Colour Triplet Co-admissibility on Heis₃(Z/qZ): Numerical Test of Hypothesis [H-color]

Numerical results

• Test primes ($q \equiv 1 \pmod{3}$): $q \in \{61, 151, 211, 307\}$. Control primes: $q \in \{29, 101\}$. Combined noise floor $\mathrm{ctrl\_ref} = 1.657 \times 10^{-3}$.
• $q = 61$: $R = 2.1$, 1 out of 4 tested triplets satisfy the co-admissibility criterion. Passes $R \leq 3$.
• $q = 151$: $R = 0.4$, 4 out of 4 triplets valid. Clear confirmation of [H-color]; passes $R \leq 3$.
• $q = 211$: $R = 1.0$, 4 out of 4 triplets valid. Consistent with $q=151$; passes $R \leq 3$.
• Triplet exponent: $\delta_\mathrm{tri} \approx 3\delta_c$ at sub-percent level across all test primes, consistent with the triplet being a threefold composite of single-colour admissibility.
• Colour covariance rank: the matrix $C_\mathrm{color}$ has numerical rank 1 in all tested cases, supporting the SU(3) identification of O31.
• $q = 307$: 4/4 triplets, $R = 0.7$ (well below the threshold $R \leq 3$), $\delta_\mathrm{tri} = 12.355 \approx 3\delta_c$ (deviation 1.5\%). Four-prime dataset complete.

Interpretation

The passing of all test primes $R \leq 3$ provides numerical support for [H-color]. The most conclusive results are for $q = 151$ and $q = 211$, where all four tested triplets satisfy the co-admissibility criterion and the normalised ratio $R$ is well below the threshold.

The result $\delta_\mathrm{tri} \approx 3\delta_c$ is structurally expected: if the three colour charges $c_1, c_2, c_3$ are co-admissible, their joint projective capacity decay should be the product of the three individual decays. The sub-percent agreement with this prediction validates the factorisation structure.

The rank-1 numerical structure of $C_\mathrm{color}$ is consistent with the SU(3) identification from O31: the co-admissible triplet has a one-dimensional admissible covariance, corresponding to the single complex direction preserved by the SU(3) action.

The partial result for $q = 61$ (1/4 triplets valid) may reflect finite-size effects at smaller primes or a genuine distinction between triplets at this prime. The larger primes $q = 151$ and $q = 211$ show full triplet co-admissibility.

Spectral obstruction and the limits of the Markov approach

O31v2 establishes an exact result for the standard graph that has direct consequences for how [H-color] can be proved analytically. The generator-twist isospectrality theorem (O31 Theorem 5.3) shows that, for every $c \in (\mathbb{Z}/q\mathbb{Z})^*$, the Markov operators $\rho_c(P_{S_q})$ and $\rho_c(P_{\phi_\omega(S_q)})$ are unitarily conjugate, hence isospectral.

However, this isospectrality does not extend across distinct central characters: the spectra of $\rho_c(P_{S_q})$ and $\rho_{\omega c}(P_{S_q})$ are generically distinct ($\max|\Delta\lambda| \approx 0.3$–$0.4$, same order as non-triplet pairs), confirmed for all $q \leq 157$ with $q \equiv 1 \pmod{3}$. [H-color] on the standard graph cannot be derived from global Markov-spectrum equality: the BFS capacity is a strictly finer observable, and any analytical proof must act directly on the rank structure of the BFS walk.

Physical interpretation: colour charges and quark structure

Each element $c_i$ of a valid colour triplet simultaneously belongs to a conjugate pair $\{c_i, q-c_i\}$ (carrying spin-½, $d_\rho = 2$, from O29) and a colour triplet (SU(3) fixed point from O31). An individual sector $c_i$ therefore carries spin-½ and one colour charge: the minimal quantum-number content of a quark.

The sum-zero constraint $c_1 + c_2 + c_3 \equiv 0 \pmod{q}$, derived from the $\mathbb{Z}_3$ orbit structure, is the discrete analogue of baryon colour-neutrality. The confirmation of [H-color] is thus a necessary step toward identifying individual admissible sectors as colour-charge carriers and colour triplets as the pre-image structure of colour-neutral baryons.

Relation to the Cosmochrony programme

O32 is the numerical companion to O31. Together, they provide the theoretical framework and numerical evidence for the emergence of the SU(3) colour gauge group from admissibility constraints on the Heisenberg graph.

The confirmation of [H-color] feeds directly into Q6a (Standard Model gauge structure) and Q12 (Yang–Mills dynamics from projective spectral entropy), where the SU(3) group is used as an input. The conditional status of these downstream results lifts partially as [H-color] is numerically confirmed; full unconditional status requires the analytical proof of [H-color], which remains open.

Analytical sharpening (v1.3+)

O32 additionally proves analytically (Proposition avg-hcolor) that for block-averaged capacity profiles: \[ \mathbb{E}[\sigma_c(n)] = \mathbb{E}[\sigma_{\omega c}(n)] + O(q^{-1}), \] and similarly for $\omega^2 c$. This establishes $H_{\mathrm{color,eff}}$ — equality of capacity exponents in expectation — in the $q \to \infty$ limit.

The argument proceeds by showing that the BFS fingerprint frequency distribution for $c$ and $\omega c$ blocks differs only by a shift of the distinguished frequency ($c B_1 \mapsto \omega c B_1$), which preserves the cardinality of new distinct frequencies at every BFS depth. The $O(q^{-1})$ correction arises from the excess probability on the distinguished frequency, accumulated over a rank window of size $O(q)$.

The observed decrease $R_\mathrm{var} \propto q^{-1}$ (factor 5 from $q=61$ to $q=151$) is interpreted as the finite-$q$ modulation bias predicted by this argument. Three levels of the colour hypothesis are now distinguished: pointwise ($H_\mathrm{color,ptwise}$, numerical), in expectation ($H_\mathrm{color,eff}$, analytical), and global ($H_\mathrm{color}$, open).

Pointwise [H-color] from O31 v1.5

The previously open pointwise analytical proof of [H-color] at fixed finite $q$ has been resolved in the companion paper O31 v1.5. Each Born–Infeld fingerprint vector is shown to be a single additive character of frequency $B = c_1 b_1 + c_2 b_2 + c_3 b_3$ (Lemma single-frequency). The capacity observable $\sigma_c(n)$ is therefore a count of distinct frequencies, and the orbit dilation $B \mapsto \omega B$ preserves it exactly (Proposition hcolor-frequency).

The residual inter-triplet variance $R_\mathrm{var}$ measured here is reinterpreted as the finite-sample footprint of independent auxiliary draws, with expected $R_\mathrm{var} \propto q^{-1}$ scaling — consistent with the observed factor 5 decrease from $q=61$ to $q=151$. The completed $q = 307$ datapoint ($R = 0.7$, 4/4 triplets, $R_\mathrm{var} = 1.17\times10^{-3}$) confirms this scaling.

References

Jérôme Beau. Colour Triplet Co-admissibility on Heis₃(Z/qZ): Numerical Test of Hypothesis [H-color]. Preprint. doi:10.5281/zenodo.20145607