Differentiable Sequence Delta (DSD)

• Differentiable Sequence Delta (DSD) is a family of operators that capture discrete change by computing differences in sequences, with formulations ranging from binary streams to real-valued and differentiable settings.
• Its methodologies include symbolic delta operations with binomial closed forms, finite-sequence calculus with precise boundary treatments, and operational-calculus techniques leveraging exponential shift operators.
• In Visual Place Recognition, DSD is implemented as a learnable temporal differencing module incorporating fixed-kernel projection and LSTM refinement to enhance sequence-level matching performance.

Searching arXiv for the cited DSD-related papers to ground the article in the current literature. Differentiable Sequence Delta (DSD) denotes a family of sequence operators centered on discrete variation, temporal differencing, or derivative propagation. In the cited literature, the term ranges from the XOR-based difference operator on binary streams, to forward and backward finite differences on finite or infinite sequences, to the operational-calculus operator $\Delta_h = e^{h\partial}-I$ for differentiable programs, to a fixed-weight temporal differencing block for sequence-level Visual Place Recognition (VPR). The common motif is the extraction or propagation of change along a sequence, but the mathematical object, codomain, and application regime differ substantially across these settings (0911.1004, Filho, 2016, Imai, 2022, Sajovic et al., 2016, Papp et al., 2024, Li et al., 19 Jul 2025).

1. Terminological scope

Across the cited arXiv literature, DSD is not a single standardized operator. It is instead a cluster of delta-based constructions whose shared role is to convert a sequence into a representation of local or structured change. This suggests a family resemblance rather than a canonical definition.

Setting	Core form	Representative source
Binary streams	$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$	(0911.1004)
Finite sequences	$DSD_B[x]_k=x_{k+1}-x_k$ on the interior, plus a boundary rule	(Filho, 2016)
Sequence calculus	$\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$ and $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$	(Imai, 2022)
Differentiable programming	$\Delta_h=e^{h\partial}-I$	(Sajovic et al., 2016)
Linear recursive sequences	derivative recurrence induced by $S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$	(Papp et al., 2024)
Visual Place Recognition	fixed anti-symmetric temporal differencing kernel, then LSTM refinement	(Li et al., 19 Jul 2025)

The variation is not merely notational. Some DSDs act on symbolic streams over $\{0,1\}$, some on $\mathbb{R}^n$, some on sequence-valued function spaces, and some on learned embeddings in $\mathbb{R}^{B\times T\times C}$. Their invariants also differ: eventual periodicity for binary streams, boundary behavior for finite sequences, algebraic closure under differentiation in programming spaces, closure of derivative sequences under recurrences, or retrieval robustness under appearance and viewpoint shifts.

2. Binary-stream delta and orbit structure

In the stream-theoretic formulation, a binary stream is a function $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$0 with $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$1, and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$2 denotes the set of all such streams. The first difference operator is

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$3

Its iterates define the $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$4-orbit

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$5

and the central theorem states that a stream $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$6 is periodic if and only if the $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$7-orbit is periodic. The result extends to block differences

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$8

with the generalized theorem: for any $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$9, a stream $DSD_B[x]_k=x_{k+1}-x_k$0 is periodic if and only if the orbit $DSD_B[x]_k=x_{k+1}-x_k$1 is periodic (0911.1004).

A structural ingredient is strong preservation of periodicity: if $DSD_B[x]_k=x_{k+1}-x_k$2 has period $DSD_B[x]_k=x_{k+1}-x_k$3 and offset $DSD_B[x]_k=x_{k+1}-x_k$4, then $DSD_B[x]_k=x_{k+1}-x_k$5 has the same $DSD_B[x]_k=x_{k+1}-x_k$6 and $DSD_B[x]_k=x_{k+1}-x_k$7. For ordinary $DSD_B[x]_k=x_{k+1}-x_k$8, iterates admit the binomial closed form

$DSD_B[x]_k=x_{k+1}-x_k$9

The parity pattern of the binomial coefficients modulo $\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$0 produces Sierpinski’s triangle. In particular, when $\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$1,

