The mathematics of PDT | Phase Differential Theory

Axioms

These are concise mathematical summaries of the axiom account in R01. Each axiom states what it supplies and what it does not.

A0

Distinction

There is a set $X$ with at least two elements.

This supplies distinctions. It does not include a distance, clock or spatial embedding.

A1

Phase

U(1) = \{\, z \in \mathbb{C} : |z| = 1 \,\}

$U(1)$ is the circle group: unit complex numbers under multiplication, with the usual topology of the circle. This phase structure is adopted in the foundations.

A2

Coherent comparison

Let $G$ be a small groupoid with object set $X$ . Its arrows $a : x \to y$ are comparisons. Every object has an identity arrow $\mathrm{id}_x$ , every arrow has an inverse $a^{-1}$ , and composition $b \circ a$ (first $a$ , then $b$ ) is associative wherever it is defined.

Let $BU(1)$ be the groupoid with a single object whose arrows are the elements of $U(1)$ . A phase assignment is a functor $\Phi : G \to BU(1)$ , so for composable arrows $a$ and $b$ :

\begin{gathered} \Phi(b \circ a) = \Phi(b)\,\Phi(a) \\ \Phi(\mathrm{id}_x) = 1 \\ \Phi(a^{-1}) = \Phi(a)^{-1}. \end{gathered}

Different full transport arrows can have the same scalar phase. Faithfulness of the whole functor is not assumed.

A3

Faithful phase encoding

\begin{gathered} U(1) \;\cong\; \mathbb{R}/2\pi\mathbb{Z} \\ e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)} \end{gathered}

Multiplication of phases corresponds to addition of angles modulo $2\pi$ . This faithfully encodes phase elements. It does not assign a distinct scalar angle to every full transport arrow.

A3 is a retained encoding convention already available from A1.

A4

Minimal real phase carrier

Within the class of faithful continuous real linear representations of $U(1)$ , select the smallest nonzero finite-dimensional carrier. The minimum real dimension is two (see the minimal phase carrier).

The representation class is an explicit choice. This is a mathematical phase carrier, not a derivation of physical spatial dimension.

A5

Retained history

\begin{gathered} h_n = (r_1, \dots, r_n) \\ h_{n+1} = (r_1, \dots, r_n, r_{n+1}) \end{gathered}

Here $r_k$ is the $k$ th record and $n$ the record count. An admissible append preserves the existing prefix. The model must specify which extensions are allowed.

Retained historical identity is a feature of the mathematical description. The physical reliability of a memory device is a separate question for its dynamics.

A6

Last coherent survivor realisation

A model specifies a finite, exhaustive family of mutually exclusive candidate records and an admissibility law determined from antecedent inputs. Let $C(s)$ be the candidates still admissible at episode parameter $s$ , and $|C(s)|$ the number remaining. Then

\begin{gathered} C(s_2) \subseteq C(s_1) \quad \text{whenever } s_2 \ge s_1 \\ s^* = \min\{\, s : |C(s)| = 1 \,\}. \end{gathered}

The minimum must exist, be attained at a finite stage and occur before extinction. The sole candidate is then taken to be realised and appended to the history, and the episode stops at that event.

The record count $n$ , the episode parameter $s$ and calibrated physical time $t$ are distinct.

Note on FCS

Finite Canonical Selection supplies a nonempty finite admissible set invariant under a specified symmetry. It is an additional assumption when used. It does not necessarily select one member.

SourceR01 paper source

Phase composition

Objects and domain

A graph with vertices in $X$ , its path groupoid, and phases $z_{x\to y} \in U(1)$ on oriented edges, with $z_{y\to x} = z_{x\to y}^{-1}$ .

Restricted case: vertex phases

If each vertex carries an angle $\theta_x$ and

z_{x\to y} = e^{i(\theta_y - \theta_x)},

then around any closed loop the angle differences telescope:

z_{A\to B}\,z_{B\to C}\,z_{C\to A} = e^{i[(\theta_B-\theta_A)+(\theta_C-\theta_B)+(\theta_A-\theta_C)]} = 1.

Worked example: a triangle

Mathematical example. Oriented A → B → C → A.

z_{A\to B} = 1,\quad z_{B\to C} = 1,\quad z_{C\to A} = e^{i\pi/3}

z_{A\to B}\,z_{B\to C}\,z_{C\to A} = e^{i\pi/3} \neq 1

The triangle is a graph-path example; its loop is not being imposed as an identity relation. Because its loop phase is not 1, no assignment of vertex angles can reproduce these edge phases.

