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

Abstract.

Quantum Traction Theory (QTT) reconstructs the foundations of classical mechanics, quantum mechanics, and general relativity from a small number of axioms about an absolute tick, a real-valued internal dial, and bundled existence at every world-cell address (framework manuscript: doi:10.5281/zenodo.17527179). The Unified Equilibrium Law (UEL) is one of the framework's central unification statements. In its established four-face form, UEL reads

E_P = m_P c² = ℏω_P = ρ_(4)(4πℓ_P⁴),

with parameter-free dimensional correctors c², ℏ, and 4πℓ_P⁴ — asserting that mass, frequency, and four-density are not independent quantities but distinct faces of one primitive capacity unit, the Planck four-cell.

This article identifies a fifth face: information capacity, expressed as the modular charge of an A7-saturated address bundle. The five-face UEL reads

E_P = m_P c² = ℏω_P = ρ_(4)(4πℓ_P⁴) = (ℏc / 2πℓ_P) Q_P,    with Q_P = 2π.

The fifth face is forced by the framework's existing axioms (A2 IR identification ℓ̃ = ℓ_P, A4 internal S¹ dial, A5 visible–hidden world-cell factorization, A6 per-address capacity, and A7 modular-bundle saturation Q_w^bundle = 2π) together with the A7 energy map; no new axiom is introduced. The dimensional corrector k_Q = ℏc/(2πℓ_P) = E_P/(2π) contains no tunable coupling: the rescaling-invariant content is the product k_Q · Q_P = E_P. Equivalently, E^bundle = E_P · S^bundle with S^bundle = 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.

Contributions of this paper.

(i) The explicit five-face form of UEL within QTT, which has not previously been written down. (ii) A Bisognano–Wichmann angular reading of the QTT charge normalization Q := 2π S, proposed here as the natural mathematical motivation for the framework's 2π factor. (iii) A unit-discipline that keeps modular charge Q distinct from Umegaki relative entropy S, with the collapsed identity E^bundle = E_P · S^bundle at S^bundle = 1 nat. (iv) The rescaling-invariant statement k_Q · Q_P = E_P, identifying what is and is not absolute in the dimensional corrector. (v) A three-layer separation of standard mathematics from QTT conventions and QTT postulates. (vi) An explicit operational-gap declaration: at present QTT does not provide an operational reconstruction protocol for Q_w from independent observables.

Connection to parallel work.

The same 2π that closes the modular bundle in A7 closes the internal S¹ dial in A4 and drives the spinor half-angle theorem of the parallel QTT structural-identification paper (forthcoming π/8 paper, Attar 2026). The two papers use the same real-dial algebra (A4) to identify two distinct structural identities and form a coordinated programme of structural identifications within the QTT ontology.

Bekenstein–Hawking calibration check.

The horizon-entropy density 1/(4ℓ_P²) integrated over the framework's area unit A_Σ = 8πℓ_P² returns 2π, consistent with QTT's calibration of the area unit to one full angular modular period: the same 2π that closes the modular bundle in A7 closes the horizon area unit in Bekenstein–Hawking accounting. Since Q = 2π·S, the saturated Umegaki relative entropy is S^bundle = 1 nat, not 2π.

What is not claimed.

No new axiom is introduced. 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 k_B T ln 2. The corollary is formulated in the regulated finite-cell algebra assumed by A5; no claim extends to continuum algebraic QFT, where local algebras are typically type III and additional structure (split property, edge modes) is required.

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 E_bundle ≠ ℏc/ℓ_P at fixed ℓ̃ = ℓ_P, would invalidate the corollary. At present QTT does not provide an operational reconstruction protocol for Q_w from independent observables; the falsifier is theory-internal until such a protocol exists.