Variational Backbone and Regime Closures IV: Observational Projection and Protocol Theorems Fiber-weight entropy, event updates, and protocol readouts

VBRC treats Part IV as the observability layer of an already licensed retained summary,not as a new footprint-generation layer. Under the Internal Invisibility Principle, unreadcontent is never exported as raw data; Part IV studies the pushforward, disintegration, event,entropy, and record structures by which an already licensed retained summary becomesobservable.Part I fixes the constitutional framework under the Internal Invisibility Principle (IIP):unread interior content does not enter observable physics as pointwise data, but only througha declared hidden-to-retained summary channel F = ΣH,P (II ), realized in the scoped operatorclass as F = DI II . Parts II–III then generate admissible laws under that fixed backboneand its declared readings. The present Part IV does not produce new laws. It studies howthose already generated retained laws become observable, namely how a declared observationprotocol reads, weights, updates, and records the retained summary channel.The minimal protocol is Inst0 = (Π, µ), refined as a representation protocol Pδ =(Πδobs, µδ,Uδ). The central result is a deterministic-vs-probabilistic dichotomy: if the retaineddeterministic update T : X → X factors through the readout, the observed update isdeterministic; if it does not, then on a positive-measure set of readouts the conditional lawof the next output is non-Dirac, equivalently assigning positive kernel weight to at least twodisjoint measurable output sets. Probability is therefore forced by the disintegration of µalong Π, rather than added as an independent axiom; it is the protocol-level form of partialobservation under IIP.From pushforward and disintegration, the following protocol outputs are obtained at theRepresentation Layer: (i) probability as the pushforward ν = Π#µ; (ii) observational entropyas fiber weight, Sδobs(y) = kB log Wδ(y), with the point-fiber form kB log µ(Π−1(y)) used onlyin finite or atomic instances; and (iii) update and event as conditionalization/disintegrationalong a declared gate, with record-entropy monotonicity for accumulated records recoveredvia the Shannon chain rule, while DPI is used only for post-processing/garbling comparisons.Observational equivalence x ∼Π x′ ⇔ Π(x) = Π(x′) defines the operational quotient Xδphys =X/∼Π; any fiber-preserving Φ with Π◦Φ = Π acts as a channel-specific operational symmetry.Concrete sector statements are recorded only as later implementations of the sameobservation-fiber/protocol structure, not as core derivations of Part IV. Part IV therefore stopsat the general protocol calculus: pushforward probability, fiber-weight entropy, conditionalupdate, event structure, record accumulation, and operational quotient/gauge. The fixedcomparison language, the background-parametrized core density, and the readings R1–R3are inherited from the law-generation papers; no metric or geometric background is varied inthis Part.