GFT as a mathematical first principle

Understanding $\log_2(m)=D_{KL}+H$ as exact conservation of the information budget.

Plain

The idea

Imagine a box with $m$ possibilities. To identify one possibility, a certain budget of distinctions is needed. GFT says this budget never disappears.

It splits into two parts: what remains structured and informative, and what spreads as noise or uncertainty. The sum always gives back the total budget.

log₂(m) = D_KL + H

Standard

Standard reading

The formula $\log_2(m)=D_{KL}(P\|U_m)+H(P)$ is an algebraic identity for any distribution $P$ on $m$ states.

Its PT strength does not come from being hard to prove, but from the role it plays: it becomes the first principle forbidding arbitrary creation or loss of persistence.

Takeaways

GFT is an identity, not a fit.
It separates persistent structure and entropy.
It provides the first principle of persistence.

Technical

Technical formulation

With $U_m$ uniform, $D_{KL}(P\|U_m)=\sum_i p_i\log_2(mp_i)$ and $H(P)=-\sum_i p_i\log_2 p_i$. Adding them cancels the $p_i\log_2 p_i$ terms.

The monograph classifies GFT-ID as an exact identity. Structural extensions keep their own status: derivation, bridge, or validation depending on the case.

Monograph: ch04_gft, ch14_thermodynamics, GFT notation.

Formulas

$\log_2(m)=D_{KL}(P\|U_m)+H(P)$

$D_{KL}(P\|U_m)=\sum_i p_i\log_2(mp_i)$

$H(P)=-\sum_i p_i\log_2 p_i$

public code

Code and scripts

The links below point to public resources or planned GitHub repositories. No local working path is exposed to the reader.

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GFT identity

Numerically checks log2(m)=D_KL+H on several distributions.

Expected: residual < 1e-12

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