HV56: BRIEF FOR AN INDEPENDENT MATHEMATICAL CHECK
Prepared 8 October 2026; accompanying the unchanged preprint v9.1.

Author: Rushikesh Murlidhar Walke
Title: Counting intersections of rational cyclic grids: an application of Smith normal form
Status: AI-assisted expository preprint; not peer-reviewed.
This brief was also drafted with AI assistance. It is a request for review, not a review certificate or a revised paper.
Please read the full v9.1 PDF alongside this brief.

READABLE MATHEMATICAL COMPANION (added 9 October 2026)
The source v9.1 PDF is unchanged. Definitions, equations (1)-(8), proof steps
and original examples are also available as selectable text:
https://rmwalke.com/hv56/read
https://rmwalke.com/hv56/HV56_Mathematics_v9_1.txt
https://rmwalke.com/hv56/HV56_Mathematics_v9_1.json
This condensed companion is not a revised preprint or a review certificate.

QUESTION AND DEFINITIONS
For positive integers n, let G_n = {j/n + Z : 0 <= j < n} in Q/Z.
Count ordered pairs (u,h) in G_d x G_b with u+h in G_y and u-h in G_k:
C(d,b;y,k) = #{(u,h): u in G_d, h in G_b, u+h in G_y, u-h in G_k}.
All grid orders are positive integers; grids are unshifted.

CLAIM IN THE PREPRINT
L = lcm(d,b), alpha = L/d, beta = L/b.
A = [[y*alpha, y*beta], [k*alpha, -k*beta]].
s1 = gcd(y*alpha, y*beta, k*alpha, k*beta).
s2 = 2*y*k*alpha*beta / s1.
J_L(s) = L / gcd(L,s).
C(d,b;y,k) = d*b / (J_L(s1)*J_L(s2)).
Equivalently C = (d*b/L^2)*gcd(L,s1)*gcd(L,s2).
The formula gives an integer count C; the prefactor d*b/L^2 need not itself be an integer.

For a union in the first input:
Q(a,b;y,k;r) = C(a,b;y,k) + C(r,b;y,k) - C(gcd(a,r),b;y,k).
Each term uses its own L, alpha, beta, s1 and s2.

PROOF ROUTE TO CHECK
1. The membership conditions become A*(i,j)^T = 0 modulo L, where i is modulo d and j is modulo b.
2. Multiplying the first column by d and the second by b gives multiples of L. Hence the map phi: Z/dZ x Z/bZ -> (Z/LZ)^2 is well defined.
3. Its image equals the image of the square-domain map A modulo L. This equality is used before applying Smith normal form; unimodular transformations are not assumed to preserve the rectangular domain directly.
4. Smith normal form has invariant factors s1,s2 since det A = -2*y*k*alpha*beta is nonzero. Its image size modulo L is J_L(s1)*J_L(s2).
5. The finite-group first isomorphism theorem gives the count d*b divided by that image size.
6. Equivalently, the square-domain kernel is normalized by its fiber size L^2/(d*b).
7. G_a intersect G_r = G_gcd(a,r); inclusion-exclusion gives the union count.

SPECIFIC QUESTIONS FOR THE REVIEWER
- Is the domain normalization and image equality fully justified for unequal grid sizes?
- Are all invariant-factor, kernel-size and integrality steps correct?
- Are unit orders, a=r and containment cases a|r or r|a handled correctly?
- Does the union formula avoid double counting?
- Are the examples, classical references and attribution accurate?
- Is the exposition understandable and useful for the intended readership?
- Please identify any missing hypothesis, invalid inference or counterexample, with the relevant page/equation.

FINITE COMPUTATIONAL EVIDENCE AND ITS PROVENANCE
The provided original brute_force_check.py was executed in this ChatGPT conversation:
- d,b,y,k in 1..12: 20,736 test tuples, 0 mismatches.
- a,r,b in 1..8 and y,k in 1..6: 18,432 test tuples, 0 mismatches.
The supplied Claude package has a separate verify_formula.py with random.seed(1) and a recorded output:
- 600 random tuples with entries in 1..40: 0 mismatches according to that supplied log.
These 600 cases were inspected in source/log, not rerun here.
The reported package total is 39,768; repeated tuples, if any, are not claimed to be distinct.
Examples: C(6,4;3,2)=2; C(6,6;3,3)=18; Q(4,4;3,6;6)=8.
Finite checks do not establish the universal theorem. Re-execution of the same checker by two AIs is not independent human peer review.

SCOPE AND DISCLOSURE
The note is an expository application of established mathematics. It does not claim a new general congruence-counting theorem or priority for the specialization.
A limited search not finding the exact framing is not a novelty proof.
The preprint reports AI-assisted drafting, editing and mathematical inspection; completed personal proof verification and independent human peer review are not claimed.
A human co-author is not required merely to provide a correctness check. Any authorship decision should reflect actual contribution and responsibility.

CLASSICAL REFERENCES
[1] A. T. Butson and B. M. Stewart, Systems of Linear Congruences, Canadian Journal of Mathematics 7 (1955), 358-368; section 5, printed pp. 360-361. DOI: https://doi.org/10.4153/CJM-1955-039-2
[2] Richard P. Stanley, Smith Normal Form in Combinatorics, arXiv:1602.00166, v2 (2016), Theorems 2.3-2.4. https://arxiv.org/abs/1602.00166

Requested outcome: a written correctness assessment and any corrections or suggestions. A private check is distinct from a journal's formal peer-review process.