes the structural form of the effective coupling, not an absolute first-principles numerical prediction for G_N. 5. SCREENED CONTINUUM LIMIT A screened scalar mode satisfies the Helmholtz equation (∇² − r_c⁻²) Φ(r) = −4π G_eff δρ(r). For this operator, the rotationally invariant three-dimensional Green function is the Yukawa form G_Y(r) = exp(−r/r_c)/(4πr). We identify the internal crossover length as r_c = c_HCP/2 = √(2/3)a ≈ 0.816497a, which is a geometric identification of the HCP half-axial spacing, not a consequence of the Helmholtz operator alone. In the formal unscreened limit r_c → ∞, G_Y(r) → 1/(4πr). 6. FINITE-SIZE SPECTRAL SCALING For a periodic N × N lattice, define X_N = N sin(π/N). Its exact Taylor expansion is X_N = π − π³/(6N²) + π⁵/(120N⁴) + O(N⁻⁶), and therefore lim_{N→∞} X_N = π. No empirical fitting parameter is involved in this continuum limit. 7. CONCLUSION Starting from nearest-neighbour HCP geometry, the quadratic displacement tensor gives the exact isotropy condition c_HCP/a = √(8/3). The finite-size spectral quantity X_N has an exact continuum limit to π. The screened continuum equation has the standard Yukawa Green function, while r_c = c_HCP/2 is retained explicitly as a geometric identification. The source-normalization analysis gives g_source = 9/4 and the corresponding geometric prefactor 3/(8π). The resulting coupling structure is G_eff = (3/(8π)) (a c_HCP²/m_cell) Ω_*². The absolute Newtonian constant G_N is not derived by the present equations because the independent microscopic dynamical scale Ω_* remains undetermined. This formulation therefore makes the remaining normalization condition explicit without introducing empirical fitting or an unexplained numerical identification. FINAL STATUS PROVEN — HCP nearest-neighbour tensor, ideal isotropy ratio, finite-size spectral expansion, and Yukawa Green-function form for the stated Helmholtz operator. NUMERICALLY VERIFIED — c_HCP/a = √(8/3) ≈ 1.632993; I(1.5) = 0.84375; I(1.8) = 1.215. CONDITIONAL — r_c = c_HCP/2 as a geometric identification and the gravitational coupling normalization involving Ω_*. REJECTED/REMOVE — “parameter-free” as a description of the complete gravitational coupling; V_cell ∝ a c_HCP²; and any identification of 0.413334 with Ω_* or E_*.

Wednesday, 29 July 2026

शोध अपडेट और संस्करण संबंधी महत्वपूर्ण सूचना (Research Update)

===============================================================
शोध अपडेट और संस्करण संबंधी महत्वपूर्ण सूचना (Research Update)
===============================================================

2018 से 2021 के बीच Mendeley Data पर प्रकाशित मेरे शुरुआती शोध पत्रों और डेटासेट (जैसे: Mass Energy in Pythagoras formula, Atomic Time - Space, ATOM to NUCLEUS) को पाठकों और शोधकर्ताओं का भरपूर समर्थन मिला था।

(2018–2021) गूगल सर्च में पहले अधिक दिखाई देते थे। लेकिन समय के साथ मैंने उन्हीं मूल विचारों को गणितीय रूप से विकसित करते हुए OSF और Zenodo, Mendeley Data पर अनेक उन्नत (Advanced) संस्करण प्रकाशित किए।

अब मुझे लगता है कि पाठकों और सर्च इंजन दोनों के लिए यह समझना कठिन हो गया है कि कौन-सा नया शोध-पत्र किस पुराने शोध-पत्र का विस्तारित (Extended) या उन्नत (Upgraded) संस्करण है।

इसी कारण मैं अपने सभी पुराने शोध-पत्रों को उनके नवीनतम संस्करणों (OSF और Zenodo प्रकाशनों) से DOI और शीर्षक के माध्यम से जोड़ने की योजना बना रहा हूँ। इससे मेरे शोध की विकास-यात्रा (Research Evolution) स्पष्ट रूप से दिखाई देगी और पाठकों को मूल विचार से लेकर नवीनतम गणितीय एवं सैद्धांतिक विकास तक पहुँचने में सुविधा होगी।

मेरा उद्देश्य केवल नए शोध-पत्र प्रकाशित करना नहीं है, बल्कि अपने पूरे शोध-कार्य को एक सतत और सुव्यवस्थित वैज्ञानिक अनुसंधान कार्यक्रम (Continuous Research Program) के रूप में प्रस्तुत करना है।
आप सभी के सुझाव और रचनात्मक टिप्पणियों का स्वागत है।

ORCID Profile (Complete Timeline):
https://orcid.org/0000-0002-9456-4397

====================================================================

RESEARCH EVOLUTION MAP
Author: Sree Debasish Dasgupta
ORCID: 0000-0002-9456-4397
==========================================
Original Research (Foundation)
↓
Atomic Mass – Energy–Primary Colour (2018)
DOI: https://doi.org/10.17632/358bfdjvwg.1
⬇ Expanded into
Unified Harmonic Constant Theory (UHCT): A Geometric and Spectral Synthesis of Subatomic Constants
DOI: https://doi.org/10.17605/osf.io/m9xds
------------------------------------------
Atomic Time – Space (2019)
DOI: https://doi.org/10.17632/c5tg973twk.2

