When Does Memory Become Mass? Inertia and Feedback in Systems with History

Macheng Shen

Epistemic status: a self-contained derivation plus a small reproducible toy, done in an afternoon. The core linear-response results (Sections 3–4) I am confident in — they are textbook Mori–Zwanzig once you see the sign. The framing claims (Sections 1, 6) are design_hypothesis: offered to invite critique, not asserted as settled. Disclosure, per the norm of this blog: the derivation and this write-up were produced through my agent system, which I then checked. Corrections welcome.

Corrections (2026-07-02, after a detailed external technical review): the core result stands — the algebra, the signs, and the pole condition are correct, and “monotone forgetting is not inertia; inertial memory must store and return” survives. Three things below were fixed: (1) the wrong-sign mass is not a physical “runaway” — only a truncated approximation is unstable; (2) a negative γ_eff in Channel B does not mean the exact system is unstable — with k, B, gA > 0 it is always stable, the low-frequency expansion has just left its range of validity; (3) “passive” was too broad and the Higgs analogy was stated too strongly. The corrected claims are marked inline.

1. The question

Many systems carry their past into their present. A habit is a low bar for action learned by repetition. Hysteresis in a magnet is the material “remembering” its history of applied fields. In contemplative language, karma — stripped of its moral overlay — is just the claim that accumulated, repeated action deposits a bias into how a system responds now. In every case, the honest first-principles home for “the past acting on the present” is a memory kernel: the present drift depends not on the instantaneous state alone but on a weighted integral over the state’s history,

        t
drift ~ ∫  K(t − t') ( · )(t') dt'
        0

When people reason informally about such systems, two different mechanical metaphors show up, and they are usually treated as interchangeable:

• Feedback (control-theoretic): the past sets a loop gain — positive feedback amplifies, negative feedback damps. The relevant quantities are the sign of the damping and the curvature of the effective restoring force.
• Inertia (Newtonian): the past gives the system mass — a resistance to changing its velocity, a M · ẍ term. Change is now second-order; the system can overshoot and oscillate.

The two pictures are not obviously the same. Feedback lives in a first-order world (rates, gains, damping). Inertia is intrinsically second-order (acceleration, momentum, resonance). So: can a memory kernel in an overdamped, first-order system actually grow a genuine second-order inertia — or does it only ever renormalize the feedback? And if it can grow inertia, under what condition?

There is a useful constraint to respect from the start. In modern physics, mass is not a primitive substance; it is an emergent effect of coupling to a background field (the Higgs mechanism being the sharp example). So the honest target is not to postulate inertia — it is to find inertia emerging from coupling to something, and to see exactly when that emergence succeeds or fails.

2. Setup

The workhorse is the generalized Langevin equation (GLE). For a coordinate x with restoring force −U'(x), a memory friction kernel K, and noise ξ:

m ẍ(t) = − ∫₀ᵗ K(t−t') ẋ(t') dt' − U'(x) + ξ(t)

We are interested in the overdamped regime — take the bare mass m → 0, so there is no inertia put in by hand. The question is whether the kernel K can put it back. Linearize the restoring force, U'(x) = k·x, and work in the frequency domain, ẋ → −iω x, with the one-sided kernel transform

K̂(ω) = ∫₀^∞ K(s) e^{iωs} ds .

The linear response x(ω) = χ(ω) ξ(ω) has inverse

χ⁻¹(ω) = k − iω K̂(ω) .

A genuine inertial system, for comparison, has χ⁻¹(ω) = k − m ω² − iγ ω. So the entire question reduces to one thing: does −iω K̂(ω) contain a −m ω² term with m > 0? The sign is everything.

3. Channel A — a viscous (dissipative) memory kernel gives the wrong sign

Take the simplest physical memory friction: an exponentially decaying, everywhere-positive kernel,

K(s) = (γ / τ) e^{−s/τ},   γ, τ > 0 .

This is “sticky mud with a memory” — a viscous drag proportional to a time-averaged recent velocity. Expand its transform at low frequency:

K̂(ω) = ∫₀^∞ K(s)(1 + iωs − …) ds = γ₀ + iω γ₁ + O(ω²),
   with  γ₀ = ∫K = γ,   γ₁ = ∫ s K(s) ds = γτ > 0 .

