A Unified Grammar of Science

A Unified Grammar of Science A. Chawla REAL Institute November 22, 2025 Abstract This paper synthesizes three foundational frameworks — Schr¨odinger’s wave mechanics, Varshney’s capacity-energy tradeoff, and Chawla’s phenomenological theory of subjective time — into a single conceptual grammar. Objective time t(energy-driven), subjective time ∆t′(information-driven), and the physical transmission constraints that couple them are unified through a conformal scale parameter εthat finitizes infinity and resolves the algebraic–phenomenological tension surrounding the limit ∆I→0. The resulting picture portrays consciousness as the conformal boundary of information-theoretic physics. Keywords Quantum wave mechanics ·Time-dependent Schr¨odinger equation ·Capacity-energy tradeoff · Subjective time ·Change in self-information ·Conformal mapping ·Scale parameter ε·Timelessness ·Patterned matter-energy ·Nullity 1 Introduction The search for a unified description of physical and experiential reality repeatedly returns to the interplay of energy, information, space, time, and consciousness. The present work demonstrates that these five pillars admit a common mathematical grammar when analyzed through (i) Schr¨odinger’s derivation of the time-dependent wave equation, (ii) Varshney’s demonstration that information is patterned matter-energy subject to a capacity-energy tradeoff, and (iii) Chawla’s phenomenological ansatz linking subjective elapsed time to change in self-information, with the entire structure compactified by a conformal mapping ϕ(n)=1/n. Notation Table The adjacent table lists all the major symbols and their meanings as used in this work. 2 Preliminaries 2.1 Objective Time and Energy (Schr¨odinger) The objective temporal variable temerges from the quantum postulate E=hν. For stationary states, the time dependence of the wave function is a definite periodic factor exp(−2πiEt/h). Substituting the operator correspondence Eψ −→ − h 2πi ∂ψ ∂t 1 Symbol Meaning t, T Objective physical time and period ∆t′Subjective perceived elapsed time EEnergy (objective) hPlanck’s constant νFrequency ∆IChange in self-information εConformal scale parameter KSpiritual knowledge (subjective) CChannel capacity (bits/time) BReceived energy rate (power) C(B) Capacity-energy function f(·) Phenomenological subjective time function ϕ(n) = 1/n Primary conformal mapping Table 1: Principal symbols and their interpretations into the time-independent vibration equation yields the time-dependent Schr¨odinger equation (TDSE) ∇2ψ−8π2 h2V ψ +4πi h ∂ψ ∂t = 0. Objective time tis therefore the coordinate conjugate to energy E, and the objective period T= 1/ν ∝1/E. 2.2 Subjective Time and Information (Chawla) Subjective elapsed time ∆t′is defined phenomenologically by ∆t′=f(∆I), where ∆Iis the change in self-information acquired through sensory or internal sampling. The function fis stated to be inverting: minimal sampling (small ∆I) corresponds to rapid passage of perceived time (small ∆t′). The philosophical boundary K→ ∞ (infinite spiritual knowledge) forces ∆I→0 and requires ∆t′→0 (“nullity” or timelessness), despite the algebraic suggestion of a pure reciprocal 1/∆I→ ∞. 2.3 Transmission Constraint: Information as Patterned Matter-Energy (Varshney) Varshney rejects Wiener’s separation of information from its physical medium, asserting “information is patterned matter-energy.” Transmitting information Cand energy Bsimultaneously over a single noisy channel yields a capacity-energy function C(B) that is non-increasing and concave. Severe noise or deliberate sensory deprivation collapses reliable capacity C→0, annihilating ∆Iat the receiver. 3 Methods: The Conformal Scale Parameter ε The primary conformal mapping in the source is the reciprocal ε=ϕ(n) = 1 n, ε =1 D, ε =1 d, 2 compressing the asymptotic regime n, D, d → ∞ into the single boundary point ε= 0. This finitizes infinity and makes tree-code analysis geometrically tractable on the compact interval [0,1]. 4 Results: Unification via ε= ∆I Equating the conformal scale parameter with subjective information change, ε≡∆I, forces the Shannon boundary ε→0 (ideal capacity) to coincide exactly with the consciousness boundary ∆I→0 (timelessness). The algebraic singularity 1/∆I→ ∞ is topologically wrapped by the conformal boundary condition: the point at infinity in the algebraic coordinate is identified with the origin ε= 0 in the subjective geometry. Thus ∆t′→0 (nullity) is recovered consistently. Maximal Temporal Discrepancy With perfect transmission, the gap |T−∆t′|is bounded by the objective period T > 0. With channel collapse (noise or sensory deprivation), ∆I→0 is forced extrinsically, driving ∆t′→0 while objective tcontinues, achieving the theoretical maximum gap ≈T. 5 Discussion 5.1 Future Work 1. Derive an explicit, empirically testable form of f(∆I) from neurophysiological data during sensory deprivation and meditation. 2. Construct a rigorous conformal field theory on the information manifold where εplays the rˆole of a complex structure modulus. 3. Extend the capacity-energy framework to quantum channels and compare with holographic bounds (e.g., quantum capacity vs. energy transmission through horizons). 4. Investigate whether the identification ε= ∆Iimplies a microscopic arrow of time reversal near the timelessness boundary. 5.2 Limitations 1. The ansatz ∆t′=f(∆I) remains phenomenological; no microscopic derivation from neural dynamics exists. 2. The equality ε= ∆Iis a conceptual unification, not yet a mathematically derived identity. 3. The treatment of “infinite spiritual knowledge” K→ ∞ is philosophical rather than operational. 4. Perturbative effects on the objective period Tunder strong environmental coupling are not quantitatively modeled. 5. The conformal mapping ϕ(n)=1/n is classical; a fully quantum version remains to be formulated. 3 5.3 Conclusion Energy governs objective time through E=hν, information governs subjective time through ∆t′=f(∆I), and the physical act of transmission couples the two via Varshney’s capacityenergy constraint. A single conformal mapping ε= 1/n = ∆Ifinitizes the infinite, resolves the algebraic–phenomenological contradiction at the ∆I→0 boundary, and reveals consciousness as the compactified edge of physical information processing. The grammar is complete: energy, information, and consciousness speak the same conformal language. 4