Entropy and Holography through Adjunctions: A Bicategorical Perspective on Landauer's Principle

Referee: [§4] §4 (Landauer adjunction and induced monad): the manuscript asserts that the Landauer adjunction induces a boundary closure monad whose Eilenberg-Moore algebras identify the visible bulk with closed boundary data, yet supplies neither an explicit formula for the monad in terms of the profunctorial feasibility relations nor a worked computation on any concrete entropy poset. Without these, the holographic claim reduces to the standard categorical fact that algebras of an idempotent monad are fixed points of the closure, rather than a derivation from thermodynamic constraints.

Authors: We acknowledge that the current manuscript presents the monad at a high level without an explicit formula or example computation. In the revised version, we will include the explicit construction: the boundary closure monad is obtained as the composite of the Landauer adjunction's left and right adjoints applied to the profunctor, specifically T(B) = {b' | exists feasible relation to bulk state closed under the round-trip}. Additionally, we will provide a worked example on a simple two-element entropy poset, computing the monad explicitly and verifying the Eilenberg-Moore algebras correspond to closed data. This will demonstrate the derivation from the thermodynamic adjunction rather than a generic categorical fact. revision: yes

Referee: [§5] §5 (holographic interpretation): the identification of bulk states with Eilenberg-Moore algebras is presented as capturing holographic bulk-boundary correspondence, but no check is performed against known holographic features such as area-law scaling of entropy or minimal dissipation bounds. The physical content therefore rests on an interpretive overlay rather than a verified equivalence.

Authors: The holographic interpretation arises directly from the Eilenberg-Moore construction applied to the induced monad, providing a categorical reconstruction of bulk from boundary. We agree that explicit verification against features like area-law scaling would enhance the physical relevance. In revision, we will add a discussion section comparing the framework to holographic models, noting that in discretized poset approximations of bulk geometries, the entropy of closed boundary data exhibits scaling consistent with area laws due to the closure operator minimizing information loss. For minimal dissipation, the quantitative enrichment already selects optimal paths, which we will link to known bounds. However, a full numerical equivalence is beyond the scope of this foundational paper and would require specific physical models. revision: partial

Referee: [§2] §2 (bicategory of open entropy systems): the claim that physical many-to-many realizations between informational and thermodynamic descriptions are faithfully modeled by profunctorial feasibility relations is introduced definitionally; no derivation from thermodynamic first principles or comparison with existing open-system models is given, leaving the weakest assumption untested.

Authors: The bicategory is defined to capture the open, many-to-many nature of physical interfaces, which deterministic maps cannot fully represent. While introduced definitionally, this choice is motivated by the profunctor's ability to encode feasibility relations between posets, generalizing binary relations in thermodynamics. In the revision, we will add a subsection comparing to existing models such as those using channel theory or stochastic matrices in open quantum systems, showing how profunctors provide a poset-enriched generalization. A derivation from first principles is not attempted as the work is a mathematical framework for unifying concepts rather than a physical derivation; we will clarify this motivation explicitly. revision: partial