The shape of the programme

The canonical sequence

1. Stage 1 · Non-equivalence

   Source–Readout Non-Equivalence: Descent and Equivariant Reconstruction Obstructions

   Establishes the foundational distinction: after declared redundancy is quotiented out, the readout presents the source exactly as a quotient while physically invariant relations can fail to descend and equivariant reconstruction can be obstructed.

   Hands the descent criterion to Paper 2 (answer maps) and Paper 3 (relation families); its equivariant obstruction returns in Paper 6's reconstruction corollary.

2. Stage 2 · Target obstruction

   Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain

   Makes obstruction target-relative: a question is answerable from the readout exactly when its correct answer never varies within a readout fiber, and every sufficient extension must separate states with different correct answers, with the joint target image as the coarsest such extension.

   Supplies the single-target completion and the flat-U(1) spectral rigidity input that Paper 3 generalizes to families of relations.

3. Stage 3 · Minimal completion

   Canonical Minimal Source Completion: The Coarsest Readout Extension on Which a Nominated Family of Source Relations Becomes Well Defined

   Constructs the canonical minimal completion: for any nominated family of invariant source relations, the joint image of readout and relations is terminal among all relation-sufficient extensions, making it the coarsest enrichment on which every nominated relation becomes well defined.

   Emits the canonical minimal completion that Paper 4 realizes geometrically; its target-relative minimality is the phenomenon Papers 5 and 6 meet again dynamically.

4. Stage 4 · Geometric realization

   Geometric Realization of Completed Source Relations: Descent, Orbit Spaces, Invariant Relations, and Variational Response in Shadow Theory

   Realizes the completion geometrically: compatible local data glue to global fields unique up to bundle isomorphism, invariant relations descend to the orbit space of physical configurations, and relation-dependent actions derive covariant responses in the Einstein, Yang–Mills, and matter equations.

   Hands closure, retention, and memory to Paper 5; its orbit-space architecture is one of Paper 6's three standard specializations.

5. Stage 5 · Projected dynamics

   Observable Quotients and Exact Projected Dynamics: Closure, Memory, Minimal Dynamical Completion, and Effective Field Operators

   Develops the observable and dynamical layer: an induced observable evolution exists exactly when the dynamics preserve the readout kernel; otherwise the exact projected law carries an unresolved-initial-state term and a memory kernel, with a minimal dynamical completion measuring exactly what must be restored.

   Provides the closure criterion, exact memory equation, and minimal dynamical completion that Paper 6 restates as its projected-law dichotomy and Paper 7 instantiates in RS2.

6. Stage 6 · Identifiability

   Non-Source Projection and Internal Identifiability

   Integrates the sequence into the non-source projection theorem: an essential non-gauge distinction inside a readout fiber proves non-source projection for the stated model and target, with exact deterministic and statistical identifiability results.

   Proves the non-source projection theorem and states exactly what a physical model must supply: a source domain, an equivalence, a readout, a target, and a witness pair.

7. Witness layer · Physical witness

   Bulk-to-Brane Projection, Dynamical Nonclosure, and Observable Residues in Randall–Sundrum Gravity

   Within RS2 gravity, Paper 7 proves that identical instantaneous brane readouts can evolve into different futures, derives the exact projected Einstein equation, and links cosmological and weak-field residues through a parameter-free relation.

   Supplies that witness in RS2 gravity, derives linked physical residues, and proves exact operational equivalence for brane-only protocols.

Downstream branch targets

Open problems attach downstream of the seven-paper foundation as branch targets. Each identifies a concrete question and the assumptions, method, support, and result a dedicated public record must establish.