$\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$2

because all interior binomial coefficients are even modulo $\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$3. The generalized operator $\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$4 is described through a $\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$5-ary Pascal triangle $\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$6 with

$\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$7

and at powers of two this simplifies to

$\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$8

The theory is accompanied by a sequence of examples. Constant streams and the alternating stream become eventually constant under iteration. For automatic and morphic sequences, the paper shows

$\Delta_R\{a_n\}=\{a_{n+1}-a_n\}$9

for the Thue–Morse stream $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$0 and the period doubling Toeplitz stream $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$1, and further gives

$\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$2

so the iterates are mutually different. For the Fibonacci stream, the orbit is not periodic, in agreement with the theorem. For the impulse stream $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$3, the orbit matrix reproduces Pascal’s triangle modulo $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$4, hence the Sierpinski pattern. The paper also identifies the empirical and then formal identity

$\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$5

linking the Sierpinski stream $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$6 and the Mephisto Waltz stream $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$7 despite their different generation rules (0911.1004).

Algorithmically, the periodicity theorem turns orbit periodicity into a decision principle for eventual periodicity. The constructive proof uses powers of two $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$8 and $\Delta_L\{a_n\}=\{a_n-a_{n-1}\}$9 with $\Delta_h=e^{h\partial}-I$0 to derive a linear recurrence of order $\Delta_h=e^{h\partial}-I$1 for $\Delta_h=e^{h\partial}-I$2, after which repeated windows force eventual repetition over the finite alphabet $\Delta_h=e^{h\partial}-I$3.

3. Finite and infinite sequence calculi

A second line of work treats DSD as a discrete derivative on finite or infinite real-valued sequences. For finite sequences $\Delta_h=e^{h\partial}-I$4, the interior index set is $\Delta_h=e^{h\partial}-I$5. The basic operators are the top operator $\Delta_h=e^{h\partial}-I$6, bottom operator $\Delta_h=e^{h\partial}-I$7, forward shift $\Delta_h=e^{h\partial}-I$8, and middle operator $\Delta_h=e^{h\partial}-I$9 on the interior. The forward difference is

$S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$0

with operator form $S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$1. A length-preserving DSD is obtained by appending a boundary rule: $S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$2 The paper lists truncated, zero-padding, hold, reflective, periodic, and operator-based boundaries. For linear boundary schemes, $S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$3 is linear, and for periodic boundary $S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$4; for reflective, zero-padding, or hold padding, the operator norm is $S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$5 (Filho, 2016).

This finite-sequence calculus recovers the standard finite-difference identities on interiors. Higher-order forward differences satisfy

$S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$6

and the discrete integral $S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$7 is defined by cumulative summation. The telescoping identity

$S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$8

is the finite-sequence form of the second fundamental theorem. Product and quotient rules are given explicitly; for instance,

$S(t;x)=P(t;x)^2/\gcd_t(P,P_x)$9

Second differences

$\{0,1\}$0

define convexity through $\{0,1\}$1, equivalently monotonicity of first differences (Filho, 2016).

A related but broader calculus of sequences defines right and left differentials

$\{0,1\}$2

together with right and left integrals

$\{0,1\}$3

These satisfy discrete fundamental-theorem identities such as

$\{0,1\}$4

Higher-order deltas are the standard forward and backward finite differences, and the framework develops sequence versions of exponentials, hyperbolic functions, trigonometric functions, Maclaurin expansions, Fibonacci generalizations, and sequence duals of factorial and Bell numbers (Imai, 2022).

The two calculi are closely aligned in their core operator algebra. The finite-sequence formulation emphasizes boundary treatment and operator norms; the infinite-sequence formulation emphasizes algebraic closure, repeated integration, and sequence analogues of classical special functions. In both, DSD is an exact discrete differential operator rather than a numerical approximation to a continuum derivative.

