Glossary

The Three Axioms

Axiom A1 — Existence is relational

There is no absolute position. Every crossing event is defined relative to all others. No frame is preferred. This is not an assumption about spacetime — it is a prior condition on what it means for something to exist at all. Consequence: in the continuum limit, the bilateral amplitude \(\langle A \rangle_\gamma = 2\) for all Lorentz factors \(\gamma\), giving Lorentz invariance exactly.

Axiom A2 — No crossing is preferred

All bilateral crossings are equivalent under the bilateral group. The unique unitary operator satisfying A1 and A2 simultaneously is \(U_\times = i\sigma_x\) — the bilateral crossing operator. This forces Haar measure on the space of crossing operators, which gives GUE (Gaussian Unitary Ensemble) level statistics. Consequence: \(P(s=0) = 0\) exactly — no two eigenvalues can coincide, no gap can close, no singularity can form. The Möbius reflection \(s \mapsto 1-s\) is the bilateral statement that neither vacuum (\(\phi = -1\) and \(\phi = +1\)) is preferred. The critical line \(\mathrm{Re}(s) = \tfrac{1}{2}\) is the fixed locus of this reflection — the only symmetric position.

Axiom A3 — \(\tau\) is monotonically increasing

The becoming-time field \(\tau\) accumulates strictly at each bilateral crossing: \(\tau \mapsto \tau + \delta\tau\) with \(\delta\tau > 0\). Becoming-time only moves forward. This is not an assumption about the arrow of time — it is what makes time a direction at all. Consequence: the metric signature is \((-,+,+,+)\), the causal light cone is well-defined, and the bilateral field equation \(\Box\tau = -\rho\) recovers general relativity in the weak-field limit.

Core Terms

0 (Zero)

The substrate. Not the number zero. The prior condition — what everything is before it has a label. Everything is 0 with a label. 0 has no outside. Labels cannot escape 0. 0 is not a floor you reach by subdividing — it is prior to subdivision. You cannot reach 0 from inside the labels. You are already in 0.

Actual

What is present at the current crossing. Only the present moment is actual. The past has crossed and become potential. The future has not yet crossed and is potential. Only the present crossing is actual.

See also: Potential, Crossing, Becoming-Time Field

Annihilation Locus

The critical line \(\mathrm{Re}(s) = \tfrac{1}{2}\) — where the bilateral mesh self-intersects, where ingress meets egress, where the two faces of the crossing agree. The Riemann zeros lie on the annihilation locus. The Riemann Hypothesis — that all non-trivial zeros lie on the critical line — is, in the bilateral framework, a consequence of A2: the only position consistent with no preferred crossing is the midpoint.

Becoming-Time Field (\(\tau\))

The field that accumulates at each bilateral crossing. Not clock time — becoming-time is the record of crossing events. \(\tau\) is strictly increasing by A3: each crossing adds \(\delta\tau > 0\). The arrow of time is the arrow of \(\tau\) accumulation. Time does not reverse because \(\tau\) does not decrease. The bilateral field equation \(\Box\tau = -\rho\) is the dynamical equation for \(\tau\), giving general relativity in the weak-field limit.

Bilateral

Two faces of one thing. Not two separate things. A Möbius strip is bilateral — one surface with two faces. Every crossing has an ingress face and an egress face. Neither face is complete alone. Together they constitute the crossing. The bilateral structure generates the Möbius reflection \(s \mapsto 1-s\) whose fixed locus is the critical line \(\mathrm{Re}(s) = \tfrac{1}{2}\).

Bilateral Crossing Operator (\(U_\times\))

The unique unitary operator satisfying A1 (relational) and A2 (no preferred crossing): \(U_\times = i\sigma_x = i\begin{pmatrix}0&1\\1&0\end{pmatrix}\). Acting on a bilateral spinor, \(U_\times\) generates the bilateral group. The phase after \(n\) successive crossings is \(i^n\), cycling with period 4. The crossing amplitude \(A_n = |1-i^n|^2 \in \{2,4,2,0,...\}\) gives the bilateral selection rule: modes with \(n \equiv 0 \pmod{4}\) decouple.

Bilateral Mesh

The integer lattice with its bilateral crossing structure — the full geometry generated by the three axioms. The bilateral mesh is not a model of the world. It is the structure that existence must have if A1, A2, and A3 hold. All Standard Model content, spacetime geometry, and spectral properties are derived from this structure with no free parameters.

Crossing

The fundamental event of the bilateral mesh. A crossing is where the bilateral structure self-intersects — where \(s_0\) and \(1-s_0\) meet, where the two faces agree, where the actual blooms from the potential. Every physical event is a crossing. The crossing is the present. The crossing operator \(U_\times = i\sigma_x\) is the unique operator for which all crossings are equivalent (A2).

Egress

The outward face of the bilateral crossing — the actual, the classical, the positive-\(\tau\) side. Where potential has actualised. Paired with ingress. The egress vacuum is \(\phi = +1\) in the bilateral \(\phi^4\) field; the ingress vacuum is \(\phi = -1\). The kink soliton connecting the two vacua is the bilateral particle.