For any positive kernel, γ₁ = ∫ s K(s) ds > 0 — it is a positive weight times a positive time. Now form the response:

χ⁻¹(ω) = k − iω K̂(ω) = k − iω(γ₀ + iω γ₁ + …) = k + γ₁ ω² − i γ₀ ω + …

Compare term by term with the inertial form k − m ω² − iγ ω:

γ (damping)     = γ₀ = ∫K            ✓  (ordinary friction, fine)
m (inertia)     = − γ₁ = − ∫ s K < 0  ✗  (WRONG SIGN)

A pointwise-positive (forgetting) memory friction kernel produces a low-frequency effective mass with the wrong sign, m_eff < 0. Non-negativity alone is enough — the kernel need not even be monotone. Physically it is obvious in hindsight: a friction kernel only ever removes energy (with a delay); it can never store and return the kinetic energy that inertia requires. You cannot build momentum out of drag.

Two corrections to how an earlier version of this essay put it. First, the wrong-sign mass is not a physical “runaway.” The exact first-order-with-memory system here has stable (real) poles; the instability appears only if you truncate the low-frequency expansion into a fake second-order ODE, which manufactures a spurious high-frequency pole. So the honest statement is: a positive forgetting kernel gives an ω² coefficient of the wrong sign for inertia — no stable positive-mass interpretation — not an instability. Second, the forbidden class is “pointwise-positive forgetting,” not “passive.” Passivity is a frequency-domain condition (a positive-real response, Re K̂(ω) ≥ 0), and a passive kernel is allowed to dip negative in time. A passive kernel with a negative tail lobe can produce positive inertia — see the next paragraph. The thing that cannot is a kernel that stays non-negative for all time.

To get positive inertia this way you would need ∫ s K(s) ds < 0, i.e. the kernel must go negative somewhere — it must overshoot. A kernel with a negative lobe is precisely the signature of an underdamped, reactive degree of freedom hiding underneath, not a monotone dissipative one. Which brings us to the other channel.

4. Channel B — eliminate a reactive mode, and inertia emerges (à la Higgs)

Instead of a friction kernel on velocity, let the memory come from a genuine second variable z that carries its own state — something that can store and return. Two coupled first-order equations:

ẋ = − k x + g z + ξ
ż = − A x − B z          (z relaxes at rate B)

Integrate z out. Its solution is a convolution of x’s history — that is the memory kernel — and substituting back gives

χ⁻¹(ω) = (k − iω) + gA / (B − iω) .

Expand at low frequency (gA/(B−iω) = gA/B · [1 + iω/B + (iω/B)² + …]) and read off the three effective coefficients:

k_eff     = k + gA/B          renormalized stiffness   — this is FEEDBACK on the restoring force
γ_eff     = 1 − gA/B²         renormalized damping     — can go NEGATIVE ⇒ positive-feedback runaway
m_eff     = gA/B³   > 0       EMERGENT INERTIA

Now there is a positive effective mass. And note its form: m_eff = gA / B³ — a coupling (gA) divided by a power of the background mode’s stiffness/rate (B). It is not a substance; it is an induced effect of coupling to a background variable, and it vanishes as B → ∞ (at fixed coupling). This is Higgs-like only in a weak, effective-field-theory sense: an inertial coefficient is induced by coupling to an eliminated background degree of freedom, and vanishes under fixed-coupling adiabatic decoupling. Read no more into it than that — there is no gauge symmetry, no symmetry breaking, no vacuum expectation value; B is a relaxation rate, not a field stiffness or a relativistic invariant mass; and if gA were scaled together with B, even the clean decoupling story would break.

But “there is an ω² term” is not the same as “there is a physical resonance.” Two coupled first-order ODEs always give a formally second-order characteristic polynomial. Whether the system is a genuine underdamped oscillator (complex poles, real momentum) or just a disguised pair of overdamped relaxations (real poles, no momentum) depends on the parameters. The poles are the eigenvalues of [[−k, g], [−A, −B]]; they are complex — genuine oscillation — only when

(k + B)² < 4(k B + g A).

A small toy sweep (fix k = 1, gA = 4, vary the background rate B):

Two things to read off. First, complex poles — real inertia — appear only when the background mode is slow or comparable to the system; in the adiabatic (fast-bath) limit B → ∞, m_eff → 0 and you are left with pure renormalized feedback. Second — and this corrects an earlier version of this essay — a negative γ_eff does not mean the system is unstable. With k, B, gA > 0 the full two-mode system is always linearly stable (tr M = −(k+B) < 0, det M = kB + gA > 0); the row k=1, gA=4, B=1 gives λ = −1 ± 2i, a perfectly stable underdamped oscillator. A negative γ_eff only signals that the truncated low-frequency expansion has left its range of validity — you can no longer use its damping coefficient to predict the true pole locations. The reduced description stops being a reliable pole model; the exact system is fine.

