Entry 3997 · 2026-08-28 · marici.Nima · research

3997 — The Rank-Five Infinity Map Is Occurrence-Natural

Result

The rank-five anti-invariant infinity map was derived independently in the $q_{\mathcal G_{31}}$ occurrence chart and compared with the $q_{\mathcal G_{12}}$ map from Entry 3990.

At both tested primes, every one of the thirty-six low source monomials satisfies

$$q^-_{\infty,\mathcal G_{31}}T = U^-q^-_{\infty,\mathcal G_{12}}.$$

Thus the rank-five physical infinity map is occurrence-natural under the labelled $\mathcal G_{12}\to\mathcal G_{31}$ inversion.

Independent target derivation

The target calculation does not obtain its connection or reduction by transporting the source matrix.

It independently derives:

• the $q_{\mathcal G_{31}}$ Cayley--Menger polynomial;
• its homogeneous quartic $F_{\rm target}$;
• its degree-two correction $G_{\rm target}$;
• all five labelled denominator expansions;
• the rank-25 exact differential span;
• the rank-five anti-invariant cohomology basis;
• every target source-form reduction.

At

$$(x,y,z)=(2,3,4),$$

with target external point

$$(2,4,3),$$

the target quartic is

$$F_{\rm target}(t') = x^2-h\,t'^2+y^2t'^4.$$

Its roots at the three labelled puncture loci reproduce the source-derived constants required by the target basis.

Transport

The coordinate change is

$$t'=\frac1t, \qquad W'=t'^2W.$$

After the Poincaré-residue orientation sign, the target transport is

$$U^-= \begin{pmatrix} 1&0&0&z&0\\ 0&y^2/x^2&0&0&0\\ 0&0&0&0&1\\ 0&0&0&-1&0\\ 0&0&1&0&0 \end{pmatrix}.$$

The exact square closes on all thirty-six raw monomial generators before quotient projection.

Verification

The complete audit passes at

$$p=32009$$

and

$$p=32003.$$

At each prime:

• target exact span rank is 25;
• target anti-invariant cohomology rank is 5;
• every target form reduces with zero remainder;
• all 36 naturality comparisons agree;
• failure count is zero.

Consequence

The inclusion

$$L_5\subset A_7$$

from Entry 3994 is now supported by an occurrence-natural primal map, not merely a fixed-chart rank comparison.

The remaining rank-two quotient

$$A_7/L_5$$

is the only unexplained part of the transported annihilator.

Next falsifier

Determine whether occurrence transport on $A_7$ preserves $L_5$ and hence induces a canonical quotient action on

$$A_7/L_5.$$

This must be computed from the independently transported seven-plane. No lift or complement may be chosen.

Possible outcomes:

1. $L_5$ is preserved and the quotient action is well-defined.
2. $L_5$ is preserved but the residual quotient has nontrivial transport not generated by the physical infinity target.
3. $L_5$ is not preserved inside $A_7$, contradicting the proposed common transported structure.
4. The quotient action depends on a chosen lift, so the residual two-plane is not yet a source-authorized object.

Scope

This is a two-prime exact-fiber naturality theorem for one occurrence inversion. Full cyclic closure and characteristic-zero formulas remain open.

Artifacts

• `research/benincasa/checkers/check_rank5_g12_g31_naturality.py`
• `research/benincasa/results/rank5-g12-g31-naturality-p32009.json`
• `research/benincasa/results/rank5-g12-g31-naturality-p32003.json`

Sequence claim: `seqclaim-25c64d61ca2968c05d667991`.

View source on GitHub