4. Operational-calculus DSD in differentiable programming

In operational calculus for differentiable programming, programs are maps $\{0,1\}$5 on a finite-dimensional vector space $\{0,1\}$6, and derivatives are represented in the tensor algebra over $\{0,1\}$7. The virtual memory space is

$\{0,1\}$8

and the differentiation operator $\{0,1\}$9 acts on programming spaces through Fréchet differentiation. The generalized shift operator is

$\mathbb{R}^n$0

Its key evaluation property is the tensor-series expansion

$\mathbb{R}^n$1

which is the Taylor expansion expressed in the programming-space language (Sajovic et al., 2016).

Within this framework, DSD is defined by

$\mathbb{R}^n$2

For unit step, $\mathbb{R}^n$3. Expanding the exponential yields

$\mathbb{R}^n$4

Acting on a program at $\mathbb{R}^n$5 in direction $\mathbb{R}^n$6, this gives the forward difference

$\mathbb{R}^n$7

If a sequence is realized by shifts of the memory state, then DSD is the discrete one-step increment in the program output. Because $\mathbb{R}^n$8 is built from powers of $\mathbb{R}^n$9, it is differentiable and compositional inside the same operator algebra.

The framework also introduces an operator form for program composition and a fractional generalization

$\mathbb{R}^{B\times T\times C}$0

A special case is ReduceSum, where summation is expressed as a sum of shifts and linked to Bernoulli-number expansions, recovering Euler–Maclaurin in the univariate setting. In this formulation, DSD is not merely a difference on a data sequence; it is an algebraic operator on programs themselves, with explicit semantics for composition, iteration, and higher-order differentiation (Sajovic et al., 2016).

5. DSD for derivative sequences of linear recurrences

For sequences of differentiable functions governed by homogeneous linear recurrences, DSD appears as a constructive map from an original recurrence to a recurrence satisfied by the derivative sequence. Let $\mathbb{R}^{B\times T\times C}$1 satisfy

$\mathbb{R}^{B\times T\times C}$2

where the coefficient functions $\mathbb{R}^{B\times T\times C}$3 are differentiable and independent of $\mathbb{R}^{B\times T\times C}$4. With characteristic polynomial

$\mathbb{R}^{B\times T\times C}$5

and coefficientwise derivative

$\mathbb{R}^{B\times T\times C}$6

the derivative sequence $\mathbb{R}^{B\times T\times C}$7 satisfies a homogeneous linear recurrence whose characteristic polynomial is

$\mathbb{R}^{B\times T\times C}$8

Consequently, there exist coefficient functions $\mathbb{R}^{B\times T\times C}$9 with $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$00 such that

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$01

for all $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$02 (Papp et al., 2024).

The construction is root-free. One forms $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$03 and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$04, computes $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$05 by the Euclidean algorithm or the Sylvester matrix, and then expands

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$06

This gives the recurrence for the derivative sequence without determining the roots of $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$07. Computationally, polynomial construction and multiplication of $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$08 require $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$09 operations, while $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$10 typically requires $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$11 to $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$12 algebraic operations, depending on implementation.

The order bound is generically sharp: if $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$13, then $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$14. When $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$15 and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$16 share factors, the order drops. For Chebyshev polynomials, where

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$17

the gcd is $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$18 for generic $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$19, so

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$20

and both derivative sequences $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$21 and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$22 satisfy the corresponding order-$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$23 recurrence. The paper also discusses Legendre, Hermite, and Laguerre polynomials, noting that $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$24-dependent coefficients place them outside the strict theorem even though derivative recurrences can still be derived (Papp et al., 2024).

6. Delta-centered continuity and compactness

A related delta-based literature studies the behavior of sequences through forward differences rather than through DSD as an operator in the narrow algebraic sense. The abstract of the paper on $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$25-quasi-Cauchy sequences defines

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$26

and calls a function $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$27 forward continuous if $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$28 whenever $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$29. It similarly introduces second forward continuity through preservation of $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$30, and defines forward compactness and second forward compactness by the existence of subsequences with vanishing first or second forward differences (Cakalli, 2010).