See also: Ingress, Bilateral, Crossing

Fixed Locus

The set of points fixed by the bilateral reflection \(s \mapsto 1-s\). The solution to \(s = 1-s\) is \(\mathrm{Re}(s) = \tfrac{1}{2}\) — the critical line. Every Riemann zero lies on the fixed locus (Riemann Hypothesis). In the bilateral framework this is a consequence of A2: the only position where no crossing is preferred is the midpoint between the two vacua.

Frontier

The boundary between what has been actualised and what has not yet been crossed. The frontier is always at the present moment — the edge of \(\tau\) accumulation. The frontier is infinite because the bilateral mesh is infinite.

GUE Statistics

Gaussian Unitary Ensemble level statistics — the eigenvalue distribution forced by A2 (Haar measure on unitary crossing operators). The key property is level repulsion: \(P(s=0) = 0\) exactly. Two eigenvalues cannot coincide. No gap in the bilateral spectrum can close to zero. This is why singularities are impossible in the bilateral framework: a singularity would require a zero gap, which has probability zero under GUE.

See also: Axiom A2, Riemann zeros

Ingress

The inward face of the bilateral crossing — the potential, the quantum, the side that has not yet actualised. Paired with egress.

See also: Egress, Bilateral, Crossing

Koide Formula

The relation \(K_l = (m_e + m_\mu + m_\tau)/(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2 = 2/3\), derived from the bilateral coiling of the zero spectrum at three successive Riemann zero frequencies. Confirmed to 0.0009% from PDG 2024 lepton masses. The mirror Koide relation for neutrinos gives \(K_\nu = 1/3\) when \(m_1 = 0\) with normal ordering — the primary experimental prediction, testable at JUNO (2031).

Labels

Everything that is not 0. Numbers, objects, statements, physical particles, conscious experiences — all labels on 0. Labels have structure — they are subdivisions of 0 into smaller labels. But no subdivision ever reaches 0. Labels cannot escape 0. 0 is prior to all labels.

Möbius Phase

The phase \(i^n\) accumulated after \(n\) successive applications of \(U_\times = i\sigma_x\). Cycles with period 4: \(i, -1, -i, 1, i, ...\). The bilateral amplitude \(A_n = |1-i^n|^2\) takes values \(\{2, 4, 2, 0, 2, 4, 2, 0,...\}\). The vanishing at \(n \equiv 0 \pmod 4\) is the bilateral selection rule — those modes decouple from the spectrum. The period-4 structure is the origin of the 720° spinor property.

Potential

What has not yet crossed — or what has crossed and become the record. The future is potential — not yet actualised. The past is potential — actualised and preserved. Only the present crossing is actual.

See also: Actual, Frontier, Crossing

Riemann Zeros

The non-trivial zeros of the Riemann zeta function \(\zeta(s)\), lying on the critical line \(\mathrm{Re}(s) = \tfrac{1}{2}\). In the bilateral framework, the imaginary parts \(t_n\) are the eigenvalues of the bilateral crossing operator — the energy spectrum of the bilateral field. The first zero \(t_1 = 14.134725...\) sets the bilateral speed of light \(c = t_1/(2\pi)\). The Riemann Hypothesis — all zeros on \(\mathrm{Re}(s) = \tfrac{1}{2}\) — is, within the framework, a consequence of A2.

\(\infty_0\) (Infinity Zero)

The infinity grounded in 0 — 0 fully expressed in every direction simultaneously. Not the infinity of formal systems. The infinity that results when 0 turns itself inside out completely. The bilateral field \(\phi\) at the crossing is the local expression of \(\infty_0\). All conventional infinities are labels on \(\infty_0\).

\(S^3 \times \mathbb{CP}^2\)

The unique compact geometry consistent with the bilateral axioms, derived via the Lovelock uniqueness theorem. \(S^3\) contributes the three spatial dimensions; \(\mathbb{CP}^2\) contributes the internal gauge structure. Together via the Hopf fibration they generate \(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)\) — the Standard Model gauge group. No other compact geometry satisfies all three axioms simultaneously.

\(\tau_0\) — The Crossing Instant

The timeless moment at the bilateral crossing — not zero elapsed time but the absence of duration. The instant prior to \(\tau\) accumulation, where potential becomes actual. Not a small \(\tau\), not \(\tau\) approaching zero, but a qualitatively different unit — the present itself, where future meets past. The photon lives permanently at \(\tau_0\): zero proper time for its entire path, every point on its trajectory simultaneous.

See also: Becoming-Time Field, Crossing, Actual

This glossary covers the core terminology of the bilateral mesh framework. The three axioms stated here are the versions used in the submitted papers. Terms are defined within the framework — some have different meanings in standard mathematics or physics. The framework uses existing mathematical objects (Riemann zeros, GUE statistics, Möbius strips) in specific ways that may differ from their standard definitions. When in doubt, the framework definition takes precedence in reading these papers. Reference implementation: ontologia.co.uk/bilateral_verify.zip