Derivation 2.3A — Structural uniqueness of the directional shell readings
theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology
Purpose
This derivation supports the claim, referenced from Foundations §2.3, that the realized directional shell readings
\[ p_k^+=M+\tfrac12 p_k, \qquad p_k^-=M-\tfrac12 p_k, \]
are not an arbitrary choice within the natural class of additive, symmetry-compatible axis-wise two-reading encodings.
The result proved here is deliberately precise in scope. We show that, under mild regularity, additivity and directional-reflection symmetry force the two readings to take an affine form
\[ F_\pm(M,p)=\alpha M \pm \beta p, \]
and that the canonical M1 realization is obtained by the normalization that identifies the recovered conserved quantities with the already named \(M\) and \(p_k\). Here \(F_+\) and \(F_-\) are proof labels for evaluations in opposed orientations, not new physical momentum variables.
This is therefore an affine uniqueness theorem. It does not prove that every imaginable nonlinear realization of the directional shell-reading structure is impossible.
It also does not cover quadratic, flux-like source-channel constructions such as null contractions of a rank-two stress tensor. Those live in a different mathematical class: this appendix constrains linear charge-like readings of conserved vector content — a class that the gravitational carrier source map of Derivation 3.3A sits squarely inside, being the shell readings themselves densitized — while the rejected null-probe diagnostic of Derivation 3.3B works with quadratic stress-tensor contractions.
Role / regime / status
- Role: derivation
- Regime: isolated systems, inertial kinematic level, one fixed chosen orientation \(\hat{k}\)
- Status: proved within the additive, symmetry-compatible, regular axis-wise class stated below
- Unresolved stronger claim: global uniqueness beyond the affine/additive class remains open
Setup
Fix an oriented unit direction \(\hat{k}\). For readability write
\[ p:=p_k. \]
Consider an axis-wise two-reading encoding that assigns to each pair \((M,p)\) two real numbers
\[ F_+(M,p), \qquad F_-(M,p), \]
interpreted as candidate evaluations in the orientations \(\hat{k}\) and \(-\hat{k}\). They are placed side by side here to test invertibility; they are not two simultaneous particle contributions within one ADMC reading.
Define the sum and difference combinations
\[ \Sigma(M,p):=F_+(M,p)+F_-(M,p), \]
\[ \Delta(M,p):=F_+(M,p)-F_-(M,p). \]
For the canonical M1 realization these become
\[ \Sigma=2M, \qquad \Delta=p. \]
The aim is to determine how much freedom is left if the encoding behaves like a genuine additive directional bookkeeping map rather than an arbitrary relabeling.
Assumptions
Assumption A1: axis-wise additivity
For each fixed orientation and each sign label,
\[ F_\pm(M_1+M_2,\,p_1+p_2) = F_\pm(M_1,p_1)+F_\pm(M_2,p_2) \]
for all pairs \((M_1,p_1)\) and \((M_2,p_2)\) in the domain.
Equivalently, \(\Sigma\) and \(\Delta\) are additive.
This is the structural assumption that the shell readings are true additive bookkeeping variables for isolated-system composition.
Assumption A2: parity exchange
Reversing the sign of the directional component exchanges the two orientation labels:
\[ F_+(M,-p)=F_-(M,p), \qquad F_-(M,-p)=F_+(M,p). \]
Equivalently, \(\Sigma(M,p)\) is even in \(p\) and \(\Delta(M,p)\) is odd in \(p\).
Assumption A3: axis-independence
The functional form of the encoding does not depend on which axis is chosen. Thus the same functions \(F_\pm:\mathbb R^2\to\mathbb R\) apply for every oriented axis.
Assumption A4: local invertibility
There exists at least one point \((M_0,p_0)\) with \(M_0>0\) such that the map
\[ (M,p) \mapsto (F_+(M,p),F_-(M,p)) \]
is locally one-to-one in a neighborhood of \((M_0,p_0)\).
This assumption excludes degenerate encodings that collapse two degrees of freedom into one.
Assumption A5: mild regularity
At least one of the additive functions \(F_+\), \(F_-\) — equivalently \(\Sigma\) or \(\Delta\) — is measurable on a set of positive measure, or continuous at one point.
This is a standard regularity hypothesis used to exclude pathological solutions to the Cauchy functional equation (Kuczma 2009).
Main theorem
Theorem 1 (affine form is forced)
Under Assumptions A1–A5, there exist constants \(\alpha,\beta\in\mathbb R\) such that
\[ F_\pm(M,p)=\alpha M \pm \beta p \]
for all \((M,p)\) in the domain.
Moreover, Assumption A4 implies
\[ \alpha \neq 0, \qquad \beta \neq 0. \]
Proof
By Assumption A1, each of \(F_+\) and \(F_-\) satisfies the additive functional equation on \(\mathbb R^2\):
\[ F_\pm\big((M_1,p_1)+(M_2,p_2)\big) = F_\pm(M_1,p_1)+F_\pm(M_2,p_2). \]
By Assumption A5, additive functions on \(\mathbb R^2\) with this mild regularity are linear. Therefore there exist constants \(a_\pm,b_\pm\in\mathbb R\) such that
\[ F_\pm(M,p)=a_\pm M + b_\pm p. \]
Now impose Assumption A2. From parity exchange,
\[ F_+(M,-p)=F_-(M,p) \]
for all \((M,p)\). Substituting the linear forms gives
\[ a_+ M - b_+ p = a_- M + b_- p. \]
Because this identity holds for all \(M\) and \(p\), coefficient comparison yields
\[ a_+=a_-=:\alpha, \qquad b_-=-b_+=:-\beta. \]
Hence
\[ F_\pm(M,p)=\alpha M \pm \beta p. \]
It remains to use local invertibility. If \(\alpha=0\), then
\[ F_+(M,p)+F_-(M,p) \equiv 0, \]
so the reading pair cannot recover any variation in \(M\); the map cannot be locally one-to-one. If \(\beta=0\), then
\[ F_+(M,p)-F_-(M,p) \equiv 0, \]
so the pair cannot recover any variation in \(p\); again local invertibility fails. Therefore \(\alpha\neq0\) and \(\beta\neq0\). ∎
Interpretation of Theorem 1
Theorem 1 is the core structural result. It says that once the encoding is required to be
- additive,
- compatible with directional reversal,
- axis-independent,
- locally invertible,
- and regular enough to exclude pathological additive functions,
then no nonlinear or asymmetrically mixed dependence on \((M,p)\) survives. The entire admissible class collapses to the two-parameter affine family
\[ F_\pm(M,p)=\alpha M \pm \beta p. \]
At this stage the coefficients are still undetermined. That remaining freedom is normalization freedom, not structural freedom.
Corollary 1 (normalization class)
Under the assumptions of Theorem 1, every admissible encoding is of the form
\[ F_\pm(M,p)=\alpha M \pm \beta p \]
with \(\alpha\neq0\) and \(\beta\neq0\).
Consequently,
\[ \tfrac12(F_+ + F_-) = \alpha M, \qquad F_+ - F_- = 2\beta p. \]
So the affine family contains a global scaling freedom for the sum combination and an independent relative scaling for the difference combination.
Proof
Substitute the form from Theorem 1 directly into the sum and difference combinations. ∎
Canonical normalization
To recover the canonical M1 variables without rescaling, one imposes the normalization
\[ M = \tfrac12(F_+ + F_-), \qquad p = F_+ - F_-. \]
Substituting the affine family into these identities gives
\[ \alpha M = M, \qquad 2\beta p = p, \]
hence necessarily
\[ \alpha = 1, \qquad \beta = \tfrac12. \]
This yields the canonical directional shell readings.
Corollary 2 (canonical realization)
If, in addition to Assumptions A1–A5, one requires that the recovered additive quantities are exactly the already named variables \(M\) and \(p\), namely
\[ M = \tfrac12(F_+ + F_-), \qquad p = F_+ - F_-, \]
then the encoding is uniquely fixed as
\[ F_\pm(M,p)=M\pm\tfrac12 p. \]
Equivalently, reinstating the directional component and physical reading labels,
\[ p_k^+=M+\tfrac12 p_k, \qquad p_k^-=M-\tfrac12 p_k. \]
Proof
This is the canonical normalization just derived, written as a uniqueness statement. ∎
Positivity and sign convention
If one additionally requires
\[ F_\pm(M,p) \ge 0 \]
on the physical domain, then positivity does not determine the coefficients by itself, but it does select the physical sign convention.
For the canonical normalization,
\[ F_\pm(M,p)=M\pm\tfrac12 p, \]
so positivity is automatic on the physical domain \(M\ge|p|\). The sign of the \(+\) reading is fixed by the convention that increasing \(p\) along the chosen orientation increases \(F_+\) and decreases \(F_-\).
What has and has not been proved
The status of the result should be stated carefully.
Proved here
Within the additive, symmetry-compatible, axis-wise class defined by Assumptions A1–A5,
- the reading encoding must be affine,
- local invertibility removes the degenerate cases \(\alpha=0\) and \(\beta=0\),
- canonical normalization uniquely fixes the coefficients to \(\alpha=1\) and \(\beta=1/2\).
Not proved here
This derivation does not establish that every sufficiently reasonable realization of the directional shell-reading structure must satisfy Assumption A1 from first principles. It also does not exclude every conceivable nonlinear encoding outside the additive class.
So the honest conclusion is:
the canonical directional shell readings are structurally unique within the natural additive and symmetry-compatible axis-wise encoding class.
That is already a strong result, but it is weaker than a global no-alternative theorem.
Result
The directional shell readings used in M1,
\[ p_k^+=M+\tfrac12 p_k, \qquad p_k^-=M-\tfrac12 p_k, \]
are the unique canonically normalized members of the affine family forced by axis-wise additivity, parity exchange, axis-independence, local invertibility, and mild regularity.
For a technically strong reader outside the project, the correct interpretation is therefore not that the form was guessed and then merely preferred, but that it is tightly constrained once one decides that directional shell readings are genuine additive conserved quantities attached to a chosen orientation.
Notes for citation in the main text
If Foundations §2.3 cites this appendix, the safest summary sentence is:
Within the natural additive and symmetry-compatible class of axis-wise opposed-reading encodings, the shell readings are forced to the affine family \(F_\pm=\alpha M\pm\beta p\), and canonical normalization fixes uniquely the realized form \(p_k^+=M+\tfrac12p_k\) and \(p_k^-=M-\tfrac12p_k\).