# cegm-x36 — companion bundle: the X(3,6) string integral through weight five

Companion release bundle for the paper "Factorization without clustering and
the six-point Grassmannian string integral" (stringintegral.pdf at the
site root), whose Section 5 (the low-energy expansion) states the results
recorded here and whose Appendix A (A.2, weights three and four; A.3, weight
five) condenses the two proof records shipped here. The bundle carries the
ancillary files of record (the exact fractions, the check ledgers, and the
machine-checked proof records) and a standalone verifier that recomputes the
documented closed forms from the exact rationals in the shipped JSON files.
The paper itself is not part of the bundle. The July 2026 note this bundle
first accompanied ("Multiple zeta values only: the X(3,6) string integral
through weight five") is superseded by the paper and is no longer shipped.

The headline artifact is the depth-two closure through weight five: at the
seed kinematic point the weight-five coefficient closes as
c5 = Q5 zeta(5) + R5 zeta(2) zeta(3) with explicit exact rationals Q5, R5
(ancillary/w5_collapse_verdict.json) and R5 nonzero; through weight four
the coefficients are single zetas times exact rationals, at two kinematic
points. The evidence for (Q5, R5), as the verdict file records it: exact
Fraction arithmetic over the fitted constants (each fit checked in its parent
chain), 768 inter-constant integer relations (each a two-precision PSLQ or LLL
candidate with a capacity bar of n*log10(2H)+30, an independent low-precision
residual and heights at most 9), and the exact census weights; no new
quadrature enters. The `evidence_class` field of that file states the same in
the working shorthand of the computation.

**Status: released 2026-07-10 under the MIT license (see LICENSE). The ancillary files
are the versions of record and are not revised; corrections are listed under Known
errata below.**

What is byte-exact and what is a public render (2026-09-03): the five JSON
files are byte-exact copies of the files of record. The three markdown proof
records (APPENDIX_ARTIFACT.md, PROOF_W4W3.md, THEOREM_W5.md) are public renders
of the records: references to the record's working directories were made
relative, and the working vocabulary of the computation was replaced by plain
words; every number, fraction, file name and claim is unchanged, and the
`core/`, `div1/`, `div2/`, `xver/` and other working files they name are not
part of this bundle. The SHA-256 pins in cegm-verify.py and MANIFEST.sha256
are those of the shipped renders. A few string fields in the JSON files (the
`evidence_class` and `gates` fields) keep the working shorthand of the
computation as it ran; the numeric and fraction fields are what the verifier
checks.

## Layout
- `ancillary/w5_collapse_verdict.json` — Q5, R5, the recorded c5 value, and
  the weight-five check ledger.
- `ancillary/t2_GATES.json` — weight 0–4 checks at the seed point (exact
  c0; the R2/Q3 closed-form names; Q4 and the recorded c4).
- `ancillary/t2_c3_final.json` — Q3 and the recorded c3 at the seed point.
- `ancillary/t2_exact_rationals.json` — R2, R3b, R3d (seed point).
- `ancillary/t2_exact_s2.json` — the second kinematic point: exact c0,
  R2, Q3 with recorded decimal values, and the E table.
- `ancillary/THEOREM_W5.md` — the weight-five structure theorem and the
  second derivation of (Q5, R5); the structure theorem is condensed in
  Appendix A.3 of the paper, which does not include the second derivation.
- `ancillary/APPENDIX_ARTIFACT.md`, `ancillary/PROOF_W4W3.md` — the
  weight-3/4 cancellation-theorem records; condensed in Appendix A.2 of the
  paper.
- `cegm-verify.py` — standalone verifier (Python 3 stdlib + mpmath).
- `MANIFEST.sha256` — SHA-256 of every other file in the bundle.
- `LICENSE` — MIT.
- `cegm-x36-w6-2026-09-28-r3.tar.gz` — the weight-six data record (added
  2026-09-11; third edition 2026-09-28; see the section below). A self-contained tarball
  with its own README, checksum list and check script; not read by
  `cegm-verify.py`.

## Running the verifier
From the bundle root:
```
python3 cegm-verify.py            # full verification at 80 working digits
python3 cegm-verify.py --selftest # in-memory mutation control
python3 cegm-verify.py --dps 120  # crank the working precision
```
The verifier pins all eight ancillary files by SHA-256 (re-checked on
every load), checks every shipped fraction for lowest terms, checks the
cross-artifact identities the bundle documents (including the exact
equality of Q5 and R5 between the collapse verdict and THEOREM_W5.md),
and recomputes every closed form the bundle documents against the recorded
decimal strings read from the pinned files — the recomputed c5 agrees
with the recorded value to 50 digits, all of its recorded digits. Exit
codes: 0 full pass; 2 pin mismatch; 3 value mismatch; 4 missing file.

## Known errata (JSON files shipped byte-exact; never mutated)

- `ancillary/t2_exact_s2.json`: the file's internal `gates` string says `@s*`, but the
  file is the bundle's **second** kinematic point (a second generic rational
  point, distinct from the seed point s*; of the bundle's two points only
  the seed point s* appears in the paper). Upstream label bug only — the values are correct and are verified
  per-point by `cegm-verify.py`.

## Weight-six data record (added 2026-09-11)

`cegm-x36-w6-2026-09-28-r3.tar.gz` (13 files, 1.2 MB) is the data record behind
the paper's weight-six statements: the two-term form
c6(s) = A(s) pi^6 + B(s) zeta(3)^2 as exact rationals at three reference
kinematic points of denominator three (the paper's base point s0 of its
Eq. (32), whose c6 row is printed in its Eq. (33) and Table 2, and two
neighbours of s0 that differ from it in one coordinate and are not printed
in the paper); the exact rational functions A(s), B(s) of the paper's
Eq. (95) over the convergence region and the weight-five functions Q5(s),
R5(s) of its Eq. (92); the 190 kinematic points on which the forms were
determined and the 90 further points, disjoint from those, on which they
were tested, each with its 68-digit value of c6 computed by two routes; the
segments on which the real-solution walls of the paper's Section 7.2 were
found, and the (c, c') and gamma twist conventions of its Section 7.4. Unpack it
anywhere and run

    sha256sum -c --quiet SHA256S.txt && python3 check_w6.py

(Python 3.8+ and mpmath): the script re-derives the three-point c6 table
exactly from the shipped forms (its s0 row is the c6 entry of the paper's
Eq. (33)), reproduces the recorded c6 values at all 190 + 90 points to
the digits the two routes certify (at least 64 at every point, 67 on the
90 test points), and exits 0 only if all six checks pass; a transcript of the assembling run is
inside. This is the record's third edition (r3, 2026-09-28): its README and
CONVENTIONS.txt now cite the equation and section numbers of the current
version of the paper and write K(3,6) for the integral; every data file other
than CONVENTIONS.txt, and the arithmetic of the check script, are byte-identical
to the second edition (r2, 2026-09-11), in which the twist data file
(twist_gamma.json) was recomputed on the two cluster-variable twist letters
written out in CONVENTIONS.txt (its own `supersedes` field names the
first-edition file by checksum). r3 replaces r2 here.
The record is covered by this bundle's MIT license.

## Scope
The bundle claims exactly what the shipped artifacts state: the closed
forms through weight four at two kinematic points, and the depth-two
closure through weight five at the seed point, with the evidence
described above and the conditionality stated in the paper and in
THEOREM_W5.md. The weight-six closure is carried by the separate data
record added on 2026-09-11 (below), which makes its own claims in its own
README; the weight-seven and weight-eight coefficients, the finite-tension
double-copy data and the solution counts are not part of this bundle.

License: MIT (see LICENSE).