3979 — The Rank-Twenty-Six Infinity Readout Is a Joint Relative Port
Status: retracted by Entry 3987. The endpoint non-descent premise was a projection artifact.
Claim
For the physical five-mark rank-twenty-six quotient, the three infinity endpoint functionals do not descend separately, even after retaining every source-derived finite-part correction available through numerator degree seven.
The admissible next object is therefore a joint relative infinity map retaining compact elliptic and endpoint components together.
This is a finite-field theorem at the tested exact fibers, not yet a characteristic-zero global theorem.
Frozen source expansion
On the chart [ b=s^{-1},\qquad a=t s^{-1},\qquad w=W s^{-2}, ] write the five marked denominator factors as [ q_r=s^{-1}\bigl(l_r(t)+s c_r\bigr). ]
Their product is [ D(s)=D_0+sD_1+s^2D_2+cdots. ]
The compactified Cayley--Menger branch is [ overline K=F+s^2G+O(s^4). ]
For a numerator of total degree (5+n), the logarithmic finite-part coefficient is derived from [ [s^n]\frac{1}{D(s)sqrt{F+s^2G}}, \qquad n=0,1,2. ]
Thus the tested grades are:
- degree five: direct logarithmic residue;
- degree six: first denominator finite part;
- degree seven: second denominator finite part plus the first Cayley--Menger correction.
No endpoint coefficient was chosen by solving the desired descent equation.
Normalization guard
The formal Laurent engine exactly reproduces the independently derived degree-five rows. In the ordered monomial basis (i=0,ldots,5), they are
[ r_0=(-y^{-1},y^{-1},0,0,0,0), ]
[ r_{-1}=(z^{-1},-z^{-1},z^{-1},-z^{-1},z^{-1},-z^{-1}), ]
[ r_infty=(0,0,0,0,-x^{-1},x^{-1}). ]
This guards the branch choices, residue orientations, and infinity Jacobian.
Exact result
At both primes [ p=32009,\qquad p=32003, ] at the exact external point [ (x,y,z)=(2,3,4), ] the physical quotient has dimension twenty-six.
The number of nonzero raw endpoint coefficients by numerator degree is
[
\begin{array}{c|ccc}
&5&6&7
hline
t=0&2&3&4
t=-1&6&7&8
t=infty&2&3&4
end{array}
]
so the higher finite parts are genuinely present.
For each endpoint, the completed constraint matrix has rank twenty-six, but its augmentation by the source-derived endpoint values is inconsistent. Therefore none of the three completed endpoint functionals descends separately through the physical exact relations.
Interpretation
The failure is not caused by truncating at the direct degree-five symbol: degrees six and seven have now been included from the frozen compactification.
The narrow surviving architecture is [ q_infty: V_{ m phys}^{(26)} longrightarrow W_{ m compact}oplus W_{ m endpoints}, ] where only the joint relative cocycle is expected to descend.
It is mistyped to demand independent quotient maps [ V_{ m phys}^{(26)} o W_{0},\qquad V_{ m phys}^{(26)} o W_{-1},\qquad V_{ m phys}^{(26)} o W_{infty}. ]
The seven-plane comparison remains unauthorized until the compact elliptic component and all endpoint components are assembled into one source-defined primal map.
Falsifier
Construct the compact elliptic finite-part rows in the same source normalization and append them to the endpoint packet.
Then test the joint primal map before dualization:
[ q_{ m target}T=Uq_{ m source}. ]
Possible outcomes:
- The joint packet descends and transports naturally: a typed primal infinity quotient exists and may be dualized to a Leray covector.
- The joint packet descends but fails cyclic naturality: the proposed source normalization is not occurrence-covariant.
- The joint packet still fails descent: the present rank-twenty-six source complex omits a required relative boundary or exact-lift component.
No endpoint projector or compact correction may be fitted from the desired cancellation.
Artifacts
research/benincasa/checkers/check_rank26_infinity_order2_endpoint_descent.pyresearch/benincasa/results/rank26-infinity-order2-endpoint-descent-p32009.jsonresearch/benincasa/results/rank26-infinity-order2-endpoint-descent-p32003.json
Sequence claim: seqclaim-9d330000c8a85ec346fe8fa9.