Forgetful Gauges in Electrodynamics & Memory

• Forgetful Gauges are abstract mechanisms that determine what is retained or decayed, playing pivotal roles in both electrodynamics and semantic information systems.
• In electrodynamics, the choice among Lorenz, Coulomb, and velocity gauges affects the causal propagation of electric fields and can introduce superluminal artifacts.
• In information systems, dynamic memory buoyancy and ForgetEval frameworks regulate visibility and deletion, ensuring context-sensitive and efficient memory management.

“Forgetful Gauges” (Editor’s term) designates two distinct technical uses of gauge-like constructs in the supplied literature. In classical electrodynamics, gauges are connections between the electromagnetic potentials, and Onoochin argues that the Coulomb and velocity gauges permit instantaneous or superluminal ingredients to survive in computed electric fields, so that only the Lorenz gauge should be used in applied calculations (Onoochin, 30 Sep 2025). In information systems, by contrast, Memory Buoyancy is explicitly presented as a dynamic “forgetful gauge” that measures each information item’s current value and drives Managed Forgetting in a Semantic Desktop (Jilek et al., 2018). A related operational perspective appears in agent-memory evaluation, where ForgetEval studies how control-plane placement determines which forgetting failure modes a system can recover across thirteen system configurations (Yang, 14 Jun 2026). The commonality is not a shared formalism but a shared role: a gauge specifies what is retained, what decays, and under what constraints.

1. Terminological scope and conceptual role

In the electrodynamic usage, a gauge is a relation imposed on the four-potential

$A^\mu = (\phi,\vec A),$

with fields recovered as

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$

The supplied material identifies three gauges that can be used to describe systems of charges and currents without restrictions: the Lorenz gauge, the Coulomb gauge, and the velocity gauge. It also states that more specific gauges are reductions of the Lorenz gauge (Onoochin, 30 Sep 2025).

In the Semantic Desktop literature, the “gauge” is explicitly metaphorical. Memory Buoyancy assigns every “thing” in a Personal Information Model a normalized score $MB\in[0,1]$ indicating its current value for a particular user. That score then drives an escalating spectrum of forgetting actions, from temporary hiding through condensation and adaptive synchronization to archiving or deletion (Jilek et al., 2018).

ForgetEval uses neither the electrodynamic nor the Memory Buoyancy terminology, but it introduces a third gauge-like mechanism: deterministic scoring of whether a memory system forgets correctly after control-plane mutations. This suggests a broader pattern in which a “forgetful gauge” is a device that makes retention and deletion operationally testable. That implication is interpretive rather than terminological, but it is consistent with the supplied descriptions of Memory Buoyancy and ForgetEval (Yang, 14 Jun 2026).

2. Electrodynamic formulation: Lorenz, Coulomb, and velocity gauges

The Lorenz gauge is defined by

$\partial_\mu A^\mu = 0,$

or, in three-vector form,

$$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$

Its potentials are retarded at the light speed $c$:

$$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$

$$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$

The fields are then

$$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$

The supplied material characterizes these fields as manifestly causal and propagating at $c$ (Onoochin, 30 Sep 2025).

The Coulomb gauge is defined by

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$0

Its scalar potential is the instantaneous solution of Poisson’s equation,

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$1

so that

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$2

The vector potential satisfies

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$3

and the fields are computed from the same reconstruction formulas,

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$4

The velocity gauge is a one-parameter family labeled by $$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$5 and defined by

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$6

This yields the decoupled equations

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$7

$$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$8

The scalar potential is retarded at speed $$\vec E = -\nabla \phi - \frac{1}{c}\partial_t \vec A,\qquad \vec B = \nabla \times \vec A.$$9,

$MB\in[0,1]$0

while $MB\in[0,1]$1 is obtained by the usual $MB\in[0,1]$2-retarded Green’s function acting on the source term above. The limiting relations are explicit: $MB\in[0,1]$3 recovers the Lorenz gauge, and $MB\in[0,1]$4 recovers the Coulomb gauge (Onoochin, 30 Sep 2025).

