The AASC–Combinatorics Sunflower Endpoint Bridge

This work presents an integrated combinatorial–AASC proof of a constant-base bound for finite three-sunflower-free uniform families. For every finite $n$-uniform family $\mathcal{F}$ containing no three-petal sunflower, the manuscript proves

∣F∣≤(8,384,512)n.|\mathcal F|\le (8{,}384{,}512)^n.∣F∣≤(8,384,512)n.

The proof combines two complementary and jointly load-bearing components. Finite combinatorics constructs cardinality-minimal blockers, private witnesses, the residual tower, branch-complete continuation paths, terminal records, finite type coordinates, collision dispositions, and a well-founded certified terminalization system. AASC supplies the necessity-rooted exhaustion theorem excluding independent standing-bearing occupants in the resulting certificate-terminal, fixed-scope endpoint package. These are completed components of one hybrid proof, not competing or partially interchangeable approaches.

Main Contributions

The manuscript establishes:

• positive finite generation of every compatible residual continuation path;
• preservation of genuine many-to-one residualization;
• an explicit triangle regression ruling out unrestricted pre-discharge root-to-seed injectivity;
• exhaustive classification of generated collisions into endpoint-neutral multiplicity, bounded residual charge, tensor or scope separation, sunflower realization, or strict reconstruction;
• well-founded certified terminalization using a four-coordinate reduction potential;
• necessity-rooted AASC governance for determinate, non-degenerate terminal incidences;
• certificate-terminal exclusion of dual independent occupancy at one fixed terminal locus;
• independent definitions of the AASC terminal relation and the combinatorial endpoint relation;

• their biconditional coincidence and the induced equivalence

  AASCEndG,U≃CombEndG,U;\mathrm{AASCEnd}_{G,U}\simeq \mathrm{CombEnd}_{G,U};AASCEndG,U≃CombEndG,U;
• quotient-final terminal-seed rigidity;
• original-edge coverage rigidity on cardinality-minimal blocker roots;
• post-exhaustion source-to-seed injectivity;

• the exact source-fibre estimate

  $$|U|\le 4094;$$

• the actual minimal-blocker estimate

  ∣B(0)∣≤2048⋅4094=8,384,512;|B^{(0)}|\le 2048\cdot4094=8{,}384{,}512;∣B(0)∣≤2048⋅4094=8,384,512;
• and the final one-point-link recurrence yielding the stated exponential endpoint.

Quotient and Hall Discipline

The argument does not assume a raw Hall matching. The rank-two triangle remains a permanent regression showing that distinct source roots can reach the same residual terminal coordinate before collision disposition.

Quotienting is used only to remove duplicate histories, route-order variation, repeated ledger presentations, and other endpoint-neutral descriptive multiplicity. It does not identify distinct coverage-bearing roots of the cardinality-minimal blocker. Private-edge minimality gives the coverage antichain law

Cov⁡F(x)⊆Cov⁡F(y)  ⟺  x=y,\operatorname{Cov}_{\mathcal F}(x)\subseteq \operatorname{Cov}_{\mathcal F}(y) \iff x=y,CovF(x)⊆CovF(y)⟺x=y,

so every coverage-preserving blocker representative is the original minimal blocker itself.

The final source-to-seed injection is therefore a post-exhaustion consequence of terminal closeout, endpoint equivalence, seed rigidity, and coverage rigidity—not a premise imposed on the raw residual relation.

Integrated Proof Architecture

The controlling dependency is

positive finite combinatorial generation
→ exhaustive collision disposition
→ certified reduction and terminalization
→ necessity-rooted AASC terminal closeout
→ AASC/combinatorial endpoint equivalence
→ quotient-final seed rigidity
→ coverage rigidity
→ source-to-seed injectivity
→ actual minimal-blocker bound
→ one-point-link recurrence
→ three-petal sunflower endpoint.

Standing, Reference, tensor content, coverage, and licensed metric description are jointly instantiated at the saturated endpoint. “Post-exhaustion” denotes logical restriction to the terminal hypotheses, not temporal succession. Cardinality reports the population of already fixed endpoint objects; it does not generate their primitive identity.

Lean Formalization

The accompanying public Lean development, AASC-Sunflower-Endpoint-Lean-Audit v0.4.0, formalizes the combinatorial and proof-theoretic component at its natural boundary. It verifies, among other results:

• generated residual and terminal-record population;
• genuine merge disposition and the triangle regression;
• failure of premature raw Hall;
• minimal-blocker coverage rigidity;
• equality of every coverage-preserving representative blocker with the original minimal blocker;
• the well-founded reduction potential;
• certified terminalization;
• necessity-rooted kernel governance;
• exclusion of the independent-authorizer branch;
• four-role terminal identity closeout;
• fixed identity and the canonical support-fibre bound;
• manuscript-to-Lean endpoint transfer;
• and exact endpoint-strength correspondence.

The manuscript supplies the sunflower-specific generated five-way collision witnesses and literal reduction certificates at their natural mathematical boundary. A typed theorem interface joins these completed components. Lean then derives terminal closeout, fixed identity, the source-fibre capacity, and the endpoint consequences.

The recorded verification includes:

• Lean 4.28.0;
• 1,143 successful public-entrypoint build jobs;
• 1,146 successful combined entrypoint-and-audit jobs;
• 250 focused #print axioms checks;
• no live project declaration using axiom, sorry, admit, or an unsafe escape.

Scope

The theorem proved here is an integrated combinatorial–AASC closure. The combinatorial–Lean component and the AASC component are complete in their assigned roles.

Replacing the terminal AASC theorem with a wholly autonomous single-language conventional reconstruction would constitute a distinct strengthening and a different proof architecture. It is neither assumed nor required by the theorem established in this work.