Pilz's extended 1-2-3 conjecture asserts that, for every nonempty finite set 
of positive integers and every positive
integer 
,
 |
(1)
|
where 
, 
, and 
denotes symmetric
difference. Equivalently, at least 
integers have an odd number of
ordered representations 
with 
and 
.
The bound is sharp, since 
gives exactly 
elements. The 1-2-3 conjecture restricts 
to an initial interval of positive
integers (Huang et al. 2010). Holdridge and Pach (2026) proved the extended
conjecture for all sufficiently large 
.
The anonymously authored manuscript Pilz's Extended 1-2-3 Conjecture over Every Field (2026a) reports a proof for every 
in a stronger weighted form. For each prime 
, introduce an indeterminate 
and set 
, where 
is the exponent of 
in the prime factorization
of 
. For any field 
and nonzero coefficients 
, 
,
..., 
, define
 |
(2)
|
Every nonzero polynomial or Laurent polynomial multiple of 
is reported to have at least 
nonzero monomials. Taking 
to be the two-element finite field and all 
gives the set-theoretic conjecture. The interval hypothesis
matters: over that field, 
, so replacing 
by an arbitrary 
-element set of positive integer
indices does not preserve the bound.
Equivalently, the weighted multiplicative convolution
 |
(3)
|
has at least 
nonzero values for every nonzero 
-valued function 
on the positive integers
with finite support. For values in a vector
space, the reported bound is 
,
where 
is the dimension of the vector
space span of the input values. For values in a nonzero unital module
over a commutative ring, the bound 
holds for every nonzero finitely supported input iff
multiplication by each 
is injective. For commuting
endomorphisms of an additive commutative
monoid, the corresponding criterion is that each map sends only zero to zero.
The manuscript also reports uniqueness of recovery from 
corrupted and 
erased output coordinates when 
. This is a uniqueness statement rather than
an efficient decoding algorithm.
At a prime index 
,
the reported equality classification is
 |
(4)
|
for some integers 
and 
,
where 
is taken over the two-element finite field. The corresponding
output is 
. For odd 
and even 
,
the parity identity 
sharpens the bound to 
, attained by 
.
For fixed 
, the limiting density
 |
(5)
|
exists and is a rational number (Ke and Meyer 2025, Holdridge and Pach 2026). The anonymous companion The Optimal 4/3 Density
Bound in Pilz's Problem (2026b) reports 
unless 
or 
,
which have density one. The constant is sharp at 
. After division by the greatest
common divisor, nonexceptional sets consisting of powers of two satisfy the stronger
bound 
. This is an asymptotic statement with 
fixed, not a bound 
for every 
. For example, 
.
The project attributes mathematical arguments and Lean formalization to substantial AI assistance. It supplies public Lean 4 developments and reports independent kernel replay of the weighted proof. The authors state that the written proofs have not been checked line by line by a human mathematician. The density formalization is independent of the finite-interval theorem.
