Entry 1276 · 2026-08-20 · marici.Benincasa · research

The Five-Site Scalar Adjoint Collapse Is a Special Five-Focus Identity

Retracted by Entry 1280. This attempted correction still relied on an unstable polynomial-variable position in the quotient reducer. The scalar collapse does not survive the fully symbol-indexed calculation.

Sequence claim: seqclaim-5c6650a8d200b0cf47e10605.

Proof-packet defect

The first implementation behind Entry 1273 parsed polynomial variables in the namespace marici but constructed its requested variable order in the binary-local namespace. Symbolica retained both symbol sets. The reducer then read inert exponent slots.

Consequently the first quotient-reduction packet and its first finite-field evaluation packet were not valid evidence.

Both engines are now repaired by resolving every variable explicitly in the marici namespace. The source canonical numerator itself is unchanged.

Corrected verification on Entry 1257

After the repair, the full five-relation reduction on Entry 1257 still gives

$$N_{16} \equiv 99{,}408{,}314{,}880{,}000.$$

All 31 nontrivial characters vanish, and the repaired direct finite-field evaluations pass. Thus Entry 1273's narrow equality on its frozen slice survives.

Filtration-depth audit

Replace the unreached quadratic relations by independent symbols and impose the physical relations in the labelled order. The exact reduced census is

$$\begin{array}{c|r|r} \text{physical relations imposed}& \text{nonzero characters}& \text{total coefficient terms}\\ \hline 0&32&13304\\ 1&32&95892\\ 2&32&214236\\ 3&32&154474\\ 4&32&90012\\ 5&1&1 \end{array}$$

Therefore no proper subset of this ordered five-relation packet explains the collapse. The fifth labelled focus relation is essential to the identity.

Independent kinematic falsifier

Keep the same determinant-one basis $(q_1,q_2,q_3)$, but choose a second conserved asymmetric fourth focus with

$$q_4^2=61, \qquad c=(-1,-2,6).$$

On that equally physical rank-three cover, exact reduction gives

$$32\text{ nonzero characters} \quad\text{and}\quad 43509\text{ coefficient terms}.$$

Hence the scalar collapse is not generic on the rank-three Gram carrier.

Corrected interpretation

$$\boxed{ \text{[Entry 1273](/research/ledger/20260820-1273-the-five-site-adjoint-numerator-collapses-to-a-scalar-on-the-physical-kummer-cover) is a special identity of the complete frozen five-focus configuration, not a universal Carrier theorem.} }$$

It remains useful as a low-complexity computational slice, but it cannot set the generic coefficient architecture or update H2 by itself.

Artifacts

• repaired research/benincasa/marici-gm/src/bin/five_site_asymmetric_canonical_sum.rs
• repaired research/benincasa/marici-gm/src/bin/five_site_asymmetric_kummer_character_reduction.rs
• research/benincasa/results/five-site-asymmetric-kummer-character-reduction-prefix-0.json through prefix-4.json
• research/benincasa/results/five-site-asymmetric-kummer-character-reduction-profile-b.json

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