Alphabeta Math

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Exponential growth of the Nyquist mode above the CFL bound

Physical scope

An exact mathematical failure mode of the scalar benchmark above its stated time-step bound; no Einstein solver or physical evolution is asserted.

Statement

Let $N \geq 4$ be even, let $h = 2 π / N$ , and choose the Courant--Friedrichs--Lewy (CFL) step ratio $λ : = k / h > 1$ . Run the periodic wave algorithm with $u_{j}^{0} = (- 1)^{j}$ and $v_{j}^{0} = 0$ , with any integer $n_{\max} \geq 1$ . For every integer $0 \leq n \leq n_{\max}$ the output has the form $u_{j}^{n} = a_{n} (- 1)^{j}$ , where, with $c : = 1 - 2 λ^{2} < - 1$ ,

a_{n} = \frac{z_{g r o w}^{n} + z_{d e c a y}^{n}}{2}, z_{g r o w} : = c - \sqrt{c^{2} - 1} < - 1, z_{d e c a y} : = c + \sqrt{c^{2} - 1} \in (- 1, 0) .

Consequently $∣ a_{n} ∣ \geq ∣ z_{g r o w} ∣^{n} / 2$ , so allowing more time steps causes exponential growth of this alternating mode. The strict hypothesis $0 < k / h < 1$ in the preceding mode theorem excludes this example; no endpoint claim at $k / h = 1$ is made here.

Statement references

A finite periodic scalar-wave benchmark

Termination and operation count of the periodic wave benchmark

Fourier-mode stability of the periodic leapfrog benchmark

Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$

Absolute value in an ordered field

Basic properties of the absolute value

Facts & Assumptions

Given: An even integer $N \geq 4$ , the benchmark step ratio $λ > 1$ , and the real initial arrays $u_{j}^{0} = (- 1)^{j}$ , $v_{j}^{0} = 0$ .

[F1]

A finite periodic scalar-wave benchmark: the initialization and periodic update are $u^{1} = u^{0} + k v^{0} + (k^{2} / 2) D_{h} u^{0}$ and $u^{n + 1} = 2 u^{n} - u^{n - 1} + k^{2} D_{h} u^{n}$ .

[F2]

Termination and operation count of the periodic wave benchmark: for every finite $n_{\max}$ the algorithm outputs every level through $u^{n_{\max}}$ .

[F3]

Fourier-mode stability of the periodic leapfrog benchmark: the preceding unit-circle result assumes $0 < k / h < 1$ and makes no claim for $k / h > 1$ .

[F4]

Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ : a positive real number has a unique positive square root.

[F5]

Absolute value in an ordered field: for real $x$ , $∣ x ∣ = x$ if $x \geq 0$ and $∣ x ∣ = - x$ if $x < 0$ .

[F6]

Basic properties of the absolute value: $∣ x ∣ \geq 0$ , $∣ x ∣ = 0$ exactly when $x = 0$ , and $∣ x y ∣ = ∣ x ∣ ∣ y ∣$ for real $x, y$ .

Verification

Given: The even periodic grid, the stated time-step ratio, and the alternating initial array.

1.1givenF1F3algebra

Because $N$ is even, both periodic neighbors of $(- 1)^{j}$ equal $- (- 1)^{j}$ . Hence $(D_{h} u^{0})_{j} = - 4 h^{- 2} (- 1)^{j}$ , and the startup in [F1] gives $a_{0} = 1$ and $a_{1} = 1 - 2 (k / h)^{2} = c$ . This mode has index $ℓ = N / 2$ ; it is outside the strict range of [F3].

2.1givenF1F4step 1.1algebra

Substitution of $u_{j}^{n} = a_{n} (- 1)^{j}$ into the update [F1] gives $a_{n + 1} - 2 c a_{n} + a_{n - 1} = 0$ . Since $c < - 1$ , [F4] supplies $r = \sqrt{c^{2} - 1} > 0$ . The two roots $z_{g r o w} = c - r$ and $z_{d e c a y} = c + r$ satisfy $z_{g r o w} z_{d e c a y} = 1$ ; also $z_{g r o w} < - 1$ and $z_{d e c a y} = 1 / z_{g r o w} \in (- 1, 0)$ .

