Conditional Reconstruction Closure B; Observation Quotients, Descent, and Computable Continuous Limits

This paper studies the conditions under which discrete closures possess continuous limits, and the demarcation line between the mere existence of a limit andits unique identifiability from observational data. The main results are as follows.First, the continuous limit is decomposed into four independent propositions—residual vanishing, limit existence, object convergence, and limit identification—and finite-dimensional witnesses are given showing that the four cannot substitutefor one another (Proposition 1.1). Second, it is proved that nonempty compactHausdorff objects with continuous transition maps yield a nonempty inverse limit,that surjectivity of the coordinate projections is equivalent to a Mittag-Leffler-typecomponentwise condition, and that this condition cannot be removed (Theorem BT1 and its supplement); moreover, it is proved that Mittag-Leffler stability togetherwith nonemptiness at each stage already implies nonemptiness of the inverse limit(Proposition 5.A, a purely set-theoretic induction that does not use compactness),so that coordinate visibility is strictly stronger than nonemptiness, and nonemptiness does not imply coordinate visibility (witness). Third, obstacle criteria fordescent gluing and higher compatibility are given: the obstacle class is nonzeroif and only if the corresponding lifting or gluing admits no global solution, andan obstacle blocks only the corresponding lifting without constituting a universalphysical anomaly (Theorem B-CT3). Fourth, an account of the extension operators from discrete to continuous and of the stability constants is established; aWhitney-form discrete de Rham complex is constructed, and the cochain-complexaxioms and the interpolation–exterior-differentiation commutative diagram (including the symbolic obligation of the normalization factor 2 in the top dimension) areverified by machine. Fifth, a conditional theorem stating that consistency, stability, and regularity imply convergence is formulated (Theorem B-CT4, anchored toexternal theorems). Sixth, four levels are distinguished—pointwise convergence,computable convergence moduli, single-instance decidability, and undecidabilityof problem families—and a computable approximation criterion is given (Theorem B-CT5). Seventh, it is proved that when the five conditions—the observationfamily separates points, the extension is uniformly stable, the discrete objects converge, Φ is a closed embedding, and observation commutes with the limit—holdsimultaneously, the limit can be uniquely identified; if any one is missing, theconclusion degrades to a family of observation-compatible candidates (Theorem BCT2). Eighth, the interchange of coarse-graining and limit-taking is registered asan independent condition, an interchange theorem under uniform convergence isgiven (Theorem B-CT6) together with a corollary that equicontinuity is a sufficient source (Corollary 4.2), and a two-sequence witness shows that the uniformcondition cannot be removed. All numerical assertions have been machine-checkedwith exact rational arithmetic (MC-B-01 to MC-B-07, all seven passing; “passing”is an execution-record convention, and consistency of recomputation is subject toindependent reruns). This paper does not claim that an arbitrary inverse system has a unique limit, does not claim that finite observations recover the entirestructure, and does not claim that general closure optimization is polynomial-time