Sunflower Endpoint Rigidity and Kernel-Forced AASC Transfer

Overview

This record contains Sunflower Endpoint Rigidity and Kernel-Forced AASC Transfer, a manuscript developing an AASC endpoint-transfer treatment of the Erdős–Rado sunflower endpoint.

The paper works on the fixed core–petal carrier for (n)-uniform families. For a family
[
\mathcal F\subseteq \binom{U}{n},
]
and each candidate core (C\subseteq U), the residual petal family is defined by
[
\mathcal F_C={S\setminus C:S\in\mathcal F,\ C\subseteq S}.
]
The sunflower endpoint is then expressed as the role-cardinality condition
[
\exists C\subseteq U\quad \nu(\mathcal F_C)\ge k,
]
where (\nu(\mathcal F_C)) is the residual matching number.

Central Contribution

The manuscript gives a kernel-forced AASC endpoint-transfer proof for the sunflower endpoint relative to calibrated proof data
[
(\mathcal C,\Complete^{\mathcal C}_{k,H_k},H_k),
\qquad
H_k\ge H_k^{\mathcal C}<\infty.
]

The proof isolates the residual no-sunflower countercase as a calibrated endpoint branch:

• the fixed core–petal carrier is preserved;

• the positive endpoint is core–petal role occupation;

• the negative branch is global non-occupation of every (k)-petal residual slot;

• bounded motif branches are preserved through a declared finite certificate language (\mathcal C);

• only the calibrated objective non-BMF residual separator is routed to the AASC no-independent-discriminator closeout.

The result is not obtained by a random-restriction, spread-lemma, or entropy-compression improvement. It belongs to the AASC proof class: fixed-carrier endpoint transfer under kernel-forced admissibility, standing, reference, and irreversibility.

Method and Proof Architecture

The proof proceeds through the following components:

• Core–petal reduction:
  A (k)-sunflower exists iff some residual family (\mathcal F_C) has matching number at least (k).

• Kernel-first dependency order:
  Determinate same-carrier endpoint or counterexample status already requires the AASC kernel:
  [
  K={\mathrm{Adm},\mathrm{St},\mathrm{Ref},\mathrm{Irr}}.
  ]

• Cost of kernel denial:
  Weakening reference, standing, admissibility, or irreversibility changes or destroys fixed endpoint status rather than producing a weaker version of the same endpoint object.

• Certificate-language layer:
  A finite certificate language (\mathcal C) records bounded motif certificates, product/factor records, endpoint-preserving injections, rank accounting, and entropy accounting.

• Calibration layer:
  The motif ceiling (H_k) must dominate the raw certified motif entropy
  [
  H_k^{\mathcal C}.
  ]
  Product transversals and (C_5)-type tensor motifs are treated as lawful negative structures, not forbidden residual separators.

• Residual separator discharge:
  A calibrated residual branch
  [
  \RBEsep^{\mathcal C}_{k,H_k}(\mathcal F)
  ]
  can stand only as an independent same-domain endpoint-status discriminator. Under local endpoint use, the AASC consequence layer excludes such a discriminator.

Lean4 Audit Support

This manuscript is accompanied by a Lean4 audit release:

• GitHub: https://github.com/somamaley-ux/AASC-Sunflower-Endpoint-Lean-Audit

• DOI: https://doi.org/10.5281/zenodo.21242337

The Lean release verifies the manuscript’s AASC endpoint-transfer proof-class spine, including:

• the fixed core–petal residual matching carrier;

• the four-role AASC kernel package;

• the calibrated certificate-language split;

• the objective non-BMF residual branch;

• local endpoint-use discipline;

• the no-independent-discriminator closeout;

• transfer from exact local countercase use to the bounded motif certificate branch.

The Lean audit is not presented as an AASC-free first-principles formalization of the classical Erdős–Rado sunflower conjecture. Its claim is sharper and bounded: it machine-checks the typed AASC endpoint-transfer mechanism and theorem-spine audit surface used by the manuscript.

Scope and Proof-Class Boundary

This manuscript does not apologize for using AASC. Its proof class is not a conventional spread-lemma or random-restriction route. The relevant correctness questions are:

• whether the fixed core–petal endpoint carrier is correctly instantiated;

• whether local exact-countercase use has determinate same-carrier endpoint status;

• whether the kernel is forced by that non-degenerate endpoint status;

• whether the calibrated residual separator performs independent endpoint-status work;

• whether the AASC no-independent-discriminator closeout applies.

An AASC-free reconstruction would require a separate finite certificate-extraction theorem producing (\BMF^{\mathcal C}_{k,H_k}) certificates directly. That is a parallel reconstruction route, not a prerequisite for the kernel-forced endpoint-transfer proof class.

Record Contents

This record includes:

• the main manuscript PDF;

• Overleaf/LaTeX source files;

• bibliography and reference metadata;

• Lean4 audit appendix;

• release references for the companion Lean repository and DOI;

• proof-class, calibration, and adversarial audit materials.