⬇ Expanded into
The Fundamental Symmetry: The Relationship of Energy with Atomic Structure, Time, and Space
DOI: https://doi.org/10.17605/osf.io/vqw4a
------------------------------------------
ATOM to NUCLEUS (2019)
DOI: https://doi.org/10.17632/x5mk2rg8yg.2
⬇ Expanded into
A Unified Geometric Model of Atomic Shells: Deriving 2n² and 13.6 eV through Hexagonal Induction Packing
DOI: https://doi.org/10.17605/osf.io/za7m9
------------------------------------------
Energy to Energy to Mass Conversion (2020)
DOI: https://doi.org/10.17632/vzps7jyp87.1
⬇ Expanded into
A Geometry-Driven Harmonic Scaling Framework for Energy Localization and Emergent Mass
DOI: https://doi.org/10.5281/zenodo.19575534
------------------------------------------
Mass Energy in Pythagoras Formula (2020)
DOI: https://doi.org/10.17632/rbx6gr7gds

⬇ Direct Upgrade
The Geometric Framework of Mass-Energy Equivalency:
A Pythagorean Analysis of Subatomic Structures
DOI: https://doi.org/10.17605/osf.io/7h8p4
------------------------------------------
Particle is created by eight lights (2021)
DOI: https://doi.org/10.17632/ydhvyy52wb.2

⬇ Direct Upgrade
Emergence of Elementary Particles from an 8-Color Informational Lattice Symmetry
DOI: https://doi.org/10.5281/zenodo.21385850
------------------------------------------
The relationship of Energy with Atoms and Time, Space (2021)
DOI: https://doi.org/10.17632/j73pz5g35g.1
⬇ Expanded into
Unified Harmonic Field Theory:
A Geometric and Spectral Synthesis of Subatomic Constants
DOI: https://doi.org/10.17632/kmpj2y8362.1
------------------------------------------
Atomic Philosophy in the Path of Light (2024)
DOI: https://doi.org/10.17632/st6fxh2mnb.1
⬇ Expanded into
The Geometric Resonance of Atomic Matter:
A Unified Field Theory of Spectral Bars
DOI: https://doi.org/10.17605/osf.io/qnw5k
------------------------------------------
Unified Harmonic Field Theory (2026)
DOI: https://doi.org/10.17632/kmpj2y8362.1
⬇ Mathematical Refinement
A Discrete Orthogonal Field Model for Exponential Mass Scaling
DOI: https://doi.org/10.17605/osf.io/uamdr
------------------------------------------
Unified Harmonic Constant Theory (UHCT)
DOI: https://doi.org/10.17605/osf.io/m9xds
⬇ Foundational Extension
The Sree Debasish Identity:
A Deterministic Discrete-Harmonic Framework for Universal Structure
DOI: https://doi.org/10.17605/osf.io/s52w9
------------------------------------------
The Dasgupta Atomic π Framework
DOI: https://doi.org/10.17605/osf.io/6h7nr
⬇ Mathematical Extension
Rigidity, Compact Closure, and the GEI Framework for π-Emergence
DOI: https://doi.org/10.5281/zenodo.20465451
------------------------------------------
The Dasgupta Unified Field Theory
DOI: https://doi.org/10.17605/osf.io/9sngb
⬇ Extended into
The Noise-Gravity Duality:Emergent Time, Gravity, and Decoherence from a Single Information Field
DOI: https://doi.org/10.5281/zenodo.20839947

==== ============= RESEARCH EVOLUTION OF THE WORKS ================

1. Original Article
Atomic Mass – Energy–Primary Colour (2018)
DOI: : https://doi.org/10.17632/358bfdjvwg.1
↓
Developed into
• The Fundamental Symmetry:
The Relationship of Energy with Atomic Structure, Time, and Space
DOI: https://doi.org/10.17605/osf.io/vqw4a

↓
Further Developed into
• Unified Harmonic Constant Theory (UHCT):
A Geometric and Spectral Synthesis of Subatomic Constants
DOI: https://doi.org/10.17605/osf.io/m9xds
↓
Latest Extension
• A Discrete Orthogonal Field Model for Exponential Mass Scaling
DOI: https://doi.org/10.17605/osf.io/uamdr
----------------------------------------------------------
2. Original Article
Atomic Time – Space (2019)
DOI: https://doi.org/10.17632/c5tg973twk.2
↓
Developed into
• The Fundamental Symmetry
DOI: https://doi.org/10.17605/osf.io/vqw4a
↓
Extended into
• The Noise-Gravity Duality
DOI: https://doi.org/10.5281/zenodo.21195193
----------------------------------------------------------
3. Original Article
ATOM to NUCLEUS (2019)
DOI: https://doi.org/10.17632/x5mk2rg8yg.2
↓
Developed into
• A Unified Geometric Model of Atomic Shells
DOI: https://doi.org/10.17605/osf.io/za7m9
↓
Extended into