3.1step 1.1step 2.1F1F2F5F6algebra∎

The sequence $(z_{g r o w}^{n} + z_{d e c a y}^{n}) / 2$ satisfies the recurrence because each root satisfies $z^{2} - 2 c z + 1 = 0$ , and its values at $n = 0, 1$ are $1, c$ . Induction identifies it with $a_{n}$ . For even $n$ the two powers are nonnegative, and for odd $n$ they are both negative; since $1 / 2 > 0$ , their half-sum has the same sign. The piecewise definition [F5] therefore gives $∣ a_{n} ∣ = (∣ z_{g r o w}^{n} ∣ + ∣ z_{d e c a y}^{n} ∣) / 2$ , while induction using multiplicativity [F6] gives $∣ z_{g r o w}^{n} ∣ = ∣ z_{g r o w} ∣^{n}$ and $∣ z_{d e c a y}^{n} ∣ = ∣ z_{d e c a y} ∣^{n}$ . Since $z_{g r o w} < - 1$ , [F5] gives $∣ z_{g r o w} ∣ = - z_{g r o w} > 1$ , while $∣ z_{d e c a y} ∣^{n} \geq 0$ by [F6]; hence $∣ a_{n} ∣ \geq ∣ z_{g r o w} ∣^{n} / 2$ .

Diagram

The following $N = 8$ picture illustrates the periodic alternating data; the last edge identifies the two ends of the grid. The proof above uses the recurrence and applies to every even $N \geq 4$ ; the drawing is not a proof.

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Zero sampled constraint residual with unresolved smooth data

Physical scope

A mathematical aliasing example for the discrete constraint diagnostic. It is not a reported simulation, a measurement, or a claim that the sampled data admit a spacetime development.

Example

Let $N \geq 4$ be even, let $N_{2}, N_{3} \geq 3$ , let $L_{*} > 0$ be a length, and choose $0 < ε < 1$ . On $T_{2 π}^{3}$ with dimensionless angular coordinates $x^{i}$ , set

\tilde{h} = L_{*}^{2} ((d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2}), \tilde{σ} = 0, τ = 0, ϕ_{N} (x) = 1 + ε \sin (N x^{1} / 2) .

Take the vacuum branch with $Λ = 0$ and $ρ = j = 0$ . Fix any $κ > 0$ with the SI units specified by the discrete diagnostic; its value drops out because $j = 0$ . The conformal data are $h = ϕ_{N}^{4} \tilde{h}$ and $K = 0$ . Use a periodic mesh with $N_{1} = N$ and counts $N_{2}, N_{3}$ in the other directions. Put $Δ_{i} = 2 π / N_{i}$ and use nodes $x_{j_{i}}^{i} = 2 π j_{i} / N_{i}$ , $0 \leq j_{i} < N_{i}$ , with indices wrapped modulo each $N_{i}$ . For $i = 1, 2, 3$ , let $e_{i}$ be the $i$ th standard basis vector of $Z^{3}$ . For a scalar grid array $f_{j}$ , use the centered periodic maps

(δ_{i} f)_{j} : = \frac{f_{j + e_{i}} - f_{j - e_{i}}}{2 Δ_{i}}, i = 1, 2, 3,

with indices reduced modulo $(N_{1}, N_{2}, N_{3})$ , and apply them componentwise. For reconstruction, partition each periodic coordinate circle into the half-open cells

C_{j} : = \prod_{i = 1}^{3} [\frac{2 π (j_{i} - 1 / 2)}{N_{i}}, \frac{2 π (j_{i} + 1 / 2)}{N_{i}}) (m o d 2 π),

and assign every component on $C_{j}$ its value at node $j$ . This specifies the piecewise-constant map $I_{Δ}$ in the constraint-residual definition. For the reported residual arrays use uniform dimensionless weights $w_{j} : = (N N_{2} N_{3})^{- 1}$ and the norm

∥ (H_{Δ}, M_{Δ}) ∥_{Δ, 2} : = {(\sum_{j} w_{j} (∣ H_{Δ} (j) ∣^{2} + h^{i k} (j) M_{Δ, i} (j) M_{Δ, k} (j)))}^{1 / 2} .

