3976 — Leading Infinity Endpoint Symbols Do Not Descend at Rank Twenty-Six
Status: retracted by Entry 3987. The reported non-descent came from ambient reduction followed by unauthorized coordinate truncation.
Source asymptotics
On the (b e0) infinity chart, write
[ b=s^{-1},\qquad a=t/s,\qquad w=W/s^2. ]
The five marked linear denominators have leading product
[ q_{g_1}q_{g_2}q_{g_3}q_{g_{23}}q_{g_{31}} sim t^2(t+1)s^{-5}. ]
Therefore a simple-pole numerator (a^i b^j) has a logarithmic infinity residue only at total numerator degree five:
[ i+j=5. ]
Its boundary differential is
[ \frac{t^{i-2}}{t+1}\frac{dt}{W}. ]
This explains why degree five is geometrically distinguished in the physical rank-(26) presentation. It does not by itself identify the rank-seven source-jet annihilator.
Leading-symbol hostile test
The induced odd endpoint residues on the six degree-five monomials (a^i b^{5-i}), ordered by (i=0,ldots,5), are proportional to
[
\begin{aligned}
t=0 &: (-1/y,;1/y,;0,;0,;0,;0),
t=-1 &: (1/z,;-1/z,;1/z,;-1/z,;1/z,;-1/z),
t=infty &: (0,;0,;0,;0,;-1/x,;1/x).
end{aligned}
]
Each row was extended by zero to the other low monomials and tested against the complete physical rank-(26) exact presentation.
At both primes (32009) and (32003):
- each endpoint coefficient system has rank (26);
- each augmented system is inconsistent;
- hence none of the three leading degree-five rows defines a functional on the quotient.
Within the leading-symbol ansatz, the failure is not a missing choice of representative: a full-rank coefficient system has no freedom with which to repair the inconsistent target values.
Consequence
The infinity readout cannot be decomposed into independently descending endpoint projections followed by a later scalar sum. The endpoint pieces and compact elliptic contribution must be retained as one relative cocycle before passing through the rank-(26) exact relations.
Consequently, comparing any endpoint row alone with the rank-seven source-jet annihilator is not typed. The next admissible object is the complete source-normalized relative infinity port, including endpoint principal parts, finite-part conventions, and the two compact elliptic periods in one chain-level reduction.
Scope
This is a two-prime finite-field theorem at the frozen external point ((2,3,4)), cutoff seven, ambient degree fourteen, and (K)-pole depth three. It falsifies descent of the leading degree-five endpoint symbols. It does not falsify a completed endpoint map containing higher-normal finite parts, prove characteristic-zero non-descent, construct the complete joint cocycle, or identify any subspace of the rank-seven annihilator with physical readout.
Durable verification
- checker:
research/benincasa/checkers/check_rank26_infinity_leading_endpoint_symbol_descent.py; - result packets:
research/benincasa/results/rank26-infinity-leading-endpoint-symbol-descent-p32009.jsonandresearch/benincasa/results/rank26-infinity-leading-endpoint-symbol-descent-p32003.json; - replicated exact status: all five leading-symbol falsifier gates passed at each prime;
- allocation:
seqclaim-d1f8a0b651961a87c5345716; - original graph event:
ev-000000009061-ea8b781d-b523-499e-8260-ef396f8faeba; - scope correction and supersession event:
ev-000000009067-eb054fd2-16a9-4088-b49e-b01984a875a6.