The period-index conjecture bounds the index of a class in the Brauer group of a field in terms of its period and the field's transcendence
degree. For 
,
its period 
is the least positive integer 
with 
. Its index 
is the square root
of the vector space dimension
over 
of the central division algebra representing
.
The period divides the index, and the two have the same prime
divisors.
For a field 
of finite transcendence
degree 
over an algebraically closed field,
the conjecture asserts
 |
Colliot-Thélène's survey records this formulation (Colliot-Thélène 2001).
Perry (2026) reported counterexamples with period 2 and index 
for every 
.
They occur over fields of rational
functions on smooth 
-dimensional projective varieties
over an algebraically closed field 
.
In characteristic 0, the construction requires
.
In characteristic 
, it requires 
, where 
is the finite field with
elements. In particular, the three-dimensional example in characteristic
0 has period 2 and index 8, whereas the conjecture would require the index to divide
4.
Perry (2026) proposes the corrected bound
 |
where 
is a factorial and 
is the exponent of the
prime 
in the prime factorization
of 
.
The correction disappears when the period is relatively
prime to 
.
The paper proves the Hodge-theoretic analog of the corrected bound on smooth proper
complex varieties. This analog concerns obstructions
from integral Hodge cycles and does not establish
the proposed bound for the algebraic index.
Perry (2026) credits ChatGPT with a flawed example that inspired the eventual construction, together with lattice theory calculations. The author found the working construction. As of Sep. 18, 2026, independent specialist verification and external peer review of the complete new results had not been reported.
