Let Y be independent bits on row ports and X independent bits on column ports. Let A(Y) and B(X) multiply the chosen lifted tensor entries on each side. Set l_0(z)=1-z and l_1(z)=z, and define F(T)=sum_{X,Y} A(Y) B(X) product_x l_(Y_x)((W(T)X)_x). Matrix multiplication here means (WX)_x=sum_y W_xy X_y. At T=0 each affine factor is one exactly when its row bit equals the matching column bit, and zero otherwise. Thus F(0) is the lifted contraction. There are M affine factors, giving degree at most M(2L-1), over K[V]/(V^ell).

Turning a lifted tensor contraction into a polynomial in one parameter

For t in S, write C(t+alpha z)=alpha z product_{q!=t}(t-q+alpha z). Each noncentral factor is coefficientwise divisible by 2^(v_2(t-q)). Their total valuation cancels the power of two in C_prime(0), leaving alpha z and an odd denominator. To obtain W_xy, omit the factor with q=y-x. If q=t, this removes alpha z and leaves a polynomial with odd-denominator rational coefficients. Otherwise alpha z remains and the omitted factor costs at most a powers of two, leaving every coefficient divisible by two. Thus W_xy(t+alpha z) has coefficients in Z_(2).

Why the wire kernel has integral local coefficients

For packet size L=2^a use S={-L,...,L-1}, C(T)=product_{t in S}(T-t), and alpha=2^(a+1). For row slot x and column slot y in one packet, set W_xy(T)=C(T)/(C_prime(0)(T-(y-x))). Here C_prime(0)=product_{q in S, q!=0}(-q). The denominator factor cancels a numerator factor because y-x belongs to S, so W_xy is a rational polynomial of degree 2L-1, not a function with a pole at y-x. Set cross-packet entries to zero and keep only real ports. At T=0, matched slots give one and all other slots give zero: W(0)=I.

AI Agents

Fix an unordered pair of symbols in each of a label coordinates. If a_prime pairs have different symbols, there are 2^(a_prime) compatible ordered label pairs. A packet of L=2^a slots takes 2^(a-a_prime) positions from each such pair. Slot bit i chooses the orientation when coordinate i has unequal symbols, or the copy when the symbols agree. Because H_pad is divisible by L, all padded positions partition into such packets. Each real wire occupies the SAME slot at its row and column ends. Unused slots have no ports. The packet count is H_pad D_arr/L, at most 3M/L.

Packet slots encode endpoint-label orientations

OpenAI. The Exponential Time Hypothesis is False. Manuscript dated October 17, 2026, revised October 25, 2026. This reference pins commit 42907c9e999cc00d6abe7a50bfc1023250edd1b9. The introduction defines the exact-counting task used by this learning path; citing it does not establish that its main theorem is correct.

https://github.com/openai/math/blob/42907c9e999cc00d6abe7a50bfc1023250edd1b9/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf

Learn Before

Learn After

Learn After