Interpretation and scope

Scalar phase records part of the transport information. Full transport can contain additional, noncommuting structure.

SourceR02 paper source

The minimal phase carrier

Objects and assumptions

Continuous real linear representations of $U(1)$ on finite-dimensional real vector spaces, required to be faithful (distinct phases act differently).

Statement

The minimum real dimension of a faithful carrier is two, realised by rotations:

\begin{gathered} R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \\ R(\theta_1 + \theta_2) = R(\theta_1)\,R(\theta_2). \end{gathered}

\begin{gathered} J = R(\pi/2) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \\ J^2 = -I. \end{gathered}

$I$ is the $2\times 2$ identity matrix. Since $J^2 = -I$ , $J$ acts like multiplication by $i$ , supplying a complex structure on the selected real plane.

Proof outline

A continuous one-dimensional real representation sends the connected compact circle group into the multiplicative group of nonzero reals. Its image is compact and connected and contains 1, so it is $\{1\}$ : every phase acts trivially and the representation is not faithful. Ordinary rotations $R(\theta)$ supply a faithful two-dimensional representation.

Scope

This establishes the minimal carrier within the stated representation class. Physical quantum states, composite systems, a Hamiltonian and physical spatial dimension require further premises.

Source · Theorem R02.2R02 paper source Theorem R02.2 statement

Last-survivor selection

Objects and assumptions

A finite candidate set with at least two members, each candidate $r$ assigned a positive finite loss time $T_r$ , and

\begin{gathered} C(s) = \{\, r : s < T_r \,\} \\ s \ge 0. \end{gathered}

Statement

A singleton occurs before extinction exactly when the largest loss time is unique. The first singleton occurs at the second-largest loss time.

Short proof

At the second-largest loss time, every candidate except the unique largest has been excluded, while the largest is still present. At every earlier stage, at least the two longest-lasting candidates remain. If the largest loss time is shared, those candidates leave together and the set passes from two or more members straight to none.

Worked example

T_A = 2, \quad T_B = 1, \quad T_C = 3

Stage $s$	$C(s)$
$0 \le s < 1$	$\{A, B, C\}$
$1 \le s < 2$	$\{A, C\}$
$s = 2$	$\{C\}$ : first singleton, A6 event, episode stops

C is the first unique survivor at $s = 2$ , the second-largest loss time. This is the same sequence as the worked example on the theory page: B leaves at stage 1, A at stage 2, and C is recorded.

Failure cases

Tie. If $T_A = T_B$ is the largest loss time, A and B disappear together; $|C(s)|$ never equals 1 and there is no A6 event.

No termination. If $C(s) = \{A, B\}$ for every finite $s$ , no singleton is reached and no record is made.

An infimum of candidate event times does not establish a first attained event unless the required minimum actually exists.

Three separate things

The theorem identifies a singleton in a supplied process.
A6 adopts that singleton as realised.
The physical explanation of the supplied loss or admissibility law is a further task for the model.

Source · Theorem R03.1R03 paper source Theorem R03.1 statement

Quantum probabilities and instruments

Objects

A finite-dimensional complex Hilbert space $\mathcal{H}$ . A density operator $\rho$ satisfies

\begin{gathered} \rho \ge 0 \\ \operatorname{Tr}(\rho) = 1. \end{gathered}

Positivity ( $\ge 0$ ) means no outcome is ever assigned negative weight; the trace $\operatorname{Tr}$ sums the diagonal entries, so total weight is 1. A measurement is a family of effects

\begin{gathered} E_r \ge 0 \\ \sum_r E_r = I \end{gathered}

where $I$ is the identity operator, which leaves every vector unchanged: the outcomes together exhaust all possibilities.

Assumption Q1

The full effect space has a physical interpretation, and the probability assigned to an effect is independent of which complete measurement contains it. Probabilities normalise across each such measurement.

Statement

p(r) = \operatorname{Tr}(\rho E_r)

This is the finite effect probability theorem used in R04, Theorem R04.1, which rests on established effect-theorem mathematics.