The accompanying exposition represents the standard ward-continuity framework around these notions. Typical results include: uniform continuity implies forward continuity; forward continuity implies ordinary continuity; on intervals, forward continuity is equivalent to uniform continuity; and forward compactness is equivalent to boundedness in $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$31. The second-order variants are described analogously, with second forward compactness likewise equivalent to boundedness. Illustrative examples include sequences such as $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$32, for which $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$33 although the sequence diverges, and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$34, for which $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$35 while $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$36 (Cakalli, 2010).

In this setting, the delta operator serves as a convergence probe rather than as a representation-learning or program-transform operator. The underlying idea is nevertheless consistent with other DSD formulations: the preservation, attenuation, or explicit modeling of discrete change determines the relevant notion of regularity.

7. DSD as a learnable temporal differencing module for VPR

In OptiCorNet, DSD is the core temporal component for sequence-level Visual Place Recognition. A sequence $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$37 is first mapped to frame-level features, producing

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$38

with $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$39 the batch size, $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$40 the sequence length, and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$41 the feature dimension. The paper allows an optional lightweight temporal encoder $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$42, but in the detailed instantiation the differencing is applied directly to the frame features. DSD then collapses the sequence into a single change vector through a fixed anti-symmetric weight vector: $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$43 The kernel is a central-difference-like $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$44-tap pattern,

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$45

centered on the mid-portion of the sequence, with all other entries zero. The differencing stage is parameter-free and fully differentiable, since gradients propagate through $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$46 and through the downstream modules (Li et al., 19 Jul 2025).

After differencing, the representation is refined by a single-layer LSTM. With $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$47, the paper reshapes it to

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$48

passes it through an LSTM to obtain $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$49, and then sets $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$50. Because the sequence presented to the LSTM is length $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$51, the LSTM functions as a gated projection that learns how to re-weight and stabilize the differenced signal. A residual path preserves semantic content: $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$52 The final DSD descriptor is $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$53, and retrieval uses Euclidean distance on these global sequence embeddings, with optional local sequence matching on per-frame descriptors for top candidates.

Training uses a quadruplet loss. For anchor $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$54, positive $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$55, and hard negatives $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$56,

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$57

where

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$58

$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$59

The reported training configuration uses mini-batches of $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$60 quadruplets, each quadruplet including $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$61 images, sequence length $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$62, kernel width $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$63, backbone NetVLAD, SGD with momentum $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$64, weight decay $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$65, initial learning rate $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$66, decay by $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$67 every $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$68 epochs, and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$69 total epochs. Extra DSD parameters are limited to the LSTM, with $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$70 weights, and the optional projection $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$71, also $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$72 (Li et al., 19 Jul 2025).

The module is computationally light. The differencing complexity is $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$73, the LSTM refinement for a single time step is $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$74 with small constants, and the end-to-end retrieval timings reported on Nordland are $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$75 ms for S1, $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$76 ms for S5, and $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$77 ms for hierarchical S5$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$78S1. On Nordland, DSD (S5) + Quadruplet attains Recall@1/5/10/20 of $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$79, while Hierarchical DSD (S5$\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$80S1) + Quadruplet reaches $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$81. The same table reports SeqNet at $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$82, MixVPR at $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$83, and EffoVPR with Recall@1 $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$84. Ablations show substantial sensitivity to the differencing stage: w/o differencing yields $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$85, w/o LSTM $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$86, w/o residual connection $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$87, and w/o residual fusion $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$88. Triplet versus quadruplet is reported as $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$89 versus $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$90 (Li et al., 19 Jul 2025).

The VPR formulation is the most explicitly differentiable and end-to-end trainable use of the term in the cited literature. Its DSD is neither the symbolic $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$91 of binary streams nor the abstract $\delta(\sigma)_n=\sigma_n\oplus\sigma_{n+1}$92 of differentiable programming, but a fixed-kernel temporal projection whose learnable capacity is deliberately shifted into LSTM refinement and residual fusion. The paper’s stated rationale is that differencing suppresses static bias, emphasizes motion- or transition-coupled cues, and allows the spatial encoder to receive gradients from a sequence-level retrieval objective (Li et al., 19 Jul 2025).