The bars around the scalar residual denote the real absolute value. For the sampled scalar and covector arrays, define the bilinear form

⟨ (H, M), (\hat{H}, \hat{M}) ⟩_{Δ} : = \sum_{j} w_{j} (H (j) \hat{H} (j) + h^{i k} (j) M_{i} (j) {\hat{M}}_{k} (j)) .

At the nodes of this example $h^{i k} = L_{*}^{- 2} δ^{i k}$ and every $w_{j} = (N N_{2} N_{3})^{- 1}$ is positive. Hence this bilinear form is positive definite on the finite-dimensional space of scalar and covector grid arrays, and the displayed quantity is its induced norm. In the fixed SI component convention, each summand has units $m^{- 4}$ , so the norm has units $m^{- 2}$ . The recorded mesh hierarchy for this diagnostic is the singleton containing this grid; no refinement or convergence comparison is made.

The discrete residuals $H_{Δ}$ and $M_{Δ}$ vanish at every grid node, so their stated weighted norm is zero. However, the continuum Hamiltonian residual at $x^{1} = π / N$ is

H (h, K) = \frac{2 ε N^{2}}{L_{*}^{2} (1 + ε)^{5}} > 0.

Thus a smooth high-frequency field can have the same samples and zero discrete residual as flat data while violating the continuum constraint between grid points. A small or vanishing grid residual alone does not certify continuum constraint satisfaction or solution accuracy.

Statement references

A finite periodic scalar-wave benchmark

A discrete Hamiltonian and momentum constraint diagnostic

Absolute value in an ordered field

Basic properties of the absolute value

Vacuum conformal constraint data

Conformal reduction of the vacuum constraint equations

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Christoffel formula for the levi civita connection

Curvature of an affine connection

Parity and the Pythagorean identity for sine and cosine

The zero sets of sine and cosine and the least positive common period 2 pi

Quarter-turn values and shifts by pi/2 and pi

The derivatives of sine and cosine are cosine and minus sine

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$

Facts & Assumptions

Given: The Axiom of Countable Choice for the exact conformal-data supplier's embedded interpretation, an even mesh count $N \geq 4$ , the stated smooth conformal data, and the vacuum convention.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ is the axiom that every countable family of nonempty sets has a choice function. It is retained because the conformal-data supplier includes the embedded Lorentzian interpretation; the algebraic grid example itself uses only the abstract data.

[F1]

Vacuum conformal constraint data: on a compact smooth three-manifold, $h = ϕ^{4} \tilde{h}$ and $K = ϕ^{- 2} \tilde{σ} + (τ / 3) h$ ; vacuum means $Λ = 0$ and zero stress-energy, and zero $\tilde{σ}$ is transverse-traceless.

[F2]

Conformal reduction of the vacuum constraint equations: for these data,

H (h, K) = ϕ^{- 5} (- 8 \tilde{Δ} ϕ + \tilde{R} ϕ - ∣ \tilde{σ} ∣_{\tilde{h}}^{2} ϕ^{- 7} + \frac{2}{3} τ^{2} ϕ^{5}) .

[F3]

A finite periodic scalar-wave benchmark: the periodic coordinate grid has nodes $x_{j} = 2 π j / N$ in its first circle factor.

[F4]

A discrete Hamiltonian and momentum constraint diagnostic: the discrete derivatives annihilate constant arrays, the signed vacuum scalar residual is $R_{Δ} + τ_{Δ}^{2} - ∣ K ∣_{Δ}^{2}$ , and the reconstruction map acts from admissible metric/extrinsic-curvature grid arrays to pointwise fields on the periodic torus without a smoothness claim.

[F5]

Christoffel formula for the levi civita connection, Curvature of an affine connection: Christoffel symbols are given by the first-derivative metric formula, and curvature is the derivative-plus- quadratic expression in those symbols.

