receiving
plain-language theorem explainer
Packages the dual of referring in seam closure: a receiving structure on which both extensions identify the two iteration orders. Category theorists and RS foundation readers cite it as the receiving-side counterpart of a referring trace. It is the exact analogue of the product of two natural-number objects being initial among bi-pointed iterations whose steps commute. Declared as a structure, not a proved theorem.
Claim. A receiving structure is data for the dual of reference along a seam: both extensions are required to identify the two iteration orders. Equivalently, it is the initiality condition for a product of two natural-number objects among bi-pointed iterations whose two steps commute.
background
The ambient module is seam-closure reference in the foundation layer. It sits beside referring traces, the refers-to relation, self-reference, step reference, and the generated substructure, and it imports the primitive recognition calculus. Reference is reduced to distinction; receiving is the dual packaging on the other side of the seam.
In that setting one has two ordered iterations (two steps). The classical categorical fact is that the product of two natural-number objects is initial among bi-pointed algebras whose two endomorphisms commute. The doc text states that receiving is exactly that analogue: both extensions must identify the two orders.
Upstream word-hits in the graph (ratio orbit two, ledger must-derive statements, hinge slot classes) are incidental token matches, not mathematical dependencies. The real prior is the referring side of the same module and the primitive recognition calculus basic import.
proof idea
No proof body: claim status is definition and the declaration is a structure. The content is the field package and the stated identification requirement (both extensions identify the two orders), recorded as the bi-pointed commuting-iteration initiality analogue. Nothing is discharged by tactics or upstream lemmas.
why it matters
Seam closure needs a receiving dual so reference is not one-sided: referring traces and generated reference reduce to distinction, while receiving records how the other side accepts the identification of the two orders. That keeps the foundation layer symmetric before higher ledger or gauge counting is attached.
No downstream uses are recorded in the graph yet, so the structure is presently a local organizing definition inside Foundation.SeamClosure.Reference rather than a cited lemma in T5–T8 or the mass ladder. It does not itself force $J$, $\varphi$, the eight-tick octave, or $D=3$; it only supplies the receiving-side type for later seam arguments.
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