prime_axis_coherence
plain-language theorem explainer
Independent prime weights always extend to a multiplicative log-character, characters that agree on all naturals agree on every prime weight, and a global power law against a fixed reference holds exactly when the prime weights are aligned to it. Phase 5 of the primitive recognition calculus packages this freedom-then-collapse story as one conjunction. Cite it when invoking prime-axis coherence or the Delta-native strong closure certificate. The proof is a three-way refine applying the three component lemmas directly.
Claim. Three claims hold at once. (1) For every weight map $a:\mathbb{N}\to\mathbb{R}$ and all nonzero $m,n\in\mathbb{N}$, the log-character of $a$ (additive extension of $a$ through prime factorization) satisfies $\chi_a(mn)=\chi_a(m)+\chi_a(n)$. (2) If two weight maps induce identical log-characters on every natural, then they agree on every prime. (3) For weight maps $a,w$, there exists a single $c\in\mathbb{R}$ with $\chi_a(n)=c\,\chi_w(n)$ for all $n\neq 0$ if and only if there exists $c$ with $a(p)=c\,w(p)$ for every prime $p$.
background
In the primitive recognition calculus each assignment of real weights to primes extends by factorization to a log-character: if $n=\prod p^{k_p}$ then $\chi_a(n)=\sum_p k_p,a(p)$. The value $a(p)$ is the log-weight on the prime axis $p$. Before any coherence constraint, these axes are free coordinates of the character.
Upstream results fix the free layer. Multiplicativity says every weight assignment extends to an additive character on products of nonzero naturals, so no relation among primes is forced yet. Faithfulness says two characters that agree on all naturals recover the same prime weights, making those weights genuine independent coordinates. A character is a power law against a reference scale $w$ when a single exponent $c$ satisfies $\chi_a(n)=c,\chi_w(n)$ for all $n\neq 0$; the prime weights are aligned to $w$ when the same form holds on primes alone, $a(p)=c,w(p)$.
The local setting is Phase 5 of the foundation layer: free characters first, then the collapse that synchronizes independent prime axes onto one common scale.
proof idea
Term proof by refining the goal into three conjuncts. The first conjunct is exactly the multiplicativity lemma for the log-character on nonzero products. The second is the faithfulness theorem: agreement of log-characters on all naturals implies agreement of prime weights. The third is the already-proved equivalence between global power-law form and prime-weight alignment. No fresh arithmetic is done; the declaration is the packaged conjunction of those three prior results.
why it matters
Phase 5 headline of the primitive recognition calculus: freedom then collapse in one statement. Independent prime axes define free multiplicative characters; coherence is their synchronization into a single scale, expressed as the equivalence of global power law and prime-weight alignment. Downstream it is consumed by the strong closure certificate in the Delta-native strong-closure module, which assembles the closed theorem surface for this foundation layer.
In the broader Recognition framework the pattern matches the forcing-chain move from free structure to unique fixed scales (T5 J-uniqueness, T6 phi, T7 eight-tick octave, T8 D=3): free axes first, then a global constraint that collapses them. The result does not itself pin a numerical exponent or identify the reference with phi; it only equates the two formulations of scale coherence that later layers can specialize.
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