G_hbar_product_eq_one
plain-language theorem explainer
Recognition Science sets the gravitational constant to phi to the fifth and the reduced Planck constant to phi to the minus fifth in its native units. This identity is cited in derivations of the Bekenstein-Hawking bound from ledger capacity on the integer lattice. The proof is a direct term reduction that unfolds the two power definitions, applies exponent addition via zpow rules, and normalizes to reach unity.
Claim. In RS natural units, where the gravitational constant satisfies $G = phi^5$ and the reduced Planck constant satisfies $hbar = phi^{-5}$, the product obeys $G hbar = 1$.
background
The module derives the Bekenstein-Hawking entropy bound from the RS ledger on Z^3, using the forced dimension D=3. G_rs is defined as phi raised to 5 and hbar_rs as phi raised to -5, so their product is the constant 1 that appears in the entropy formula S = A/(4 G hbar). Upstream results supply the positivity of phi and the general expression for G in terms of the recognition length and Planck constant, which reduce to the phi powers once c=1 and hbar=phi^{-5} are imposed.
proof idea
The term proof unfolds the definitions of G_rs and hbar_rs. It rewrites the positive exponent via zpow_natCast, combines the exponents with zpow_add_0 (using phi_ne_zero), and finishes with norm_num to obtain phi^0 = 1.
why it matters
The identity is invoked directly by bekenstein_hawking_from_rs to replace 4 G hbar with 4 and by area_not_volume_certificate to obtain S = A/4. It realizes the RS-native unit choice that closes the ledger-to-holography step in brain_holography_fully_forced, linking the constant derivations to the T8 forcing of three spatial dimensions.
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