Naive Quartic Reduction Does Not Lift Carrier Reduction
Record
Status: first-order obstruction to lifting Entry 466 by coefficientwise reduction modulo the full Cayley--Menger quartic.
Entry 466 constructs the soft-fiber carrier reduction because every frozen exact image is divisible by (a^4). The simplest proposed lift would replace (a^4) by the full monic relation
[ K=a^4+u,a^2(1-b^2) \qquad (\bmod,u^2) ]
and reduce every exact-form representative coefficientwise modulo (K).
Finite falsifier
Use the source-defined sector
[ (s_a,s_b)=(1,1), \qquad f=1, ]
with
[ L_1=b+1-u, \qquad L_2^-=a-\frac u2. ]
Its (q)-exact operator is
[ q
-L_1K -\frac32L_1L_2^-K_a. ]
Modulo (K), the first term vanishes. To first order,
[ K_a=4a^3+2ua(1-b^2). ]
Using the quartic relation
[ a^4=-u,a^2(1-b^2) ]
gives
[ \boxed{ q equiv 3u(b+1) left[ a^2(1-b^2)+a^3
ight] pmod{K,u^2}. } ]
This polynomial is generically nonzero. Its six monomials all have (a)-degree two or three, so no residual factor (a^4) remains.
Therefore coefficientwise reduction does not send the complete exact image to zero:
[ \boxed{ C_{ m full}^{ m Rees}longrightarrowmathbb Q[u,a,b]/(K) } ]
is not a chain map in this naive form.
Interpretation
The obstruction is source-derived and occurs before any endpoint issue. Entries 483--484 close the two boundary components of the odd-tail comparison, but they do not supply the interior homotopy required here.
This is exactly the distinction anticipated in Entry 447: the quartic module is flat, but coefficientwise (partial_u) does not descend. A lift of carrier reduction must be made in the relative de Rham complex and include a Gauss--Manin/Koszul correction for the (K_a) and (K_b) terms.
No new carrier component is indicated. The failed object is a naive coefficient map, not the shared-carrier hypothesis.
Classification
- quartic carrier family: existing and flat;
- naive coefficientwise reduction: falsified as a chain map;
- obstruction class: interior relative-de-Rham coefficient data;
- endpoint support: already closed by Entries 483--484;
- new carrier datum: none.
Next falsifier
Construct the canonical conormal/Koszul homotopy for the monic quartic relation and apply it to all exact sectors. Test whether it cancels the displayed obstruction and makes carrier reduction a chain map through first order in (u). If a residual survives, compute its filtered rank and provenance before considering higher Rees order.
Evidence
research/benincasa/marici-gm/src/bin/soft_axis_naive_carrier_lift.rs;- Entries 447, 466, and 483--484.