Invariant inclusive single-particle/jet cross sections in p–p collisions can be factorized in terms of two separable <inline-formula></inline-formula> dependences, a <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>p</mi><mi>T</mi></msub><mo>−</mo><msqrt><mi>s</mi></msqrt><mo stretchy="false">]</mo></mrow></semantics></math></inline-formula> sector and an <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>x</mi><mi>R</mi></msub><mo>−</mo><msub><mi>p</mi><mi>T</mi></msub><mo>−</mo><msqrt><mi>s</mi></msqrt><mo stretchy="false">]</mo></mrow></semantics></math></inline-formula> sector. Here, we extend our earlier work by analyzing more extensive data to explore various s-dependent attributes and other systematics of inclusive jet, photon and single particle reactions. Approximate power laws in <inline-formula><math display="inline"><semantics><mrow><msqrt><mi>s</mi></msqrt></mrow></semantics></math></inline-formula>,<inline-formula><math display="inline"><semantics><mrow><msub><mi>p</mi><mi>T </mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>x</mi><mi>R </mi></msub></mrow></semantics></math></inline-formula> are found. Physical arguments are given which relate observations to the underlying physics of parton–parton hard scattering and the parton distribution functions in the proton. We show that the <inline-formula><math display="inline"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><msqrt><mi>s</mi></msqrt><mo>,</mo><msub><mi>p</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> function, introduced in our earlier publication to describe the <inline-formula><math display="inline"><semantics><mrow><msub><mi>p</mi><mi>T</mi></msub></mrow></semantics></math></inline-formula> dependence of the inclusive cross section, is directly related to the underlying hard parton–parton scattering for jet production, with little influence from soft physics. In addition to the <i>A-</i>function, we introduce another function, the <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msqrt><mi>s</mi></msqrt><mo>,</mo><msub><mi>x</mi><mi>R</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> function that obeys radial scaling for inclusive jets and offers another test of the underlying parton physics. An application to heavy ion physics is given, where we use our variables to determine the transparency of cold nuclear matter to penetrating heavy mesons through the lead nucleus.