Configurational Mass: Two Readings of the Same Integer
by Pedro Henrique Corrêa Garcia · Opus 4.7 · 5 pp
Page 1 of 5· 5 stated
Configurational Mass:
Two Readings of the Same IntegerPedro Henrique Correa Garcia
Joinville, Santa Catarina, Brazil
garcia.pedro.wow@gmail.com
· April 2026Abstract
The rung-number of a configuration admits two simultaneous
readings: an additive reading as a count, and a
multiplicative reading as an archival coordinate (prime
factorization). The two readings are not alternatives. They are
simultaneous faces of the same integer, and the consequences of
holding them together are qualitatively different from the
consequences of taking either alone. This article is an
arithmetic statement about pairs of readings on a single
positive integer. We call the pair configurational mass
and we develop its immediate structural consequences. We make no
empirical claim.
1IntroductionEvery positive integer A admits two readings. The first is
additive: A is a count, a number of unit constituents,
reachable from zero by A applications of the successor map.
The second is multiplicative: A is the product of a unique
sequence of primes, reachable from the empty product by a finite
tower of prime applications. Both readings exist for every
positive integer, and both are forced by the Fundamental Theorem
of Arithmetic, which asserts that the two factorizations
(additive into units, multiplicative into primes) are each unique
up to reordering.In the framework of this series, the two readings are not just
two ways of writing the same integer. They are the two
projections of a rung Rn onto the two layers of the
manifold. The additive reading projects onto the local layer of
Article 5; the multiplicative reading projects onto the archival
layer of Article 6. We call the pair configurational mass
of the rung and we show that both projections are necessary.Unrecognized environment: keyideaAn integer has two faces: an additive face that counts and a multiplicative face that locates. The framework uses both.RemarkThe additive/multiplicative distinction drawn here is not
peculiar to rung arithmetic. It is the ladder-level appearance
of a deeper split in the Dedekind lattice: the counting layer
and the gear layer, which are forced apart at
D(3) = 20 = P12 · P3 by the first non-squarefree
Dedekind factorization (see Article 2, the second-invariant
section on the ratio 1/5 at D(3), and the parallel remark
in Article 9). The multiplicative form lives in the
gear layer, which admits only coprime rotor combinations. The
additive form lives in the counting layer, which
accommodates non-coprime stackings such as P12,
P13, or P22. The reason both forms are needed
is that the Dedekind lattice forces both layers to exist from
k = 3 onward, and any integer with a non-squarefree
factorization necessarily participates in both.2The additive form: DefinitionThe additive reading of a positive integer A is the
non-negative integer obtained by counting the unit contributions
that reach A from zero under successor. We write this reading
as (A) = A and understand it as the local-layer projection
of A.The additive form is local (it depends only on the configuration
itself), it is measurable by counting alone, and it is linear
(it adds under direct sum of configurations). These are the
properties that make the additive form computationally easy to
work with. They are also the properties that make it
incomplete: the additive form sees no prime structure, no
polarity matrix, and no ceiling.3The multiplicative form: