Non-History Temporal Superposition Acceleration

• Non-history temporal superposition acceleration algorithm is a method that replaces sequential time-history accumulation with recursive updates using compact auxiliary variables.
• It achieves significant computational gains by reducing the per-step workload from quadratic to linear scaling in time steps.
• The approach is versatile, with extensions to quantum gate ordering and non-Markovian PDE surrogates, offering broad applicability in simulation and modeling.

“Non-History Temporal Superposition Acceleration Algorithm” denotes a family of time-advancement strategies that accelerate long-horizon evolution by replacing explicit accumulation of full temporal history with a more compact representation of past influence, or by exploiting coherent superposition over alternative temporal orders or propagators. The phrase is used most precisely in heat-transfer simulation for borehole heat exchangers and finite line source models, where “non-history dependent” refers to a marching scheme whose update depends only on previously stored auxiliary variables and the current load, not on the full load record. In other settings, related constructions include quantum superposition of gate orders, superposition of unitaries, Laplace-domain temporal superposition, and time-parallel decomposition. The unifying theme is the replacement of naive sequential accumulation by structured temporal composition, but the literature also shows that genuine history elimination is not universally possible, especially for non-Markovian dynamics (Lazzarotto et al., 27 Jan 2025, Basquens et al., 24 Jul 2025, Taddei et al., 2020, Shikhman, 10 May 2026).

1. Terminology and scope

Across the literature, the same phrase refers to distinct technical objects. In heat-transfer modeling, temporal superposition means convolution of loads with analytical source responses; in quantum information, it means coherent superposition of gate orders or propagators; in reduced-history solvers, it means reconstruction of a time-domain state from transform-space components or time-slab decompositions.

Domain	Temporal superposition object	Status of “history”
Finite line source heat transfer	Convolution or exponential-kernel superposition of source responses	Explicit load history replaced by recursive internal states
Quantum switch and related protocols	Coherent superposition of gate orderings or event times	Not a history-state encoding
Non-Markovian PDE surrogates	Shift-append decomposition in lifted history space	History remains necessary, but deterministic parts are not relearned

The most literal usage occurs in “A non-history dependent temporal superposition algorithm for the finite line source solution” (Lazzarotto et al., 27 Jan 2025). That paper defines a history dependent method as one whose current update explicitly depends on the full past load sequence, and a non-history dependent method as a marching scheme in which the update from one time level to the next depends only on the previous state and the load over the current interval. The same terminology is preserved and extended in the later causality-inspired acceleration method for finite line source solutions (Basquens et al., 24 Jul 2025).

By contrast, quantum-information papers do not use the phrase itself, but they instantiate the “temporal superposition” part directly. “Computational advantage from quantum superposition of multiple temporal orders of photonic gates” uses the quantum $N$-switch to place multiple gate orders in coherent superposition (Taddei et al., 2020). “Indefinite Causal Orders from Superpositions in Time” treats event time as a quantum variable and derives a SWITCH-like indefinite order from superposed event times rather than from a history-state construction (Felce et al., 2021). This suggests that the expression names a methodological family rather than a single standardized algorithm.

2. Heat-transfer origin: finite line source temporal superposition without explicit history

The classical formulation arises in subsurface heat-transfer simulation, especially for fields of borehole heat exchangers. There the temperature response is written by Duhamel superposition as a time convolution between the load and an impulse response. In nondimensional variables, the governing representation is (Lazzarotto et al., 27 Jan 2025)

$$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$

A naive discrete implementation revisits all earlier time steps at every update, giving $\mathcal O(N_t^2)$ work. For decade-scale hourly simulations, the paper explicitly identifies this as the temporal-superposition bottleneck.

The non-history dependent method replaces that explicit time-history sum by an exponential-kernel representation of the impulse response,

$$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$

and introduces exponentially weighted memory variables

$$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$

For piecewise-constant loads, the update becomes

$$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$

This is the core non-history recursion: each $\zeta$-node stores one scalar state, and the entire past is compressed into those states. The update uses only the previous state and the current load.

The method separates temporal and geometric content. Writing

$$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$

the recursion for $\hat F$ carries the load history, while $v(\zeta)/\zeta^2$ carries the geometry. For the point source, the paper derives the closed-form factor

$$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$0

Finite line source variants are then built by integrating the point-source representation along one or two line segments, yielding segment-to-point and segment-to-segment geometry terms without direct inverse Laplace transformation of the finite line source kernel.

The computational consequence is the central result of the paper: standard temporal superposition scales as $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$1, whereas the proposed recursion has per-step cost $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$2 and total cost $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$3, hence linear in $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$4 (Lazzarotto et al., 27 Jan 2025). The same work reports that near double precision can be achieved by refining the $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$5-quadrature, line quadrature, and truncation cutoff $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$6, and that simulations with $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$7 time steps can run in a fraction of a second. In this literature, “non-history” does not mean absence of memory; it means that memory is represented by fixed-size recursively updated state variables rather than by an ever-growing explicit history.

