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Actual middle spectator transport

Abstract

Actual middle spectator transport

Theorem 1.1 (Two actual channels with the same numerator).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorFourFieldMiddleSpectator.actual_four_field_middle_spectator_transport

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorFourFieldMiddleSpectator.actual_four_field_middle_spectator_transport (✓ std3). ∎

Source. Repository-derived.

Commentary.

For arbitrary lattice data, four labelled charges alpha,beta,gamma,eta and oscillator descendants u,v,t,q, one locality order k is chosen from all six original pairs before the arbitrary fifth charge delta and state w. The same original bounds a and generally infinite four-variable numerator P retain all four canonical-on-w truncations, all twenty-four signed labelled product laws and the previous actual inner-channel identity u_x(v_y((t_b q)_z w)) at P(x,y,b+z,z). With these exact witnesses, at every signed e, polynomial action by Rmid on the actual channel u_x((v_b t)_z(q_y w)) equals signedExtend of the native substitution P[X0,X1+X2,X2,X3]. In output coordinates x,b,z,y, Rmid is x^a0 (b+z)^a1 z^a2 y^a3 (x-b-z)^k (x-z)^k (x-y)^k b^k (b+z-y)^k (z-y)^k. The actual outer canonical mode has charge alpha and input charge (beta+gamma)+(eta+delta); the middle mode has charge beta+gamma and input eta+delta; the two inner modes have inputs gamma and delta. Each mode index is minus its coordinate minus one. Three finite spectator expansions use shifted natural bounds a1+p,a2+r,a3+s derived from the original bounds. Original v/q locality supplies the balanced Borcherds reconstruction; original t/q locality and t-on-w truncation make its residue finite before the outer u mode is mapped. Only finite sums are exchanged. The resulting five cross factors are compared with the original product law on the fixed original u,q slice, identifying the same P at x,b+z,z,y for negative as well as nonnegative indices. No independently selected witnesses, enlarged k or a, extra cutoff on q_y w or a fused state, finite numerator, rank, Gram, domain, energy, positivity or charge restriction is assumed. This is an algebraic middle collision together with the previous inner collision; a common identity for all five trees, the pentagon, sewing, analytic convergence, physical realization and full string theory remain outside this conclusion. The original Bakalov–Kac, Matsuo–Nagatomo and Carnahan attributions are retained.

References