Physical Formulas Are Not Written Into Nature Quantities and Formulas Both Emerge From Comparing Binary Referential Relations: Measurement, Comparison and Formula in RECT — Distance Is Not Special, and the Relational Foundation Is Invariant

A common picture of physics treats its formulas as laws already written into nature.On this picture physics only has to read the laws off. Physical quantities areproperties carried by systems, and distance is the special quantity supplied by ametric on a pre-given spatial or spacetime manifold. Reference Object EntropyCycle Theory (RECT) reverses both orders. Every physical quantity is a reading of arelation between reference structures, and distance is not a privileged case. Everyphysical formula is obtained by comparing such quantities. Formulas are notpresupposed.The paper develops one account in three layers. (1) Measurement. Referencestructures measure one another. No prior choice of a pair or a standard is needed.Each structure can measure the shared relation at the same time, taking itself as itsown baseline. Because the measured relation is one and the same, comparing thereadings gives the conversion relation between the standards, . Ameasured number together with its unit is a physical quantity. (2) Comparison.Horizontal comparison across the dimensional expressions of the relations findswhich relations stay invariant and which change, and how their changes gotogether. (3) Formula. When the corresponding comparison relations areorganized, a structure forms naturally; that structure is the formula, and it is whatwe call a mathematical structure or mathematical formula. If speed is found to stayconstant, the change of distance can be known from the change of time, and follows. Other formulas are obtained in the same way, and so are the constants theycarry. Logic itself also arises in the comparison of different physical quantities.Since comparison needs at least two quantities, logic, and therefore every formula,requires .The measurement layer rests on four results. First, the relation count is complete.For reference structures, every binary relation that can be specified between twochosen structures is an element of the complete pair domain, of size. Second, distance is not special. Different physical quantities aredifferent dimensional expressions of the same binary relation, and dimension isequivalent to physical quantity. Third, volume emerges, and area and length are itsscales, emerging together with it. "Volume first" is a mathematical order ofdescription, and physically the three are the same thing seen from different angles.Two reference structures already give volume, area and length, and the number ofreference structures is not a dimension, . Fourth, when the geometricexpression changes, the relational foundation does not change, and the values of therelations are settled by the metric, that is, by measurement.The stability of effective quantities is a question of which dimensional expressionsthe dynamics of the referential system selects and maintains. Through comparison,that stability and change become knowable. Concepts, definitions, laws andconstants all originate in comparison. Mainstream physics calls the definitionof speed. RECT adds that even a definition comes from comparison, because withoutcomparison time, distance and speed could not be distinguished as concepts at all.Constants in formulas, such as and , likewise come from comparison. Oneelaboration is proposed for the author's confirmation: dimensional constants areread as conversion coefficients between standards, which is how comparison yieldsthem. The official structural-constraint axiom, , is kept. The paper states itsstrongest prediction: by comparing physical quantities with one another, newphysical relations and physical quantities that are as yet unknown will necessarilybe obtained. It ends with a proposed formal formulation, limitations, conditionalchecks and a list of open decisions.