3. Causality-inspired refinement: blockwise acceleration of finite line source superposition

The later refinement, “A causality inspired acceleration method for the fast temporal superposition of the finite line source solutions,” preserves the non-history recursion but reorganizes its precomputation stage (Basquens et al., 24 Jul 2025). Its starting point is that the original non-history implementation had shifted the dominant cost from online time stepping to offline precomputation, especially for line-to-point and line-to-line interactions involving highly oscillatory integrals.

The new idea is approximate causality. The heat equation is formally non-causal in the strict sense that the fundamental solution is nonzero everywhere for any $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$8, but the paper shows that one can define an influence region by fixing an acceptable error tolerance $$T(\tilde t)-T_0 = \frac{1}{k_g}\frac{r_b^2}{\alpha} \int_0^{\tilde t} \tilde q'(\tilde \tau)\, \frac{d\tilde h}{d\tilde t}(\tilde t-\tilde \tau)\, d\tilde \tau .$$9. Contributions below that tolerance are discarded. This yields interaction-dependent delays $\mathcal O(N_t^2)$0 and a block decomposition of source history. The block radii are grown geometrically,

$\mathcal O(N_t^2)$1

with the paper recommending $\mathcal O(N_t^2)$2.

The block source functions are

$\mathcal O(N_t^2)$3

and they satisfy a blockwise recursion. The important structural point is that older blocks carry an additional Gaussian factor $\mathcal O(N_t^2)$4. This damps high-$\mathcal O(N_t^2)$5 oscillations and shortens the effective integration range in $\mathcal O(N_t^2)$6. Simultaneously, for line-source geometries, the source domain itself is truncated to the portion that can influence the target above tolerance. The combined effect is a much better behaved integrand.

This change in integrand regularity allows the paper to replace the Bakhalov–Vasil’eva oscillatory integration method, for $\mathcal O(N_t^2)$7, by a hybrid of adaptive Gaussian quadrature and an asymptotic method for highly oscillatory integrals. The asymptotic treatment is applied to integrals of the form

$\mathcal O(N_t^2)$8

with error controlled by explicit bounds. The interval is split at a point $\mathcal O(N_t^2)$9 determined by the asymptotic error estimate, so that the final evaluation is

$$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$0

The resulting online temperature contribution for interaction $$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$1 is then assembled as

$$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$2

The reported gains are concentrated in precomputation. For a single interaction, the paper states that point-to-point precomputation is typically $$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$3–$$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$4 faster than the original non-history method, line-to-point is faster by up to $$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$5 orders of magnitude, and line-to-line by $$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$6 to $$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$7 orders of magnitude. A concrete example given in the text is line-to-line precomputation at $$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$8, reduced from $$\frac{d\tilde h}{d\tilde t}(\tilde t) = \int_0^\infty e^{-\zeta^2\tilde t} v(\zeta)\,d\zeta ,$$9 to $$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$0 (Basquens et al., 24 Jul 2025). The same paper states that for all tested tolerances the maximum absolute error satisfies

$$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$1

The principal limitation is equally explicit: the first block $$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$2 does not benefit from Gaussian suppression and still requires the original specialized oscillatory treatment.

4. Quantum temporal superposition: order, event time, and superposed propagators

In quantum information, temporal superposition refers not to thermal convolution but to coherent superposition of dynamical orderings or propagators. The most direct algorithmic example is the quantum $$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$3-switch (Taddei et al., 2020). There a control register coherently selects one of several gate permutations,

$$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$4

with $$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$5 the product of $$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$6 unknown gates in the $$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$7-th order. Under a superposed control, the target undergoes a coherent superposition of temporal orders. For the Hadamard promise problem, the ordered products satisfy

$$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$8

where $$F(\zeta,\tilde t) = \int_0^{\tilde t} e^{-\zeta^2(\tilde t-\tilde \tau)} \tilde q'(\tilde \tau)\,v(\zeta)\,d\tilde \tau .$$9 are entries of a known Hadamard matrix, and the decoding identity is

$$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$0

The paper attributes the advantage to query complexity: the $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$1-switch solves the problem with $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$2 oracle queries, whereas known exact fixed-order methods require $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$3 queries for $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$4, and the demonstrated $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$5 case uses $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$6 rather than $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$7. It also removes the earlier target-dimension bottleneck, reducing the required target dimension to $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$8 regardless of $$F(\zeta,\tilde t+\Delta\tilde t) = e^{-\zeta^2\Delta\tilde t}F(\zeta,\tilde t) + \tilde q'(\tilde t)\left(1-e^{-\zeta^2\Delta\tilde t}\right)\frac{v(\zeta)}{\zeta^2}.$$9 (Taddei et al., 2020). In this setting, “non-history” is apt only in the sense that the resource is coherent temporal order rather than a history-state encoding.

