The Exact Perimeter of Semantic Energy: What I(Y;Z) Does and Does Not License — A Scoped Property Inventory for E_sem — Dissipation, Conservation, No-Cloning, the Failure of Extensivity, and the Estimation Tax

Abstract

The MOBIUS corpus defines semantic energy as E_sem := I(Y;Z), the Shannon mutual information between a semantic source Y and an interpreting internal representation Z. The reverse-derived cosmology (MRDC) identified this as the *only* quantity surviving unchanged from the prior corpus under its one-way-valve triage — and it is notable that the sole survivor is a physical-sounding metaphor attached to a classical quantity. Before any instrumentation is built on it, the metaphor's load rating must be fixed. This paper does exactly that, giving the failure results equal billing with the positive ones. We assemble and prove, from standard information theory: E_sem is bounded (L1), dissipative — never increasing — along processing chains (L2), conserved exactly under sufficient transformations — invertible recoding is the universal special case, but sufficiency does not require invertibility, and everything Y-irrelevant may be discarded at zero semantic cost (L3, correcting an earlier invertible-only framing after external review), additive over independent channels (L4a), and uncopyable — the joint energy of all copies is capped by the source (L6); governance interventions on the representation can only spend it, never mint it, making semantic cost of governance a well-defined nonnegative quantity (L5). Against this: additivity fails in general — the XOR construction gives two channels each carrying zero energy that jointly carry one full bit, so per-channel energy accounting can be arbitrarily wrong in both directions (L4b) — and finite-sample measurement is biased with a support-determined sign: the plug-in estimator's leading bias is (m_YZ − m_Y − m_Z + 1)/(2n ln 2) bits in the occupied-support counts (P7) — under full joint support this is (|Y|−1)(|Z|−1)/(2n ln 2) > 0, an overstatement worst exactly in the large-alphabet regime language models occupy; under thin dependent support (Y = Z uniform on k symbols) it is −(k−1)/(2n ln 2) < 0, an understatement. Theorem 8 assembles the perimeter: "energy" is warranted in the dissipative-resource sense and unwarranted in the extensive/conserved-quantity sense. Numerical demonstrations (frozen seeds) confirm every check: closed-form vs. numeric Gaussian agreement to 1.2 × 10⁻¹³ bits, zero data-processing violations in 15/15 grid cells and 1,000/1,000 masking draws, a plug-in bias table matching the Miller–Madow prediction within 3.5% in the asymptotic regime while exhibiting its documented breakdown at n not dominating the alphabet size, a Y = Z demonstration confirming the predicted negative bias under thin support, and a non-invertible sufficient-statistic projection conserving I(Y;Z) to machine precision in 1,000/1,000 draws. A verified fact anchors the paper's stance: no MOBIUS code computes mutual information today. T5 is therefore specification-first — the contract, including mandatory bias correction, that any future E_sem instrumentation must satisfy. Theory paper T5 of the MOBIUS 2026-08 theory series (six papers, T1–T6). Version 0.1, deposited as a preprint; journal submission of a revised version is planned, and the journal version may differ. AI co-observer: Claude Fable 5 (Anthropic), working method only; the registered author is the human author alone.