Written derivation, self-review; fresh independent audit pending. No novelty claim. The modification is allowed in the exact-complex-field DFT model. It is not automatically valid in the earlier Gaussian-dyadic integer-multiplication model, because it uses1/3.
Claim and assumptions
Take the pinned OpenAI triple network described in ../dft/PROOF.md and raw ../dft/sources/network.tex, collection commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Fix h>=21, v=binom(h,3), d=binom(h-3,3)+3(h-3), m=h³. There is a version with
W'=2v³+3v²(vd+h), Delta'=2v²(v-3h²).
Let P' be the smallest power of two at least W', a'=log2(P'), lambda'=m-Delta'/P', and theta'=log_m(lambda'). The arbitrary-role tensor bound is O_h(2^k(k+1)^theta'). With the same upstream exact-width compiler and all-length transfer it gives O_h(n(log n)^theta'(loglog n)^(4-theta')) DFT cost, including preparation and logarithmic-word addressing, with one supplied specified root of order<1024n³. Every h is fixed before the input. No numerical stability, practical crossover, or bit-complexity assertion is made.
At h24, theta'<1-52/10^11, so the pure-log bound is O(n(log n)^(1-51/10^11)). The fixed-side-correction central rank is exactly h, proving that fewer central coordinates cannot represent that same central matrix via a linear gather/scatter factorization.
The new scalar identity
Let B be the v×h triple-incidence matrix B_(S,j)=1 if j belongs to S, else0. The published gather is G=B^T with one additional total-sum row. But each triple has cardinality3, hence 1_h^T G=3·1_v^T. The extra gather row is redundant.
Delete the extra central wire. For c in C^h define
(R'c)S=(sum(j in S)c_j-(1/3)sum_(j=1)^h c_j)/2.
The coefficient of x_T in R'Gx is (|S intersection T|-1)/2, exactly the old central matrix M. Keep the old side injection JV, so JV+R'G=I_v. The eight-row sequence is now y-=JA, y-=R'c, A+=Vx, c+=Gx, y+=R'c, y+=JA, c-=Gx, A-=Vx.
Its net change is y+=(JV+R'G)x=x, and it restores A and c. This identity holds for arbitrary original c; it does not assume that c satisfies any relation to a deleted coordinate. Thus we are defining a new h-coordinate circuit, not restricting the old circuit to a subspace of initial states. The reverse sequence with negated signs is its inverse. The same three-stage shear composition sends (X,Y) to(-Y,X) and restores auxiliaries.
The new scatter uses all h central values, but the published central gate already includes all targets and central wires. The gather/scatter remain fixed finite linear maps, so the unchanged common-frame commutation argument applies. The rational1/3 scalar is prepared once by dividing1 by3, whose nonzeroness is explicit. It requires no new root or operation outside the DFT cost model.
Labels and directional budget
Keep all data and side labels, and the published label path on each of the surviving h central wires. Deleting one identically labeled central role does not alter an inclusion or an orthogonality relation. The geometric label dimension is still h: reducing the number of central roles is not reducing the ambient binary label space.
All residual complements have an orthonormal binary basis exactly as in the audited variable-h derivation: h>6 leaves coordinate units outside supports of at most six; tensor complements also have norm-one witnesses. The stage-two reversed schedule has the same physical touch pattern as before. Its linear maps changed but the wires touched and their common labels did not.
There are3v² invocations, each with vd side roles and h central roles. This proves W'. The signed dimension budget is W'm-2v³. There are3v²h central decreasing edges, each losing h dimensions. Thus
s'=W'm-2v³+6v²h²=W'm-Delta'.
Every surviving auxiliary role has source frame0 and final full frame; hence all roles undergo the same tensor transform, including arbitrary initial auxiliary data. No hidden zero initialization has replaced that requirement.
The strict saving condition is binom(h,3)>3h², equivalently h²-21h+2>0 for h>0. It fails at h20 and holds at h21, after which the polynomial increases. For h>=21, 0<Delta'<2v³<=W'<=P', so1<lambda'<m. Padding and exact batching are unchanged with threshold K'=m(a'+1). The recurrence t(k)<=A_h+lambda' t(floor(k/m)) yields the stated tensor exponent. Downstream compiler, synchronization and root-order arguments use only C, theta'>0 and fixed constants, so they preserve the DFT conclusion.
Exact h24 certificate
The independently reproducible rational checker verifies
v=2024, d=1393, m=13824, W'=34666930287616, P'=35184372088832,
Delta'=2425172992, s'=479235641870830592,
epsilon'=Delta'/(mP')=2368333/474989023199232.
It checks epsilon'>(52/10^11)(191/20) and sum_(j=0)^39 (191/20)^j/j!>13824. The positive exponential series implies log(m)<191/20, so lambda'<m^(1-52/10^11). Absorb the fixed loglog power in (log n)^(1/10^11) for sufficiently large n. This proves the final exponent1-51/10^11 without floats.
Optimal central rank for this fixed side correction
The required central matrix is M=(BB^T-J_v)/2. Since B1_h=3·1_v,
M=B H B^T, where H=(I_h-(1/9)J_h)/2.
For h>=4,
B^T B=alpha I_h+beta J_h,
alpha=(h-2)(h-3)/2>0 and beta=h-2>0. This follows by counting triples containing a given point or pair. Therefore B has full column rank. H has eigenvalue1/2 on the sum-zero subspace and (9-h)/18 on constants. Consequently rank M=h except at h9, where rank M=h-1. One may also derive its nonzero spectrum from HB^T B: (h-2)(h-3)/4 with multiplicity h-1 and (9-h)(h-1)(h-2)/12 once; the remaining v-h eigenvalues are zero (with one more zero at h9). At h24 these are231/2 and-1265/2; their multiplicities give trace2024, agreeing with the diagonal entries1.
Any factorization M=RG through c central coordinates has rank M<=c. Thus c>=h for h>=21, and the new factorization achieves the minimum. This only concerns this fixed side correction and this linear gather/scatter interface. It does not rule out different side matrices, scalar networks, label families, nonlinear algorithms, or lower DFT complexity.
Evidence and independent review
verify.py reconstructs both old and new central maps over rationals, checks their equality on arbitrary inputs, executes the new schedule and inverse with nonzero auxiliary values, verifies incidence Gram/rank formulas, and rejects deliberate missing-total-correction and wrong-budget variants. The finite tests do not prove the full label theorem; the symbolic argument above carries that step. Raw pinned sources remain in the sibling directory and are read-only. Prior-art classification of the original h24 tuning is in ../dft/LITERATURE.md; this elementary central elimination has no established novelty status.