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Mathematics · A guided explanation

Lin with Hyra: claimed corrected partial-Hadamard asymptotics above the quadratic scale

Claimed a uniform asymptotic count of n-by-m sign matrices with orthogonal rows when m is divisible by four and m/[n² log(2πm)]→∞, retaining the essential exp[−n(n−1)(2n−1)/(24m)] correction to the standard main term. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.

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Count patterns that cancel

Imagine a table filled only with plus ones and minus ones. We want different rows to cancel when multiplied entry by entry and added. This zero total is called orthogonality. The question is how many such tables exist for a chosen number of rows and columns. The paper claims a counting formula that reaches substantially narrower tables than earlier asymptotic results, while retaining a correction that becomes essential there.

A small table you can check

Use the rows (1, 1, 1, 1) and (1, 1, -1, -1). Their dot product is 1 + 1 - 1 - 1 = 0. Changing the last minus sign to a plus sign makes the dot product two and breaks the condition. This classroom example teaches the constraint only. A large-table asymptotic formula is an approximation in a specified limit; it is not automatically an exact count for this two-row example.

Moving below the cubic scale

Let n be the number of rows and m the number of columns. The new claim requires m to be divisible by four and m divided by n squared times log(2 pi m) to tend to infinity. For example, take n = k squared and m = 4k to the fifth power. Then m/n cubed tends to zero, while the new required ratio tends to infinity. That exhibits a regime below the cubic scale without reaching the quadratic endpoint.

The correction is part of the result

The formula includes an exponential correction involving n(n-1)(2n-1)/(24m). Below the cubic scale, dropping it can change the predicted count enormously. This is why saying only that the old formula now works for smaller tables would lose the contribution. The counting assembly in the supplied Lean material assumes an analytic row estimate called hstep. Checking the assembly does not independently supply that missing analytic proof, and WRS has not replayed the formalization.

A useful first contribution

Ask your assistant to verify the toy dot products and the two ratios in the k example. Keep elementary checks separate from the manuscript claim. A more advanced reader can compare the near-quadratic statement with Davis's cubic-regime theorem, preserving the correction and quantifiers. Record exactly which lemma is established by which source. The printed human author is Haowei Lin; source-reported Hyra assistance does not replace that authorship.

Scientific scope

Source-claimed proof, not independently verified by WRS. The result covers each fixed power-law regime m/n^(2+η)→∞ with η>0, not the endpoint m/n²→∞. The supplied Lean counting assembly assumes the unformalized analytic row estimate hstep; the exponent module proves a sufficient-regime bridge, not impossibility of the counting formula at smaller exponents.

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People and systems

Haowei Lin

Named source authors and agent execution are recorded separately; individual effort shares are unknown.

Hyra · Tencent · UNKNOWN
Source-reported proof-strategy development and initial detailed mathematical draft; subsequent human audit and repairs; the exact underlying model identifier is unknown. Source-reported attribution; provider attestation is unavailable.

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