A second line of work derives indefinite causal order directly from superpositions in time. “Indefinite Causal Orders from Superpositions in Time” treats the time of an event as a quantum variable with amplitudes $\zeta$0 and $\zeta$1, yielding a state of the form

$\zeta$2

Under the symmetry assumptions stated in the paper, this reproduces the standard quantum SWITCH structure (Felce et al., 2021). The construction is foundational rather than complexity-theoretic, but it makes explicit that coherent temporal order can emerge from superposed event times themselves.

A third quantum variant studies superposition of unitaries rather than superposition of orders. “Enhanced non-macrorealism: Extreme violations of Leggett-Garg inequalities for a system evolving under superposition of unitaries” defines an effective propagator

$\zeta$3

implemented by ancilla-controlled branches and postselection (Chatterjee et al., 2024). The paper does not claim computational speedup, but it does show that this temporal superposition can drive Leggett–Garg violations beyond the usual qubit temporal Tsirelson bound $\zeta$4, with ideal behavior approaching the algebraic maximum $\zeta$5, and with experimental values $\zeta$6 and $\zeta$7. This suggests that temporal superposition can serve as a dynamical primitive even when the target is not algorithmic acceleration in the strict complexity sense.

5. Reduced-history temporal acceleration outside the finite line source setting

Several classical time-dependent solvers exhibit the same architectural pattern—replace sequential history accumulation by a global superposition formula or by time-domain decomposition—without using the exact heat-transfer terminology.

The asynchronous Laplace-transform method reconstructs a solution at time $\zeta$8 from independent Laplace-space solves,

$\zeta$9

rather than by step-by-step time marching (Magoules et al., 2019). In the linear case, this is essentially non-history in time: the full intermediate time trajectory need not be computed. The asynchronous variant removes synchronization barriers among processors, but the paper is explicit that for quasilinear problems the method still depends on previous nonlinear iterates through a frozen coefficient $$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$0, and it does not provide a rigorous convergence proof.

Paraexp-based acceleration of direct–adjoint looping takes a different route. For linear systems, it decomposes the global solution into local inhomogeneous solves on time subintervals and homogeneous propagations of interface data, with final reconstruction by superposition (Skene et al., 2020). The direct solution on each slab is written as a local particular solution plus sums of homogeneous continuations. The same idea is applied to the adjoint, whose linear or linear time-varying structure makes Paraexp applicable. The method is plainly temporal superposition, but it is not history-free in the strong sense: segment boundary states still carry past influence, and adjoint computation still requires access to the direct trajectory, whether by storage, checkpointing, or reconstruction.

These methods broaden the meaning of the expression. A plausible implication is that “non-history” often means “non-sequential in explicit time stepping” rather than “memoryless.” The distinction is important, because the mathematical object carrying past influence may shift from a long load record to transform samples, interface states, or recursively updated auxiliary variables.

6. Limits, misconceptions, and the non-Markovian counterexample

A common misconception is that temporal acceleration in a system with memory can always be made history-free. Work on non-Markovian PDE surrogates shows the opposite. “HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations” argues that if the instantaneous field $$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$1 is not a sufficient state, then a current-state autoregressive operator is structurally misspecified (Shikhman, 10 May 2026). In the paper’s finite-dimensional analysis, if a compressed representation $$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$2 identifies two distinct histories $$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$3 with different futures, then any deterministic predictor based only on $$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$4 incurs a minimum conditional squared error

$$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$5

The implication stated by the paper is that some representation of relevant history is fundamentally necessary.

HS-FNO therefore replaces “non-history” by structured history. The lifted state is

$$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$6

with natural state space $$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$7. The exact update is decomposed into a learned predictor for the newly exposed slice and an exact shift-append transport for the known part of the history window:

$$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$8

$$F(\zeta,\tilde t) = \hat F(\zeta,\tilde t)\,\frac{v(\zeta)}{\zeta^2},$$9

This is a superposition of exact deterministic transport and learned irreducible novelty, not elimination of memory. The reported aggregate rollout error drops from $\hat F$0, $\hat F$1, and $\hat F$2 for current-state, lag-stack, and unconstrained history-to-history baselines to $\hat F$3, while the default HS-FNO uses $\hat F$4 parameters rather than $\hat F$5 for the unconstrained History2History baseline (Shikhman, 10 May 2026).

This counterexample clarifies the limits of the term. In Markovian or effectively compressible settings, explicit history can often be replaced by recursive state variables or coherent control structures. In genuinely non-Markovian settings, a truly history-free algorithm is generally impossible unless the missing information is restored through an equivalent enlarged state, a sufficient latent representation, or a controlled approximation. The strongest general interpretation of the phrase is therefore not “memory can be abolished,” but “temporal dependence can be reorganized so that only its irreducible component is carried forward.”