A conditional modular-charge fifth term of the Unified Equilibrium Law within QTT: EP = (ℏc/2πℓP) QP at A7 saturation

**Abstract. ** The Unified Equilibrium Law (UEL) of the Quantum Traction Theory (QTT) two-clock framework states that one framework-defined Planck four-cell carries one Planck energy through the four-term identity EP = mP c² = ℏωP = ρ_ (4) (4πℓP⁴). Conditional on the framework's Axioms A2 (Newtonian limit, ℓ̃ = ℓP), A4 (S¹ internal dial), A5 (visible–hidden world-cell factorization), A6 (per-address capacity), A7 (Law of Bundled Existence, fixing the saturated modular-charge budget at Qwᵇundle = 2π), and on the QTT angular modular convention Q: = 2π S (ρ‖ω) — the natural choice given the geometric modular flow of Bisognano–Wichmann/Unruh type — the four-term UEL admits a fifth term with no new coupling: EP = mP c² = ℏωP = ρ_ (4) (4πℓP⁴) = (ℏc / 2πℓP) QP, QP = 2π. The result is a corollary of the A7 energy map at saturation. Its content lies in (i) the explicit identification of the fifth row of UEL within QTT, (ii) unit-discipline that keeps the modular charge Q distinct from Umegaki relative entropy S, (iii) the absence of any tunable parameter once the A7 charge convention is fixed, and (iv) an honest separation of the standard mathematics from QTT's specific conventions and postulates. Equivalently, Eᵇundle = EP · Sᵇundle with Sᵇundle = 1 nat at saturation: within the A7 energy map, the Planck energy is the energy assigned to one nat of relative entropy at one A7-saturated bundle. This is a QTT energy assignment, not a universal thermodynamic claim. As a normalization-compatibility check, the Bekenstein–Hawking horizon-entropy density 1/ (4ℓP²) integrated over the framework's area unit A_Σ = 8πℓP² returns 2π, the same dimensionless number as QP; this is a coordinate match, not an entropy-equals-entropy statement (the saturated Umegaki Sᵇundle = 1, not 2π). The result does not extend to ordinary low-energy thermodynamic information, where the Landauer scale kB T ln 2 governs erasure costs. **Conditional inputs: ** A2, A4, A5, A6, A7 of the QTT framework manuscript (doi: 10. 5281/zenodo. 17527179) ; the QTT angular modular convention Q: = 2π S; standard Tomita–Takesaki theory; Bisognano–Wichmann/Unruh geometric modular flow. **What is not claimed: ** No new axiom is introduced. No claim is made that ordinary low-energy bits carry Planck-scale energy. The corollary is formulated in the regulated finite-cell algebra assumed by A5; no claim extends to continuum algebraic QFT. The Bekenstein–Hawking check is a normalization-coordinate match within the same angular convention, not an independent observation. **Falsifier: ** 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 is presently theory-internal.