IndisputableMonolith.Mathematics.ProjectionMultiplicityMethod
Formalizes classical extremal problems as visible objects, a counted relation, and two readouts: visible size and event count. Packages lift data, finite windows, and projection-multiplicity certificates that beat a linear baseline by a polynomial factor. Extremal combinatorics and RS discrete-geometry arguments cite the certificate and gain lemma. Structure is mostly definitional, with one certificate-to-gain theorem.
claimA classical extremal problem consists of a class of visible objects $V$, a relation $R$ whose instances are counted, and two numerical readouts: visible size $|V|$ and event count $e(R)$. The projection multiplicity method supplies lift data and finite windows yielding a certificate that the construction beats the linear baseline by a polynomial factor.
background
The module sits in the Mathematics domain and imports only Mathlib. Its opening notion is a classical extremal problem: visible objects, a relation whose instances are counted, and two scalar readouts (visible size and event count).
Around that interface it introduces lift data (how a lower-dimensional or projected configuration is raised), finite windows (local counting regions), and the comparison predicate that a construction beats the linear baseline by a stated factor. Projection-multiplicity candidates and certificates package the combinatorial hypotheses needed to turn those ingredients into a gain bound.
No upstream RS forcing lemmas are wired here; the module is self-contained combinatorial scaffolding.
proof idea
Primarily a definition module: it declares the classical extremal problem record, lift data, finite windows, the beats-linear-by relation, projection-multiplicity candidates, and certificates. The main theorem-shaped result is a one-step implication that a valid certificate yields polynomial gain over the linear baseline. No deep tactic scripts; the argument is packaging plus that certificate-to-gain lemma.
why it matters in Recognition Science
Supplies the combinatorial language for projection-multiplicity arguments inside the RS mathematics layer. Downstream use is not yet wired in the graph (no used_by edges), so the module is infrastructure: later discrete or geometric bounds can cite the certificate and the polynomial-gain lemma rather than redeveloping the extremal interface. It does not itself touch T0–T8, RCL, or the phi ladder; it is pure counting setup.
scope and limits
- Does not prove any concrete extremal number or RS physical constant.
- Does not connect to the forcing chain T0–T8 or the Recognition Composition Law.
- Does not assert uniqueness of the projection or optimality beyond the stated polynomial gain.
- Does not supply numeric windows or explicit lift maps; those are parameters.
- Does not yet appear as a dependency of downstream RS theorems in the graph.