3. Causality critique and the claim that Coulomb and velocity gauges are unusable in applications

The central controversy in Onoochin’s treatment is that, although it is commonly accepted that the Lorenz, Coulomb, and velocity gauges are equivalent in the sense that they yield identical electromagnetic fields, the Coulomb and velocity gauges are said to produce solutions corresponding to superluminal propagation of the electric field (Onoochin, 30 Sep 2025). The paper’s conclusion is categorical: because such propagation has not been observed experimentally and is forbidden by special relativity, calculations in the Coulomb and velocity gauges may yield incorrect results and therefore cannot be used in applied electromagnetic calculations.

For the Coulomb gauge, the scalar potential is described as an instantaneous “action-at-a-distance” functional of $MB\in[0,1]$5 via the Poisson Green’s function

$MB\in[0,1]$6

Hence $MB\in[0,1]$7 carries information instantly across space. The source term $MB\in[0,1]$8 in the wave equation for $MB\in[0,1]$9 is likewise non-local and instantaneous, and the supplied account states that the net effect is that both pieces in $\partial_\mu A^\mu = 0,$0 contain parts that depend on $\partial_\mu A^\mu = 0,$1 and $\partial_\mu A^\mu = 0,$2 at the same time $\partial_\mu A^\mu = 0,$3 at arbitrarily large separation (Onoochin, 30 Sep 2025).

A concrete example is given for a point charge $\partial_\mu A^\mu = 0,$4 at rest until $\partial_\mu A^\mu = 0,$5 at a point $\partial_\mu A^\mu = 0,$6 a distance $\partial_\mu A^\mu = 0,$7 from a detector $\partial_\mu A^\mu = 0,$8, after which it suddenly moves with velocity $\partial_\mu A^\mu = 0,$9. For $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$0, the light front from $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$1 has not reached $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$2, so a causal electric field should vanish. In the Coulomb gauge, however,

$$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$3

even for $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$4. The logarithmic term is said to arise from the volume integral of $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$5 over the sphere of radius $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$6, which is presented as an explicit demonstration of “superluminal” or “instantaneous” arrival (Onoochin, 30 Sep 2025).

For the velocity gauge with finite $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$7, the scalar potential is retarded at a superluminal speed $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$8. When $$\nabla\cdot \vec A_L + \frac{1}{c}\partial_t \phi_L = 0.$$9 is built by integrating its nonlocal source $c$0 over the $c$1-retarded light cone, contributions arise from space-time points satisfying

$c$2

which can lie outside the true light cone $c$3 if $c$4. The supplied summary states that a power-series argument shows that the net $c$5 contains terms proportional to $c$6 that do not cancel $c$7, and hence remain nonzero for spacelike separations (Onoochin, 30 Sep 2025).

The paper ties this directly to special relativity: no signal or physical influence can propagate faster than $c$8, and electromagnetic fields carry energy and momentum. It therefore treats the uncanceled instantaneous or $c$9-retarded pieces in $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$0 and $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$1 as physically unacceptable. A common misconception addressed by the paper is that formal gauge equivalence is automatically sufficient for applied work; the argument presented is that, in these gauges, the gauge-dependent pieces no longer cancel exactly at superluminal separations, so practical calculations can acquire spurious field impulses before the true signal arrives (Onoochin, 30 Sep 2025).

4. Memory Buoyancy as a dynamic forgetful gauge

In the Semantic Desktop framework, every e-mail, file, calendar entry, contact, Web page, or annotation is represented as a “thing” in a Personal Information Model connected in a rich semantic graph. Memory Buoyancy is the core mechanism by which the Semantic Desktop continuously measures each information “thing’s” short-term relevance and thereby drives Managed Forgetting. It is explicitly described as a dynamic “forgetful gauge” that tells the system which items to keep in view and which to let sink below the user’s radar (Jilek et al., 2018).

