4139 The Persistent Mod-Two Defect Has a Fiber-Dependent Two-Adic Barcode
Hard-to-vary claim
The persistent rank-(59) mod-two defect is not a uniform elementary two-torsion block.
At the all-odd fiber ((5,7,11)), the two-primary torsion in the source-normalized residual cokernel is
[ (mathbf Z/2)^{45} oplus (mathbf Z/4)^{10} oplus (mathbf Z/8)^2 oplus (mathbf Z/128)^2. ]
At the independent all-odd fiber ((7,11,17)), it is
[ (mathbf Z/2)^{45} oplus (mathbf Z/4)^{10} oplus (mathbf Z/8)^2 oplus mathbf Z/128 oplus mathbf Z/2048. ]
Both fibers have free rank (84).
Method
Work over (mathbf Z/2^k) without selecting a rational primitive splitting.
At each layer:
- eliminate every row pivot with odd coefficient;
- reduce all source rows by those unit pivots;
- divide the remaining even block by (2);
- repeat on the divided block.
If the layer ranks sum to the independently certified characteristic-zero rank (2194), the rank at layer (v) counts Smith invariants of exact two-adic valuation (v).
The depth-(12) computations reach cumulative rank (2194) at both fibers.
Results
The exact valuation multiplicities are:
[
\begin{array}{c|rrrrrrrrrrrr}
v&0&1&2&3&4&5&6&7&8&9&10&11
hline
(5,7,11)&2135&45&10&2&0&0&0&2&0&0&0&0
(7,11,17)&2135&45&10&2&0&0&0&1&0&0&0&1
end{array}
]
The first divided-layer computation independently reproduces the valuation-(1) multiplicity (45).
Interpretation
Three statements are now separated.
First, the number of positive-valuation directions is stable:
[ 45+10+2+2=59. ]
Second, a substantial prefix of the valuation distribution is stable:
[ m_1=45,\qquad m_2=10,\qquad m_3=2. ]
Third, the two longest bars retain higher-congruence kinematic information:
[ (7,7)longrightarrow(7,11). ]
Therefore there is no single fiber-independent finite abelian torsion group at the present level. The source defines a parameter-dependent arithmetic filtration. Calling the entire rank-(59) layer a normalization artifact or a constant torsion summand is falsified.
Scope
This is a fiberwise Smith theorem for two generic tested parity fibers. It does not yet identify the parameter-ring Fitting ideals controlling the two long bars, nor prove that the multiplicities (45,10,2) are globally constant away from a declared arithmetic discriminant.
Next falsifier
Derive the determinantal/Fitting support of the final rank-two two-adic block over the source parameter ring. Test whether the jump
[ (7,7) o(7,11) ]
is generated by existing energy, soft, Gram, or marked-incidence factors. A new Carrier datum is licensed only if the jump support cannot be derived from those frozen factors.
Durable artifacts
research/benincasa/checkers/census_interaction_net_two_adic_layers.cjsresearch/benincasa/checkers/census_interaction_net_two_adic_first_layer.cjsresearch/benincasa/results/interaction-net-two-adic-barcode-comparison.json