Signatures and Algebraic Behavior

The signature of \(Cl(p, q, r)\) defines the square of each generating basis vector:

\[ e_i^2 = \begin{cases} +1 & i < p,\\ -1 & p \le i < p + q,\\ 0 & p + q \le i < p + q + r. \end{cases} \]

These signs propagate through every Clifford product. They are separate from the positive coefficient geometry that clifra also exposes for losses, diagnostics, and stable optimization.

Signed forms and lane energy answer different questions

For compact coefficients \(x_i\), lane energy is

\[ E_{\mathrm{lane}}(x) = \sum_i x_i^2. \]

It is positive definite and independent of metric signs. It measures the size of a coefficient tensor in its declared layout. lane_energy, lane_norm, lane_distance, and the per-grade lane functions use this geometry.

A Clifford form is built from algebraic products. For example, a signature norm squared is a scalar projection such as

\[ Q(x) = \langle \widetilde{x}x\rangle_0, \]

with signs determined by the signature and blade grades. Depending on the form and convention, its value can be positive, negative, or zero for a nonzero multivector. Null directions in a degenerate signature make the last case fundamental rather than exceptional.

Quantity	Signature-sensitive	Positive definite	Typical use
Lane energy	No	Yes	Coefficient scale, regularization, stable distances
Per-grade lane energy	No	Yes	Grade distribution and diagnostics
Signature or conjugate scalar form	Yes	No in general	Algebraic invariants and metric-aware constraints
Magnitude derived with abs	Yes, before abs	Nonnegative, but not a norm in every signature	Stable scale based on a signed form

Calling both quantities a norm without qualification hides an important design choice. A signed form represents the algebraic metric. Lane energy represents a Euclidean geometry on the coordinate lanes used by the optimizer.

Signature determines product behavior

A planned product stores fixed left-lane positions, right-lane positions, output positions, and coefficients. The coefficients include permutation signs, metric signs, and zero products from null basis directions. Planning removes interactions that are known to be zero and execution performs the remaining gather, multiply, and reduction operations.

The resulting tensor program is signature-specific even though its runtime operations look like ordinary PyTorch arithmetic. Changing \(Cl(3, 0, 0)\) to \(Cl(1, 2, 0)\) changes more than coefficient names: it changes basis products, signed forms, exponential behavior, and potentially executor eligibility.

Forward and backward use the same algebra

Autograd differentiates the actual tensor program used in the forward pass. For an exact planned product with fixed coefficients,

\[ y_k = \sum_{i,j} c_{ijk} a_i b_j, \]

the backward pass contains the same signature-dependent coefficients \(c_{ijk}\):

\[ \frac{\partial L}{\partial a_i} = \sum_{j,k} c_{ijk} b_j \frac{\partial L}{\partial y_k}. \]

Backpropagation retains the algebraic signs from the forward calculation; the gradient follows that signed tensor program, with no substituted positive metric.

Using lane energy as a loss or regularizer introduces a positive Euclidean objective on coefficient space. That choice changes the optimization geometry and therefore the path taken by learning, while leaving the model's Clifford product unchanged. An indefinite algebra can therefore use a positive coefficient-space objective and still retain its original signature.

Algebra and optimization geometry

Keep three layers of claims distinct:

1. Algebraic definition. Exact product, reverse, projection, and exact exponential routes implement the declared Clifford operations up to ordinary floating-point error.
2. Numerical method. Clamps, null fallbacks, filtered eigengradients, and spectral truncation define how difficult cases are computed stably.
3. Learning objective. Lane losses, signed invariants, regularizers, and domain constraints define what the model is trained to prefer.

A positive coordinate-space loss leaves the Clifford product in the forward program unchanged. An invariant form of the Clifford algebra and an optimization geometry on its coefficients are different definitions and should be presented accordingly.

The same distinction applies to numerical safeguards. Taking the absolute value before a square root yields a stable magnitude-like scalar while the underlying form remains indefinite. A Euclidean fallback near a null vector avoids division by zero without certifying unit signed norm.

Scope of approximate exponentials

For spectral-local bivector exponentiation, the backward pass is the derivative of the implemented local computation. Near repeated eigenvalues, a filtered eigendecomposition suppresses unstable inverse spectral gaps. If plane truncation occurs, both forward and backward belong to the truncated model.

This remains a valid differentiable numerical method, but identities that depend on the exact full exponential must be validated to the required tolerance for that method. The algebraic target and its numerical approximation remain distinct.

Choosing a quantity for learning

Use lane geometry when the intended statement concerns coefficient scale, regularization, or comparison in a stable positive space. Use a signed Clifford form when the intended statement concerns the metric, an algebraic invariant, or a geometric constraint. In an indefinite or degenerate signature, decide how negative and null cases should affect the application rather than applying abs without interpreting it.

For a complete model, verify both levels independently:

• compare planned algebraic operations with an exact reference where feasible;
• test the model-level invariant or scientific constraint actually claimed;
• monitor lane-scale diagnostics for numerical conditioning;
• characterize approximation and gradient behavior in the operating regime.

The separation preserves the meaning of the declared signature while permitting a positive coordinate-space objective.