IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PrimeAxisCoherence
Defines the log-character built from prime weights by additive extension through prime factorization, and proves prime-axis coherence: the character is a pure power law if and only if the weights are aligned. Downstream Delta-native analysis and the objecthood registry import this module. The argument is algebraic: multiplicativity of the character, comparison with real logarithms, and an iff between power-law form and weight alignment.
claimFor weights $a$ on primes, the log-character $\chi_a$ is the unique completely additive map on positive integers with $\chi_a(p)=a(p)$ on primes. Write $\mathrm{IsPowerLaw}(\chi_a)$ when $\chi_a(n)=c\,\log n$ for some constant $c$, and $\mathrm{WeightsAligned}(a)$ when the prime values $a(p)$ are proportional to $\log p$. Prime-axis coherence asserts $\mathrm{IsPowerLaw}(\chi_a)\Leftrightarrow\mathrm{WeightsAligned}(a)$, with the character realized as a real power $n\mapsto n^{r}$.
background
In the Primitive Recognition Calculus, discrete axes are indexed by primes. A weight assignment $a$ on primes is extended to all positive integers by complete additivity along prime factorization: if $n=\prod p_i^{e_i}$ then $\chi_a(n)=\sum e_i,a(p_i)$. That extension is the log-character of $a$; on the unit one has $\chi_a(1)=0$, and $\chi_a$ is a group homomorphism from $(\mathbb{N}_{>0},\cdot)$ to $(\mathbb{R},+)$.
Two structural predicates organize the theory. $\mathrm{IsPowerLaw}$ says the character coincides with a constant multiple of the ordinary logarithm. $\mathrm{WeightsAligned}$ says the prime values themselves sit on a single logarithmic ray, $a(p)=c\log p$. Faithfulness records that distinct weight systems yield distinct characters. The module sits in the Foundation layer and is imported only by Mathlib plus three sibling calculus modules.
proof idea
Definition block first: logChar as the additive extension, with elementary lemmas logChar_one, logChar_mul, logChar_prime fixing the homomorphism laws and the prime values. Faithfulness is the injectivity statement for the extension map. IsPowerLaw and WeightsAligned are Prop-valued predicates; powerLaw_iff_aligned is the biconditional linking them. Comparison lemmas logChar_log and character_is_rpow identify power-law characters with real $r$-powers $n\mapsto n^r$. The capstone prime_axis_coherence packages the iff and the power realization. No deep analysis: pure multiplicative number theory and uniqueness of additive extensions.
why it matters in Recognition Science
Prime-axis coherence is the bridge from discrete prime weights to continuous scaling in the recognition calculus. Once characters are forced to be pure powers, native $\Delta$-analysis can treat axis increments as logarithmic derivatives rather than arbitrary additive noise. The module is imported by DeltaNativeAnalysis, DeltaNativeStrongClosure, and ObjecthoodRegistry, so every later claim that a recognition object has a coherent prime axis, or that a native defect closes under strong limits, rests on this identification. In the broader RS forcing chain it supplies the arithmetic side of self-similar scaling (the same logarithmic structure that later meets $\varphi$ and the eight-tick octave), without yet fixing the golden ratio or spatial dimension.
scope and limits
- Does not fix the numerical constant $c$ or identify it with $\log\varphi$.
- Does not treat non-completely-additive characters or Dirichlet $L$-functions.
- Does not prove existence of aligned weights; only the power-law $\Leftrightarrow$ alignment iff.
- Does not address continuous groups, $p$-adics, or archimedean places beyond $\mathbb{R}$.
- Does not derive physical constants, mass rungs, or the eight-tick period.