The score is normalized,

$$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$2

and indicates an item’s current value for a particular user. When $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$3 falls below configurable thresholds, the system may hide items from search results, demote them in sidebars, synchronize them less eagerly, or eventually delete them. The supplied material emphasizes that this avoids a rigid keep-or-delete paradigm (Jilek et al., 2018).

The initial formulation separates a static, time-independent component from a dynamic, time-dependent component. The static part is

$$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$4

where $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$5 is the resource, $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$6 the user, $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$7 the context, $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$8 a base weight by resource type, $$\phi_L(\vec r,t)=\frac{1}{4\pi}\int \frac{\rho(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r',$$9 the number of semantic links in the PIMO, and $$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$0 a tuning constant. For stimulation events at times $$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$1, the saturating stimulation function is

$$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$2

and the piecewise decay is

$$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$3

with $$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$4. The total initial value is

$$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$5

Implementation follows the same decomposition: the static part is updated only on graph changes, while the dynamic part is computed on demand when an application requests MB (Jilek et al., 2018).

The three years of daily use reported in the supplied material yielded both success and failure cases. As a success, workshop artifacts faded gracefully: immediately after a meeting, related e-mails, tasks, and slides all showed $$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$6; eight months later only project people and photos still appeared; two years later only the top-level project node and frequently revisited photos remained visible. As failures, the semantic-search “MB threshold” slider was habitually dragged down to $$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$7, bypassing forgetting, and documents relevant in one context resurfaced in another because the initial model ignored user context (Jilek et al., 2018).

To address these limitations, the advanced version introduced Local MB, Global MB, and Group MB. Local MB measures an item’s relevance for a user within a context, and when the user switches context the old local values are frozen and do not decay further until that context is revisited. Global MB is a context-free aggregate shielded from sudden context-switch perturbations. Group MB summarizes relevance across all users in the Group Information Model and drives shared-memory aging and archival. Each variant reuses the same static-dynamic architecture but filters stimulation events and decay behavior according to context or group membership. The supplied interpretation is that this restores coherent “golden threads” across switches (Jilek et al., 2018).

5. ForgetEval and control-plane gauges for agent-memory forgetting

ForgetEval studies forgetting not as passive decay but as correctness of control-plane mutation. Its core claim is that where intelligence is placed in the pipeline—deterministic primitives only, an LLM at inscribe time, or an LLM at mutation time—shapes which forgetting failure modes are recovered (Yang, 14 Jun 2026).

The benchmark is deterministic and substring-match based. Each test case $$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$8 contains setup_facts, control-plane calls (supersede, release, purge), a final query $$\vec A_L(\vec r,t)=\frac{1}{4\pi c}\int \frac{\vec J(\vec r',\,t-|\vec r-\vec r'|/c)}{|\vec r-\vec r'|}\,d^3r'.$$9, a must_contain set $$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$0, and a must_not_contain set $$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$1. For top-$$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$2 retrieved hits

$$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$3

the indicators are

$$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$4

A case passes iff

$$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$5

and over $$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$6 cases the accuracy is

$$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$7

The suite uses $$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$8 on the deterministic template suite and $$\vec E_L=-\nabla\phi_L-\frac{1}{c}\partial_t\vec A_L,\qquad \vec B_L=\nabla\times\vec A_L.$$9 on the adversarial layer. Lift from baseline $c$0 to enhanced system $c$1 is defined as

$c$2

An example reported in the supplied material is the $c$3 pt joint-placement lift on the external 77-case subset (Yang, 14 Jun 2026).

The six-method Adapter Protocol keeps recall LLM-free. A backend implements reset, inscribe, and recall_texts, with optional supersede, release, and purge. If a primitive is absent, NotImplementedError yields an honest “N/A”; main tables exclude N/A cases from the denominator, while a strict-denominator score also treats N/A as failures (Yang, 14 Jun 2026).

