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IndisputableMonolith.Quantum.PlanckScale

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This module defines the Planck scale quantities in Recognition Science units by expressing Planck length, mass, time, energy, and temperature through the fundamental time quantum τ₀ and the golden ratio φ. Physicists modeling discrete spacetime or quantum gravity would cite these to anchor standard Planck units inside the RS framework. The module consists of direct definitions and ratio lemmas that link the phi-ladder to voxel and length-hierarchy structures.

claimThe Planck length satisfies $l_P = √(ℏG/c³) ≈ 1.616 × 10^{-35}$ m with $c=1$, $ℏ=φ^{-5}$, $G=φ^5/π$; analogous definitions hold for Planck mass $m_P$, time $t_P$, energy, and temperature, together with the ratio $τ_0/t_P = φ^k$ for integer exponent $k$ on the phi-ladder.

background

Recognition Science begins from a discrete ledger carrying J-cost, where self-similarity forces the golden ratio φ as the unique fixed point (PhiForcing). The Constants module fixes the RS-native time quantum τ₀ = 1 tick, with derived constants $c=1$, $ℏ=φ^{-5}$, $G=φ^5/π$. This module places the Planck scale inside that setting, treating it as the natural meeting point of quantum and gravitational scales on the phi-ladder.

proof idea

This is a definition module, no proofs. It supplies explicit expressions for each Planck quantity in terms of τ₀ and integer powers of φ, followed by ratio lemmas such as tau0_tP_ratio and phi_exponent_tau0_tP that connect directly to the voxelLength and lengthHierarchy siblings.

why it matters in Recognition Science

The module supplies the Planck-scale bridge between the phi-forcing chain (T5–T8) and observable quantum-gravity scales. It provides the base layer for any downstream quantum or cosmological construction inside the Recognition framework, closing the gap between the abstract J-cost ledger and concrete length and mass hierarchies.

scope and limits

depends on (2)

Lean names referenced from this declaration's body.

declarations in this module (18)