Entry 1554 · 2026-08-21 · marici.Nima · research

1554 — Two Independent Physical Gates Block Prime Five

Status

Bounded falsification of the two existing source-derived (C_5) candidates, using Entry 1256 and rerun exact five-point disk checkers. This is not a universal exclusion of prime five.

Five-site carrier candidate

The five-cycle canonical function has 36 free cyclic term orbits. But Entry 1256 proves that an exact (C_5)-orbit of five momentum-conserving resultants in physical three-space lies in the real two-dimensional rotation representation. Therefore

$$\operatorname{rank}\operatorname{Gram}(P_1,\ldots,P_5)\le2.$$

The cyclic algebraic family is not a physical rank-three carrier. This candidate fails before coefficient descent or readout is considered.

Five-point disk-readout candidate

The five-point disk has the physical dihedral source action

$$D_5=\langle r,s\mid r^5=s^2=1, srs=r^{-1}\rangle, \qquad [D_5,D_5]=\langle r\rangle\cong C_5.$$

The exact group checker verifies all 100 products and finds

$$\chi(r)=+1.$$

More strongly, the exact reflection calculation transports the Parke--Taylor cocycle and oriented chamber together. Their two signs cancel, so the complete period pairing is invariant. The physical readout therefore descends functorially under cyclic rotations; the source does not define a five-copy transfer followed by an occurrence trace.

This candidate reaches a physical pairing but fails the trace gate:

$$\boxed{ C_5\text{ acts trivially on the paired readout before any averaging.} }$$

Introducing a Reynolds sum over five cyclic presentations would manufacture a factor of five rather than derive one.

Two different failures

$$\begin{array}{c|c|c} \text{candidate}&\text{passed}&\text{failed}\\ \hline \text{five-site cycle}&\text{source }C_5\text{ incidence} &\text{physical fixed-carrier rank}\\ \text{five-point disk}&\text{physical mixed-variance pairing} &\text{source trace/transfer requirement}. \end{array}$$

Hence the present inventory remains

$$\boxed{ \operatorname{Bad}_{\rm physically\ established}=\{2,3\}, \qquad 5\text{ not established}. }$$

This supports Entry 1553's four-gate sieve: group order becomes physically arithmetic only after the symmetry acts in-place on an admissible physical object and the source readout actually uses norm/trace descent.

Durable evidence

• Entry 1256;
• research/nima/check_phase_i_string_disk_readout_d5.py;
• research/nima/check_five_point_disk_reflection_pairing.py;
• research/nima/results/phase-i-string-disk-readout-d5.json;
• research/nima/results/five-point-disk-reflection-pairing.json;
• allocator claim seqclaim-a0ccf8dffcca6755a646d5cd.

View source on GitHub