# Verification report

Paper: An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four.
Author: Alper Ferudun. Date: 8 September 2026.

Status: complete proof accepted after the originating researcher's detailed
self-audit. No independent mathematical review, external peer review, or
proof-assistant formalization has occurred. The author's removal of an
outside-review approval gate does not alter the mathematical scope.

## Theorem covered

The displayed Faran map and the linear embedding are joined through proper
rational maps B^2 -> B^4, jointly continuous on the closed ball, with each
slice extending holomorphically past that ball and of rational degree at most
three. The general AIM request for homotopy invariants remains outside this
theorem. A wholly polynomial homotopy and joint C1 boundary regularity are
not claimed. The particular path explicitly fails C1 boundary convergence.

## Load-bearing steps checked

1. Original source: the middle coordinate is sqrt(3)zw, of degree two. The
   exact displayed AIM pair and D'Angelo-Lebl Definition 2.2 match the theorem.
2. The four polynomial components use nonnegative coefficients with a
   denominator at least two for the entire parameter interval.
3. Five rational identities, including AB=CD, prove the sphere equation
   after retaining all mixed Hermitian terms. No sample interpolation is used.
4. The initial coordinate change is an exact unitary identity over
   Q(sqrt(2),sqrt(3)); P_1=R_0 holds exactly.
5. The rational denominator is nonzero on the closed ball for a<1. The
   a=1 endpoint is separately defined as the cancelled polynomial.
6. The written estimates prove sup ||R_a-R_1|| <= 4 sqrt(1-a), including
   the boundary corner. Pointwise convergence alone is not used.
7. The maximum principle and compactness prove properness of each slice.
   Exact interior-defect identities supply a corroborating algebraic check.
8. The final target unitary and Whitney path have the stated endpoints.
   Pasting five continuous pieces keeps four target coordinates throughout.
9. A common-denominator representation proves the rational degree bound.
   The coefficient lift is continuous when the endpoint pair is left unreduced.
10. The consequence for all four Faran representatives follows by combining
    the main theorem with the explicitly cited D'Angelo-Lebl paths. It is not
    a classification theorem for all maps into B^4.

## Executable evidence

- check_faran_homotopy.py: exact rational-function and Hermitian polynomial
  identities for the complete parameter range, plus endpoint cancellation.
- check_faran_endpoints.py: exact starting polynomial/unitary identity and
  separately labelled numerical sphere/interior/near-corner regressions.
- check_faran_rank_and_defects.py: exact coefficient-rank, full interior-defect
  and affine-chord identities in independent formal polynomial variables.

Each program completed successfully under Python 3.12 and Python 3.13 on
8 September 2026. Outputs are retained in reproducibility/. These are two
runtime executions of the author's checkers, not two independent reviews.
The manuscript supplies the analytic arguments the algebraic scripts do not
establish. No proof assistant is used and no absolute novelty conclusion
follows from these checks.

The earlier research dossier retains additional normal-form, coefficient-lift
and failed-search material. Those supplementary claims are not needed for the
present proof and are not silently counted as additional publications.

## Presentation checks

The final PDF is five pages. All five rendered pages were inspected for
clipping, overlap, equation breaks, reference rendering and author layout.
The visible byline contains only Alper Ferudun; affiliation/contact details
are footnoted. The AI disclosure is ordinary prose. The final TeX build has
no unresolved citations or references, missing glyphs, overfull boxes, or
remaining TeX warnings. The package manifest records exact artifact hashes.

Known prior work and bounded novelty-search limitations are documented in
source_and_novelty_review.md. Publication does not constitute peer review.