The three placement regimes have complementary behavior. Deterministic primitives are strong on lexical and temporal correctness—substring_trap ≥ 89 %, temporal_qualifier = 100 %, negation_trap ≥ 95 %—but weak on canonicalization, with identifier_obfuscation ≤ 5 % (2/38) and cross_lingual_identifier = 0 % (0/38). They also fail intent-aware deletion, with compound_fact = 0 % and prefix_collision reported as 82→0 % with no LLM. Inscribe-time LLM systems recover canonicalization entirely—identifier_obfuscation = 100 % and cross_lingual_identifier = 100 %—but still fail deletion-precision and intent-aware tasks, with prefix_collision = 0 % and compound_fact = 0 %. Mutation-time LLM hooks recover canonicalization to 92–100 %, prefix_collision to approximately 79 %, and compound_fact to 78–85 %, with overall ex-compound_fact 93.3–94.2 % (345 evaluable cases) and including compound_fact: 91.7–93.2 % (385 cases) (Yang, 14 Jun 2026).

The benchmark also quantifies operational trade-offs. On the 385 adversarial cases, the mutation-time hook with DeepSeek-V3 costs approximately $0.17. Reported latency per case on a single CPU is ~ 74 ms for Lethe, ~ 64 ms for LangGraph, ~ 191 ms for MemPalace, ~ 514 ms for Mem0 with vector plus Qdrant cold start, and ~ 2.3 s / case for the hooked mutation-time approach, while the recall path remains unchanged. The external 77-case subset reports Letta only: 52.5 % pass (32/61 evaluable) and Letta+LLM: 80.3 % pass (49/61), yielding the stated +27.8 pt lift (Yang, 14 Jun 2026).

A misconception explicitly addressed here is that memory benchmarking is equivalent to recall benchmarking. The supplied material states that production failures are predominantly forgetting failures rather than recall failures, citing cases such as rotated credentials still being suggested and GDPR-deleted records still being retrieved. ForgetEval is presented as filling that gap by decomposing five primitive families and ten adversarial categories under deterministic scoring (Yang, 14 Jun 2026).

6. Comparative interpretation across the three usages

Across the supplied sources, the gauge concept serves different technical purposes. In electrodynamics it constrains the potentials from which $c$4 and $c$5 are reconstructed; in Semantic Desktop systems it measures short-term relevance and triggers forgetting actions; in agent-memory evaluation it operationalizes whether forgetting mutations behave correctly under adversarial conditions (Onoochin, 30 Sep 2025).

The most substantive commonality is selective suppression. In the electrodynamic account, the issue is whether gauge-dependent pieces cancel so that only causal propagation survives. In Memory Buoyancy, the issue is whether semantic items sink below visibility thresholds without destroying future recoverability. In ForgetEval, the issue is whether supersede, release, and purge remove or preserve the right records under lexical, canonicalization, and intent-aware perturbations. This suggests that “forgetful gauge” is best understood as an abstract editorial umbrella for mechanisms that regulate disappearance under constraints, rather than as a single established term spanning the three literatures.

The principal divergence is normative. Onoochin’s argument is exclusionary: Coulomb and velocity gauges are said to be pathological for real, time-dependent sources, while the Lorenz gauge alone “remembers” light-speed propagation and is therefore the only safe choice for applied electromagnetic problems (Onoochin, 30 Sep 2025). Memory Buoyancy is adaptive rather than exclusionary: forgetting is a design goal, provided that it preserves trust, coherence, and context sensitivity (Jilek et al., 2018). ForgetEval is diagnostic: it does not prescribe one storage substrate, but it does show that architectural placement of LLM intelligence matters more than whether an LLM is present at all, because different placements recover different forgetting failure modes (Yang, 14 Jun 2026).

Taken together, the sources indicate that a gauge becomes “forgetful” when it mediates attenuation, suppression, or deletion in a technically consequential way. In one domain this is treated as a source of unphysical artifacts; in the others it is treated as the basis of controlled memory management. The shared vocabulary is therefore structural rather than doctrinal, and the differences in what counts as acceptable forgetting are central to the topic rather than incidental.