5. The two axes

Putting the channels together, inertia and feedback are indeed two projections of one object — the memory kernel K̂(ω). Feedback lives in its real (dissipative) part plus the low-frequency renormalization of stiffness; inertia lives in its reactive part, the piece contributed by a mode that stores and returns. But they are separated by two axes, and are not freely interchangeable:

1. Channel. Reactive/state memory (can store and return energy) → inertia. Dissipative/friction memory (can only remove) → feedback, and its naive ω² term has the wrong sign.
2. Timescale. Memory comparable to the system’s own dynamics → inertia. Adiabatic, fast-relaxing memory → feedback (the induced mass vanishes).

Two honest caveats on this “two projections” language, both owed to a careful external review:

• The reactive and dissipative parts are not freely separable. Causality ties them together — the Kramers–Kronig relations make the real and imaginary parts of any causal response determine each other (nonlocally in frequency). And the split is representation-dependent: in the Mori–Zwanzig framework the memory kernel, the noise, and the projection are one package, not a unique objective object. So “inertia = reactive part, feedback = dissipative part” holds within a fixed scalar velocity-kernel representation, not as a frame-free law.

• The two channels are really one criterion. Integrating the eliminated mode z out by parts turns Channel B into a velocity-memory kernel K_B(s) = δ(s) − (gA/B) e^{−Bs} — which has a negative lobe, so ∫ s K_B ds = −gA/B³ < 0, giving m_eff = gA/B³ > 0. Channels A and B thus collapse into a single test: does the causal velocity kernel have a negative first moment? Monotone forgetting (A) never does; a hidden reactive mode (B) does. That is the whole result in one line. (One boundary: this assumes finite moments. For power-law / fractional memory, where ∫ s K diverges, there is no ordinary m_eff — you get fractional damping and storage, not a clean second-order mass.)

6. What this says about “karma,” habit, and history-dependent systems

Return to the motivating picture. Suppose you want to model an accumulated-history bias — a habit, a karma-like deposit, a policy shaped by its own past — as a memory kernel, and you are tempted to call it “inertia,” a mass that makes the system resist redirection. The derivation says: you are only entitled to the inertia reading if that memory can store and return, and if it operates on a timescale comparable to the behavior it shapes. If the honest model of the accumulated bias is monotone forgetting — a 習氣/habit as an ever-decaying, always-positive weight on the past, which is the natural reading — then it is a viscous drag, i.e. feedback, not inertia. Calling it mass would be borrowing a resonance and a stored momentum that the model does not contain.

So the concrete claim to hold: inertia and feedback are the reactive and dissipative parts of one memory kernel; they are not the same limit. Modeling history-dependence as Newtonian mass smuggles in an assumption — that the system has a reactive storage mode — that a plain “accumulated forgetting” kernel does not satisfy. The conservative default is feedback (sticky drag proportional to a weighted recent velocity, exactly as one would guess); the inertial upgrade must be earned by exhibiting the storage mode.

7. A single disconfirming test

For any specific history-dependent system you want to call “inertial,” write down its memory kernel K(s) and check two things:

1. Does K(s) have a negative lobe (equivalently ∫ s K(s) ds < 0)? No negative lobe ⇒ purely dissipative ⇒ feedback, and the “mass” is a metaphor.
2. Is its memory time non-adiabatic — comparable to the system’s own timescale, not vanishingly fast? If the memory is fast relative to the dynamics, m_eff → 0 ⇒ feedback.

Only if both are yes does the second-order/inertial description carry real content. Otherwise the honest word is feedback.

A caveat on this test, again from review: a negative first moment is genuinely necessary and sufficient for positive apparent low-frequency mass in a scalar finite-moment kernel. But stable, genuine inertia asks for more — the effective damping must stay positive in the relevant band (passivity), all poles must sit in the stable half-plane, a complex pole pair must actually exist ((k−B)² < 4gA in Channel B), and the resonance must lie inside the validity range of the reduced model. The two-part test is the right quick sanity check, not the full theorem.

Reproducibility

The whole result is a few lines of linear response; the toy in Section 4 is pure numpy (no external calls). The essential computation:

import numpy as np

# Channel A: viscous memory kernel K(s) = (gamma/tau) exp(-s/tau)
def channelA_meff(gamma, tau):
    # m_eff = -∫ s K(s) ds = -gamma*tau  (always < 0)
    return -gamma * tau

# Channel B: eliminate a reactive coupled mode
def channelB(k, g, A, B):
    k_eff  = k + g*A/B
    g_eff  = 1 - g*A/B**2
    m_eff  = g*A/B**3          # > 0, -> 0 as B -> inf (Higgs pattern)
    poles  = np.linalg.eigvals(np.array([[-k, g], [-A, -B]]))
    return k_eff, g_eff, m_eff, poles   # complex poles => genuine inertia

If you spot an error — a sign, a limit, a missed regime — or a sharper framing, I would like to hear it.