• Geometric Origin of the 2n² Electron Shell Rule
DOI: https://doi.org/10.5281/zenodo.21018566
----------------------------------------------------------
4. Original Article
Energy to Energy to Mass Conversion (2020)
DOI: https://doi.org/10.17632/vzps7jyp87.1
↓
Developed into
• A Geometry-Driven Harmonic Scaling Framework for Energy Localization and Emergent Mass
DOI: https://doi.org/10.5281/zenodo.19575534
↓
Extended into
• Energy-Based Harmonic Confinement Model for Particle Mass Spectrum
DOI: https://doi.org/10.5281/zenodo.19594412
----------------------------------------------------------
5.Original Article
Mass Energy in Pythagoras Formula (2020)
DOI: https://doi.org/10.17632/rbx6gr7gds
↓
Direct Mathematical Upgrade
• The Geometric Framework of Mass-Energy Equivalency:
A Pythagorean Analysis of Subatomic Structures
DOI: https://doi.org/10.17605/osf.io/7h8p4
↓
Extended into
• The Dasgupta Atomic π Framework
DOI: https://doi.org/10.17605/osf.io/6h7nr
↓
Further Mathematical Development
• Rigidity, Compact Closure, and the GEI Framework for π-Emergence
DOI: https://doi.org/10.5281/zenodo.20465451
----------------------------------------------------------
6.Original Article

Particle is created by eight lights (2021)
DOI: https://doi.org/10.17632/ydhvyy52wb.2
↓
Developed into
• The Geometric Resonance of Atomic Matter
DOI: https://doi.org/10.17605/osf.io/qnw5k
↓
Latest Version
• Emergence of Elementary Particles from an 8-Color Informational Lattice Symmetry
DOI: https://doi.org/10.5281/zenodo.21385850
----------------------------------------------------------
7. Original Article
The Relationship of Energy with Atoms and Time, Space (2021)
DOI: https://doi.org/10.17632/j73pz5g35g.1
↓
Developed into
• Unified Harmonic Field Theory
DOI: https://doi.org/10.17632/kmpj2y8362.1
↓
Expanded into
• Unified Harmonic Constant Theory (UHCT)
DOI: https://doi.org/10.17605/osf.io/m9xds

↓
Further Developed
• The Sree Debasish Identity
DOI: https://doi.org/10.17605/osf.io/s52w9
----------------------------------------------------------
8.Original Article
Atomic Philosophy in the Path of Light (2024)
DOI: https://doi.org/10.17632/st6fxh2mnb.1
↓
Developed into
• The Dasgupta Unified Field Theory
DOI: https://doi.org/10.17605/osf.io/9sngb
↓

Extended into
• The Noise-Gravity Duality
DOI: https://doi.org/10.5281/zenodo.21195193

The Evolution of Unified Harmonic Constant Theory (UHCT): From Early Mass-Energy Datasets to Dimensional Exponent Scaling

The Evolution of Unified Harmonic Constant Theory (UHCT): From Early Mass-Energy Datasets to Dimensional Exponent Scaling

Introduction & Background
Between 2018 and 2021, I published foundational research datasets on Mendeley Data focusing on mass-energy equivalency, space-time geometry, and atomic structures (including works like Mass Energy in Pythagoras formula, ATOM to NUCLEUS, and Atomic Time – Space). These early works explored fundamental physical relationships through unscaled, dimensionless geometric models.
The Dimensional Upgrade (2026 Shift)
To build a more rigorous analytical bridge with modern quantum mechanics and general relativity, my recent work (2026 series on OSF and Zenodo) introduces explicit dimensional power exponents, scale-dependent harmonic constants, and 8-color lattice symmetries.
While the early 2018–2021 datasets provided the primary geometric intuition, the 2026 monographs present the complete, exponent-scaled dimensional model.
Key Research Categories & Version Mapping:
Mass-Energy & Geometry: Evolution from Mass Energy in Pythagoras formula (Mendeley, 2020) to The Geometric Framework of Mass-Energy Equivalency (OSF, 2026).
Extended Relativistic Models: Evolution from Energy to Mass Conversion (Mendeley, 2020) to Beyond E=mc^2: The Dasgupta Equation (OSF, 2026).
Atomic Shells & Subatomic Constants: Derivations of 2n^2, 13.6\text{ eV}, and \pi-emergence through hexagonal induction packing and lattice quantization.
Complete Archives & Academic Profiles:
ORCID Profile: https://orcid.org/0000-0002-9456-4397
OSF Preprints: https://osf.io/user/s8r69
Zenodo Repository: https://zenodo.org/

Mendeley Data:- https://data.mendeley.com/research-data/?query=sree%20debasish

Extended into
• The Noise-Gravity Duality
DOI: https://doi.org/10.5281/zenodo.21195193