# Verification of the null chain theorem

The eight-page manuscript proves the explicit rotation and closure criterion
in Theorem 1, the algebraic-parameter classification in Theorem 2, and the
nonsemialgebraicity and algebraic Zariski-density consequences in Corollary 3.
This is originating-researcher self-audit, not independent or formal review.

## Geometric and analytic checks

The parametrization covers the entire open Lorentz belt, not just one
separation chart, and is regular at the equator. Only a homeomorphism is
used at the degenerate tropics. The conformal cylinder has circumference
4A and boundary levels plus/minus B. Both an equator return and one
tropic-to-tropic leg advance 2B, but the latter changes tropics. Thus the
physical chain has an additional even-leg condition.

The upper-half-plane branch signs give A+B=pi, not A-B=pi. Endpoint
indentations vanish. The contour calculation takes imaginary parts before
the large-radius limit and never separately integrates divergent real tails.
The derivative of A in c is strictly negative; existence, uniqueness and
bounds are proved analytically for all positive parameters.

## Arithmetic checks

The curve has four distinct branch points when a and b differ. The two
punctures and the simple-pole differential are defined over the algebraic
numbers, with residues minus one and plus one. The torus restriction of
the associated invariant form is dt/t, fixed by the primitive puncture loop.
The chosen A-cycle has nonzero projection to the elliptic Lie algebra.

A second genuine closed cycle has a positive real period. It excludes an
elliptic subgroup annihilated by the invariant form, since such a subgroup
would force all periods to be rational multiples of 2*pi*i. The rational and
irrational algebraic cases use the analytic subgroup theorem with all field,
dimension and nonzero-period hypotheses stated. For the latter case the
one-dimensional torus kernel has an integer slope. The theorem permits
nonzero vectors with exponential equal to the identity.

The rotational degeneration is treated separately. The semialgebraic claim
is restricted to algebraic coefficients and uses real-closed-field transfer.
The stronger polynomial-vanishing obstruction uses specialization at an
algebraic pair of horizontal parameters.

## Reproducibility and document checks

The supplementary checker verifies ten exact symbolic identities and the
third-kind residue signs. Nine 55-digit quadrature cases check complementary
periods, axis symmetry, scaling and monotonicity, and four bisections check
closing examples. No floating calculation is an interval certificate or
evidence of transcendence. Normal and optimized fresh-process runs are
compared to the retained outputs after extracting the publication archive.

The built-in LaTeX compiler reports success. All eight final pages were
visually inspected, including the author footnote, formulas, subgroup proof,
numerical table and bibliography. The final exported PDF has no undefined
references, missing characters, overfull or underfull box warnings. Detailed
hashes and replay receipts accompany the sealed package.