[F6]

Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Quarter-turn values and shifts by pi/2 and pi: $∣ \sin x ∣ \leq 1$ , $\sin (π j) = 0$ for integer $j$ , and $\sin (π / 2) = 1$ .

[F7]

The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : the second derivative of $x \mapsto \sin (N x / 2)$ is $- (N / 2)^{2} \sin (N x / 2)$ .

[F8]

Absolute value in an ordered field: for real $x$ , $∣ x ∣ = x$ if $x \geq 0$ and $∣ x ∣ = - x$ if $x < 0$ .

[F9]

Basic properties of the absolute value: $∣ x ∣ \geq 0$ and $∣ x ∣ = 0$ exactly when $x = 0$ ; in particular, $∣ 0 ∣ = 0$ .

Verification

Given: The mesh, parameter $L_{*}$ , amplitude $ε$ , and the vacuum conformal data and discrete reconstruction in the Example.

1.1givenA1F1F6algebra

Since $N / 2$ is an integer, $ϕ_{N}$ is $2 π$ -periodic. By [F6], $ϕ_{N} \geq 1 - ε > 0$ . The flat seed metric is smooth positive-definite; $\tilde{σ} = 0$ is symmetric, trace-free and divergence-free, and $τ = 0$ is constant. Thus the data satisfy [F1] and $h = ϕ_{N}^{4} \tilde{h}$ , $K = 0$ .

2.1givenF3F4F6F8F9step 1.1algebra

At each first-coordinate grid node $x_{j}^{1} = 2 π j / N$ , [F6] gives $ϕ_{N} (x_{j}) = 1$ . The sampled arrays therefore have $h_{i j} = L_{*}^{2} δ_{i j}$ , $h^{i k} = L_{*}^{- 2} δ^{i k}$ , $K_{i j} = 0$ , $ρ = j = Λ = 0$ . Since the discrete derivatives annihilate these constant arrays, the discrete connection and curvature formulas of [F4] give zero; [F4] then gives $H_{Δ} = M_{Δ} = 0$ at every node. By [F8, F9], each scalar term $∣ H_{Δ} (j) ∣^{2} = ∣ 0 ∣^{2} = 0$ , and the covector term is zero because $M_{Δ} = 0$ . The induced norm of this zero array is therefore zero. The specified piecewise-constant reconstruction of these samples is flat.

2.2givenF2F5F6F7step 1.1algebra

In the lifted angular coordinates the coefficients of $\tilde{h}$ are constant. The Christoffel formula in [F5] therefore gives ${\tilde{Γ}}^{k}_{i j} = 0$ , and the curvature formula in [F5] gives $\tilde{R} = 0$ . Thus $\tilde{Δ} ϕ_{N} = L_{*}^{- 2} \partial_{1}^{2} ϕ_{N}$ . By [F7], $\tilde{Δ} ϕ_{N} = - \frac{ε N^{2}}{4 L_{*}^{2}} \sin (N x^{1} / 2) .$ At $x^{1} = π / N$ , [F6] gives $ϕ_{N} = 1 + ε$ and $\sin (N x^{1} / 2) = 1$ . Substituting $\tilde{σ} = 0$ , $τ = 0$ and this Laplacian into [F2] gives $H (h, K) = 2 ε N^{2} / [L_{*}^{2} (1 + ε)^{5}] > 0$ .

3.1step 2.1step 2.2F3F4∎

Step 2.1 and step 2.2 exhibit identical grid data with zero discrete residuals and a smooth continuum representative with a nonzero exact Hamiltonian residual. The reconstruction and finite differences cannot identify the unresolved sine mode from these samples; the residual therefore does not certify continuum constraint satisfaction.

Numerical geometry and constraint monitoring — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Exponential growth of the Nyquist mode above the CFL bound

Statement

Statement references

Facts & Assumptions

Verification

Diagram

Zero sampled constraint residual with unresolved smooth data

Example

Statement references

Facts & Assumptions

Verification

Sources