Patch Cover Inventory¶
Decomposition registry — instantiates Patchwise Global Certification
Enumerates and justifies the set of local patches, and pins the domain they are meant to fill, so a whole can be certified piece by piece.
The Patch Cover Inventory turns "the whole" into a named, justified list of local patches and declares the domain those patches are supposed to fill. It answers one question and only one: what are the pieces, and why do we believe they are meant to tile the thing we intend to certify? It does not check whether any piece passes — that belongs to the local witness — and it does not prove the pieces actually leave no gap — that belongs to the audit. Its whole job is to fix the cover itself, the collection {Uᵢ}, and to bound the domain X those patches claim to span, so that everything downstream has an explicit, auditable decomposition to work against instead of an implicit one.
Example¶
A national mapping agency wants to certify that a new topographic dataset covers an entire national forest at 1:24,000 scale. The forest cannot be surveyed as one undivided object, so the certification will have to be assembled from local pieces. The Patch Cover Inventory is where those pieces are fixed. Cartographers divide the forest into standard quadrangle sheets, and build a registry: forty-two rows, each a sheet with an ID, its corner coordinates, the neighboring sheets it abuts, and a 200-metre overlap collar with each neighbor. Above the list sits an explicit scope statement: "the certified region is the forest boundary as of this survey, tiled by these forty-two quadrangles, in this projection." Not a single sheet's accuracy has been checked yet, and no one has proven the forty-two truly leave no hole. The inventory's work is done the moment the pieces and the whole they claim to fill are both on paper — a decomposition anyone can inspect and argue with.
How it works¶
- Fix the domain first. State exactly what is in scope and where the boundary of
Xfalls, before any patch is drawn. - Choose a cover geometry. Partition into disjoint tiles, an overlapping cover with collars, a nested cover, or a sample — and name which one you picked, because it decides what claims the cover can later support.
- Enumerate each patch. Give every patch an ID, a precise extent, and its adjacency to neighbors, including where overlaps physically fall.
- State the spanning argument. Record why the union is intended to equal the domain — an argument, not a proof; the proof is the audit's.
The inventory records that overlaps exist and where their geometry lies, so downstream tests have seams to examine, but it never checks whether the patches agree there.
Tuning parameters¶
- Patch granularity — few large patches are cheap to list but hard to witness; many small ones are each trivially checkable but multiply seams. Set it against how expensive a single local check is.
- Overlap width — wider collars give downstream stitching more shared context but add redundant patch area to check.
- Cover type — partition vs. overlapping vs. sampling. A sample is cheap and forfeits any genuine completeness claim; only say "cover" if you mean to span.
- Scope tightness — how conservatively the domain boundary is drawn; a tighter domain is easier to fill but certifies less.
When it helps, and when it misleads¶
Its strength is that it forces the decomposition into the open before any verdict is formed, so the question "what did we actually claim to cover" always has a written answer. Every later step — witnessing, seam-checking, aggregation — hangs off this explicit list rather than a tacit mental picture of the whole.
Its failure mode is a cover that quietly samples the domain while presenting itself as one that spans it: the registry looks complete, but the union has holes it never advertises. The classic misuse follows directly — treating a tidy patch list as if it were proof of coverage, and certifying the whole on the strength of an inventory that was only ever meant to name the pieces. The guarding discipline is to keep the inventory honest about its cover type (sample or span) and to hand the union-equals-domain question to the audit, never answering it here. The mathematical shadow of this object is an open cover.[n1]
How it implements the components¶
local_cover_or_patch_set— its primary output: the enumerated, geometry-tagged, justified collection of patches{Uᵢ}.global_verdict_target— it declares and bounds the domainXthe cover is meant to fill; it fixes the scope half of the target that every later step certifies against.
Does not implement coverage_completeness_map — proving the union actually equals the declared domain with no gaps is the job of Coverage Completeness Audit, its nearest twin. The inventory lists the patches; the audit proves they leave nothing out.
Related¶
- Instantiates: Patchwise Global Certification — the inventory supplies the cover and scope every other mechanism operates on.
- Sibling mechanisms: Local Witness Checklist · Overlap Compatibility Test · Gluing or Recomposition Workflow · Obstruction Register Review · Coverage Completeness Audit · Global Certificate Template · Local-to-Global Dashboard
Editorial Notes¶
Form Classification¶
Form family: Record, Log & Register
Rationale: Patch Cover Inventory operates as a persistent ledger, log, register, or case record that preserves history and traceability because it enumerates and justifies the set of local patches, and pins the domain they are meant to fill, so a whole can be certified piece by piece.
Independent corroboration: The frozen evidence defines Patch Cover Inventory as 'Enumerates and justifies the set of local patches, and pins the domain they are meant to fill, so a whole can be certified piece by piece', so its operative form is Record, Log & Register.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Single lineage
Present-day reach: Multi-domain
Rationale: Patch Cover Inventory is rooted in mathematics: The mechanism directly operationalizes topology's open-cover requirement as an auditable patch inventory.
Review resolution: Both blind reviewers agree that mathematics is the primary origin. Reconciliation resolves reported_ambiguity, domain_reach_disagreement. Formative alternate lineages are not added; later breadth of use is recorded separately as domain_reach=multi_domain, while origin_mode=single_lineage describes the relationship among origin lineages.
Attribution caveat: The inventory format is an encyclopedia-specific operationalization of mathematical cover arguments.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Reconciled after independent review; medium confidence.
Notes¶
[n1] In topology, an open cover of a space X is a collection of open sets whose union contains X. The inventory is the working analog: it names the sets and asserts their union is meant to be the whole — the very assertion the audit must later verify rather than assume. ↩