A modular-charge fifth face of the Unified Equilibrium Law in Quantum Traction Theory: EP = (ℏc/2πℓP) QP at A7 saturation

Abstract. The Unified Equilibrium Law (UEL) is the central capacity-equilibrium identity of Quantum Traction Theory (QTT). It is the framework's load-bearing spine: mass, frequency, four-density, and modular bundle capacity are read as distinct faces of one primitive Planck capacity rather than as independent primitives. In its established four-face form, UEL reads EP = mP c² = ℏωP = ρ_ (4) (4πℓP⁴), with parameter-free dimensional correctors c², ℏ, and 4πℓP⁴. The fifth face was already in the book v6 back in 2025 in different section about quantum capacity. However, This paper makes explicit to make fifth UEL face clear: information capacity, expressed as the modular charge of an A7-saturated address bundle. The five-face UEL reads EP = mP c² = ℏωP = ρ_ (4) (4πℓP⁴) = (ℏc / 2πℓP) QP, with QP = 2π. The fifth face is a QTT-native structural corollary of the framework's existing axioms (A2 IR identification, A4 internal S¹ dial, A5 visible-hidden factorization, A6 per-address capacity, and A7 modular-bundle saturation Qwᵇundle = 2π) together with the A7 energy map; no new axiom is introduced. Its content is not a fitted Planck-scale constant but the closure of modular capacity into the same primitive unit that already carries mass, frequency, and four-density. The dimensional corrector kQ = ℏc/ (2πℓP) contains no tunable coupling; the rescaling-invariant content is kQ · QP = EP. Equivalently, Eᵇundle = EP · Sᵇundle with Sᵇundle = 1 nat at saturation: within QTT, the Planck energy is the energy assigned to one nat of Umegaki relative entropy at one A7-saturated bundle. Version 3. 7 update. This version does not change the equations or the core result. It clarifies the status of the result: the fifth face is a QTT-native structural corollary of A6--A7 and a central UEL face, while preserving the scope limits on low-energy Landauer physics, continuum algebraic QFT, and the still-open operational reconstruction of Qw. Scope. No claim is made that ordinary low-energy bits carry Planck-scale energy; everyday bits in computers, detectors, or low-energy cold-atom systems are governed by the Landauer scale kB T ln 2. The corollary is formulated in the regulated finite-cell algebra assumed by QTT A5; no claim extends to continuum algebraic QFT without additional structure. Falsifier and operational gap. A reproducible determination of a per-address modular-budget normalization Q* ≠ 2π in the QTT angular convention, or an independent measurement of a saturated-bundle energy Ebundle ≠ ℏc/ℓP at fixed ℓ̃ = ℓP, would invalidate the corollary. At present QTT does not provide an operational reconstruction protocol for Qw from independent observables; the falsifier remains theory-internal until such a protocol exists.