3807 — The Generic and Physical Rank-26 Source Orbits Have Different Word Relations
Question
The generic twist gamma = 5 and physical twist gamma = -1/2 each carry a
rank-26 marked-relative quotient. Does identity on derivative words define a
canonical specialization between their source-generated covariant orbits?
Frozen comparison
Use the G12 residue chart at (x,y,z)=(2,3,4), pole depth three, and the
complete labelled word family in the three external derivatives through word
length four:
W_4 = union from r=0 to 4 of {0,1,2}^r.
Thus dim W_4 = 121. No word is removed after inspecting either fiber.
Let
ev_5 : W_4 -> H_5,
ev_half : W_4 -> H_half
be the source-derived covariant-word evaluation maps followed by the complete twist-specific reduction.
Exact finite-field result
At both primes 32003 and 32009:
rank(ev_5) = 26
rank(ev_half) = 19
dim ker(ev_5) = 95
dim ker(ev_half) = 102
dim(ker(ev_5) intersect ker(ev_half)) = 86
The physical orbit therefore has codimension seven in its rank-26 quotient, but the relation discrepancy is not exhausted by seven additional physical relations. The two kernels contain twist-specific directions on both sides. In particular,
ker(ev_5) is not contained in ker(ev_half).
Identity on labelled derivative words consequently does not descend from the generic source orbit to the physical source orbit.
The mismatch is already visible in the 40-word prefix through length three: both prefix evaluations have rank 19, both kernels have dimension 21, and their intersection has dimension 20. It is not produced only by the seven generic directions appearing at length four.
Narrow conclusion
Equal quotient dimension does not provide a canonical generic-to-physical comparison. The physical half-twist is not the identity specialization of the generic derivative-word module, and its source orbit is only rank 19 while the generic orbit reaches rank 26.
A comparison now requires an independently derived twist-changing morphism or a homotopy that transports the full relation defect. Choosing convenient independent words in each fiber would only choose bases and would not supply that missing morphism.
This result does not alter the rank-26 geometric closure and does not obstruct
constructing the physical Leray covector directly at gamma = -1/2.
Artifacts
research/benincasa/checkers/check_rank26_twist_word_kernel_transport.pyresearch/benincasa/results/rank26-twist-word-kernel-transport-k3-p32003.jsonresearch/benincasa/results/rank26-twist-word-kernel-transport-k3-p32009.json
Allocator claim: seqclaim-9174febfb2a06b7df26b00de.