Instruments

For a supplied instrument with Kraus operators $K_{r\alpha}$ , where $\alpha$ indexes operators associated with the same recorded outcome $r$ and $\dagger$ denotes the adjoint (conjugate transpose):

\begin{gathered} \mathcal{I}_r(\rho) = \sum_\alpha K_{r\alpha}\,\rho\,K_{r\alpha}^{\dagger} \\ E_r = \sum_\alpha K_{r\alpha}^{\dagger}K_{r\alpha} \end{gathered}

\begin{gathered} p(r) = \operatorname{Tr}[\mathcal{I}_r(\rho)] \\ \rho_r = \frac{\mathcal{I}_r(\rho)}{p(r)} \quad (p(r) > 0). \end{gathered}

The effects determine outcome probabilities. The instrument also specifies the conditional state after an outcome.

Worked example (illustration of the specified quantum model)

|\psi\rangle = \tfrac{\sqrt{3}}{2}\,|0\rangle + \tfrac12\,|1\rangle

\begin{gathered} p(0) = \left(\tfrac{\sqrt3}{2}\right)^2 = \tfrac34 \\ p(1) = \left(\tfrac12\right)^2 = \tfrac14 \\ \tfrac34 + \tfrac14 = 1. \end{gathered}

Under the projective instrument in this basis, the conditional states are $|0\rangle$ and $|1\rangle$ .

Reversible dynamics

Theorem R04.2: a channel with a completely positive, trace-preserving inverse on the same finite sector is unitary. A map is completely positive when it keeps states positive even with an untouched ancillary quantum system included alongside.

Under the additional continuous-time premise, with an identified physical clock and action scale $\hbar$ , and with $\hat H$ the Hermitian Hamiltonian (distinct from the Hilbert space $\mathcal{H}$ ):

\begin{gathered} U(t) = e^{-i\hat{H}t/\hbar} \\ i\hbar\,\partial_t|\psi\rangle = \hat{H}|\psi\rangle. \end{gathered}

Scope

The physical Hilbert space, Hamiltonian and instrument are supplied through the model’s premises.

Source · Theorems R04.1, R04.2R04 paper source R05 paper source R06 paper source Theorem R04.1 statement Theorem R04.2 statement

The exclusion construction

Objects and assumptions

Nonzero mutually orthogonal projectors $P_j$ with $\sum_j P_j = I$ . For an active label set $S$ with $m \ge 2$ members, $P_S = \sum_{j\in S} P_j$ .

Construction

For states supported in $P_S$ , the operator recording exclusion of $r$ is

\begin{gathered} K_{S,r} = \frac{P_S - P_r}{\sqrt{m-1}} \\ \sum_{r\in S} K_{S,r}^{\dagger}K_{S,r} = P_S. \end{gathered}

With current populations $p_r = \operatorname{Tr}(\rho P_r)$ , the probability of excluding $r$ is

\Pr(\text{exclude } r) = \frac{1 - p_r}{m - 1}.

Every positive-probability history remains nonempty and reaches one survivor after $m-1$ exclusions. The terminal operation for survivor $j$ is $\rho \mapsto P_j\rho P_j$ , and the survivor probability is $\operatorname{Tr}(\rho P_j)$ .

Why the operators sum to the projector

Each $P_S - P_r$ is a projector, so $K_{S,r}^\dagger K_{S,r} = (P_S - P_r)/(m-1)$ . Summing over the $m$ labels gives $(mP_S - P_S)/(m-1) = P_S$ .

Worked example: three labels

\begin{gathered} p_A = \tfrac12 \\ p_B = \tfrac13 \\ p_C = \tfrac16 \end{gathered}

First excluded	Then	Survivor	Probability
B	C	A	$\tfrac14$
C	B	A	$\tfrac14$
A	C	B	$\tfrac16$
C	A	B	$\tfrac16$
A	B	C	$\tfrac1{12}$
B	A	C	$\tfrac1{12}$

\tfrac14+\tfrac14+\tfrac16+\tfrac16+\tfrac1{12}+\tfrac1{12} = 1

Survivor totals: A $\tfrac14+\tfrac14=\tfrac12$ , B $\tfrac16+\tfrac16=\tfrac13$ , C $\tfrac1{12}+\tfrac1{12}=\tfrac16$ , reproducing the initial populations.

