# Hosting notes — Boundary reconstruction (2026-09-22.r2)

Note dated 21 September 2026. Supplementary remarks and checker revision dated
22 September 2026. Prepared for Matthew Alan Newman. Not a source paper. Not a
joining theorem of Papers I–VII or paper 8. Does not enlarge Paper VI. Frozen
G₆ unchanged (RELEASE.json digest f63ae1c2d595b4805ba9d36687a828fd87200febe01079b184723f7d5eaf746c).

The nine-page PDF is the written proof. Reconstruction is restricted to the
stated field spaces H_A and F_A = H_A ⊕ ℝJ. Twelve aligned samples do not
determine the cubic coefficient; thirteen measurements give an explicit
symmetric decoder, not a minimality claim.

## Original proof artifacts (unchanged)

- Boundary_Reconstruction_Transport_Formal_Proof.pdf — 344230 bytes, SHA-256
  a8c9e39e29f283272b3c7dc2933c7dad1a02bc2651ce0341508317de3d295571.
  Rendered equation (1.2) contains −τ^{-2} with a raised exponent. Flattened
  extraction of that superscript is a viewer artifact, not a document defect.
- pasted_tex_as_received.txt — 27552 bytes, SHA-256
  7db0d779b8e975fcf9a02f37d6d0895beedaf7f9049b5272afe7e6e3e04a3ba4.
- Boundary_Reconstruction_Transport_Formal_Proof.tex — one leading backslash
  restored (27553 bytes, SHA-256
  fcaadf4b55fc24fae1c8cfb1d8839d22a95eea27d3124fbe19002c50cce6c89c).

τ = 1/(φ+π) = 1/Ω₀ in this note. It is not the contraction λ = φ/Ω₀.

## Checker revision 2026-09-22.r2

The 21 September hosted checker compared 1/21 with itself and never computed
the spherical mean of J². That program (9033 bytes, SHA-256 acc774f6…) is
superseded. Its stdout happened to be the same 897-byte transcript as the
repaired program; an output hash alone does not identify which program ran.

Current checker: verify_boundary_reconstruction.py, 10661 bytes, SHA-256
ce3e5ea29ca8de60e27e45164903065f340e54826935f88045b0ed715b5e3efe. The
J-squared spherical mean is computed exactly with fractions.Fraction from the
expanded polynomial. Remaining groups are numerical NumPy corroboration.

## This-archive replay

Python 3.10.21 with NumPy 2.2.6, captured 2026-09-22T12:35:49Z in this archive
build/reproduction environment (distinct from the ChatGPT audit environment
Python 3.12.14 / NumPy 2.3.5):

```bash
python3 verify_boundary_reconstruction.py
python3 -O verify_boundary_reconstruction.py
```

Both executions exited 0 with empty stderr. Stdout is byte-identical, 897
bytes, SHA-256 60ad47fa29f0c7ada869555a393e497411f9c8b08344b2850b71c84a840890b1,
and ends with TOTAL 30 PASS. The ENVIRONMENT.json record binds that transcript
to checker hash ce3e5ea2….

Temporary copies (not the accepted source) with expected mean 1/22, and with
the xyz coefficient of J changed from 1 to 2, each exit 1 in both normal and
-O modes with FAIL J2_spherical_mean_one_twenty_first and no TOTAL 30 PASS.

## Remarks

REMARKS.md (22 September 2026) contains the written arguments for the two-step
identity, even interpolants E_j, three-axis independence, the 160/60 antipodal
criterion, the transported surviving kernel with
widetilde E_j = E_j ∘ A^{-1}, and the exact spherical mean of J². Those remarks
are not pages of the original PDF.

## What this is not

Not an additional source paper, not peer review, not a proof-assistant
certificate, not an exact_finite_certificate for the whole NumPy checker, not
a 13-point degree-six cubature, not a blowup theorem, and not an enlargement
of Paper IV, Paper VI, paper 8, or /extension. Alignment of the icosahedron is
required for J(n_i)=0. Independent third-party reproduction is none recorded.