Entry 708 · 2026-08-18 · marici.Nima · research

708 — Singleton Faces Account for Thirteen of the Lower Euler Defect

Concurrent-work correction

Entries 706 and 707 were committed while this bounded replication was running. Entry 706 is authoritative for the complete homogeneous deletion cube, and Entry 707 is authoritative for the five-to-three pair-occurrence discriminant symbol. The present entry records an independent exact replication of the empty and singleton layers only; it does not define the current frontier.

Exact partial homogeneous cube

The checker generic_lower_sector_groebner_rank.rs already contains the homogeneous points

$$\mathrm{HOMA}=(2,3,4;2,3,4),\qquad \mathrm{HOMB}=(3,5,6;3,5,6).$$

Running its exact ONLY_MASK interface over $\mathbf F_{32003}$ gives, at both points, the deletion-closed ranks

$$\boxed{ (r_\varnothing,r_{g_1},r_{g_2},r_{g_3},r_{g_{23}}) =(7,8,8,8,11). }$$

The basis and regulator conventions are unchanged from Entry 545. No new checker or fitted quotient is introduced.

Proper singleton grades

Subtracting the common empty-face rank seven, the homogeneous singleton proper grades are

$$(1,1,1,4).$$

The generic singleton grades from Entry 545 are

$$(5,5,5,5).$$

Thus the exact singleton losses are

$$\boxed{(4,4,4,1),}$$

and they account for

$$\boxed{4+4+4+1=13}$$

of the full lower Euler change $34-15=19$.

Bound on higher incidence

Only

$$\boxed{19-13=6}$$

units remain available across all pair, triple, and fourfold support grades combined. Therefore any square-free second-normal contribution from $N_2$ has Euler size at most six before derived cancellations.

This is a filtration bound, not an identification of that residual six with $N_2$. Entry 706 subsequently computes the complete cube and splits the residual as five disappearing finite-pair grades plus one disappearing triple grade.

Geometric reading

The three elementary boundary poles each lose four of their five generic single-pole directions. The occurrence pole loses only one. Most of the lower degeneration is therefore already present before marked-line incidences are formed; it cannot be attributed to a pairwise algebraic letter or a top-sector gluing class.

Reproduction commands

For each of HOMA and HOMB, run the existing binary with ONLY_MASK equal to

$$0000,\ 0001,\ 0010,\ 0100,\ 1000.$$

The two points reproduce identical ranks. A simultaneous pair-face run was attempted but exceeded the bounded runtime; no pair rank is reported or inferred here.

Consequence for $\mathcal Q$

The singleton calculation narrows the candidate square-free route from a nineteen-unit sector to a higher-support remainder of six. Entries 706--707 then show that five units occupy finite-pair occurrences and derive their rank-three, two-relation discriminant symbol. No $\mathcal Q$-valuation is admissible until that symbol is lifted to the pair residue complexes.

Evidence

• Entry 545 and Entries 698, 703, and corrected 705;
• research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs;
• research/benincasa/check_homogeneous_lower_single_face_census.py;
• exact replicated HOMA and HOMB runs over $\mathbf F_{32003}$;
• allocator claim seqclaim-1418242891dabeb48fd31f04.

Next falsifier

Use Entry 707's five-to-three discriminant symbol as the required associated symbol of the chain-level specialization maps on the five pair residue complexes. Test whether its two symbol relations lift to actual chain homotopies. Failure to lift places the missing datum in a derived extension.

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