Example: the history B then C

Excluding B first has probability $(1-\tfrac13)/2 = \tfrac13$ . The populations in $\{A, C\}$ become $\tfrac34$ and $\tfrac14$ , so excluding C next has probability $(1-\tfrac14)/1 = \tfrac34$ . The history probability is $\tfrac13\cdot\tfrac34 = \tfrac14$ .

Scope

This construction represents a supplied quantum instrument through a last-survivor process. Its agreement with the supplied outcome probabilities does not independently select nature’s instrument.

Theorem R05.4: arbitrarily small unrestricted full-rank perturbations can remove exact zero effects. Exact exclusion and operationally reliable records therefore require different claims and tests. A small positive population rounded to zero is not a physical realisation.

Source · Theorems R05.1, R05.4R05 paper source Theorem R05.1 statement Theorem R05.4 statement

Physical geometry and gravity

Geometry

An internal positive metric on a mathematical carrier is distinct from a physical Lorentzian spacetime. The comparison seed and the physical displacement interpretation are supplied premises.

SourceR07 paper source

Gravity: premises

G1: a common four-dimensional Lorentzian metric $g_{\mu\nu}$ .
G2: a local diffeomorphism-invariant pure-metric action with at most two derivatives in total, modulo boundary terms, and a positive kinetic coefficient.
G3: a matter action and total stress coupled to that metric.

Statement (units with c = 1)

Theorem R10.1 classifies the actions in this declared class. The resulting field equations are

\begin{gathered} G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa\, T_{\mu\nu} \\ \nabla^{\mu} T_{\mu\nu} = 0. \end{gathered}

$g_{\mu\nu}$ is the metric; $G_{\mu\nu} = R_{\mu\nu} - \tfrac12 R\, g_{\mu\nu}$ is the Einstein tensor built from its curvature; $\Lambda$ is the cosmological constant; $\kappa$ is the gravitational coupling; $T_{\mu\nu}$ is the total stress tensor; $\nabla^\mu$ is the covariant divergence.

$\kappa$ and $\Lambda$ are not numerically selected by this classification. A claimed common matter, record and gravity model must account for apparatus, reservoir and record contributions to $T_{\mu\nu}$ consistently.

Source · Theorem R10.1R10 paper source Theorem R10.1 statement

Cosmology

R11 supplies a homogeneous geometry, matter content and initial data. Dilution and expansion claims require a defined stress tensor. Under separately conserved, isotropic, traceless stress, a component behaves like radiation; that same model cannot simply be renamed cold dark matter or dark energy.

SourceR11 paper source

Joint physical history

R12 requires one common model. Separate solutions in different sectors do not by themselves establish a shared physical solution.

SourceR12 paper source

Clocks and calibration

Record order alone does not determine durations or rates. A physical clock requires a clock map, identifying a process with measured duration, together with calibration inputs. The record count $n$ , episode parameter $s$ and clock time $t$ remain distinct.

Using the electron mass as an empirical anchor fixes a scale. It does not predict that mass or determine dimensionless constants.

SourceR08 paper source

Sources and verification

Every result above summarises the R00 to R13 current foundations, which give the complete arguments. Theorem identifiers cited: R02.2, R03.1, R04.1, R04.2, R05.1, R05.4 and R10.1. Theorem links lead to the result statement on each foundational paper page. Full proofs are in the manuscripts. Manuscript PDFs for every paper in the suite are available on this website. Open the manuscripts.

The worked examples were checked by direct calculation: the vertex-phase loop product is 1; the triangle loop product is $e^{i\pi/3}$ ; the rotation matrices compose and $J^2 = -I$ ; the loss-time example selects C at stage 2; the tie and non-terminating cases produce no A6 event; the qubit probabilities sum to 1; the six exclusion histories sum to 1 and reproduce the survivor totals. These checks verify the examples only. They are not experimental confirmation or formal proof-assistant verification.

A $\text{conditional theorem}$ is proved to hold whenever its stated assumptions hold; the assumptions themselves are not thereby shown to describe nature.

Current foundations Glossary Concept explorer

The mathematicsof PDT.

Axioms

Distinction

Phase

Coherent comparison

Faithful phase encoding

Minimal real phase carrier

Retained history

Last coherent survivor realisation

Phase composition

The minimal phase carrier

Last-survivor selection

Quantum probabilities and instruments

The exclusion construction

Physical geometry and gravity

Clocks and calibration

Sources and verification

The mathematics
of PDT.