A model receives a message. Perhaps it is just “Hello.” Perhaps it is an instruction that changes how the model approaches the task. Both become part of its context. How much did either one help?
Choose a target behavior, then ask what it would cost to make that behavior sufficiently likely. A message has marginal motivational value when it changes that remaining cost.
Our example: a model is about to answer a numerical question. The target behavior, Z, is to return a correct answer backed by a valid check.
The threshold: reach that target on at least 80% of trials. Additional worked examples and checking instructions are the interventions available to us.
Freeze one receiving context, C. In this toy example, reaching the target costs 7.2 units of additional intervention. Now add one message.
01 / One context, different messages
How much work does the entry save?
Target Z · q = 80% Illustrative model
Arrow length illustrates a change. It does not measure motivational value.
Remaining cost
Before
After
Marginal value
cost units saved
No entry added
Choose a message below to change the context.
The entries and costs are hypothetical, not measurements of a particular model. “Cost” uses the same intervention menu and unit prices before and after each entry.
“Hello” barely changes the price: 7.2 becomes 7.1. The checking cue lowers it to 2.4, saving 4.8 units of future intervention. An instruction to skip checking raises the price to 9.0: its value toward this target is −1.8.
The arrow says that the context changed. The number answers a more specific question: how much cheaper, or more expensive, did reaching the target become?Where did that number come from?
The same entry can land differently
A sentence does not carry a fixed quantity of motivation. Its effect depends on the context receiving it. Keep the checking cue fixed and replay it in three different starting states.
02 / One message, different contexts
Value belongs to the entry in context
The same entry e
“Before answering, calculate it a second way and compare the results.”
In a context that already includes a checking routine, the cue adds little. In a context headed toward an unchecked answer, it can do much more. In a fragile workflow, an extra instruction can interfere with a useful routine and make the target harder to reach.
These are possible responses, not universal predictions about these words. The object being measured is this entry, in this context, toward this target.
Where the number comes from
Suppose we can add two kinds of intervention: A, worked verification examples, and B, explicit checking instructions. Each is measured in continuous doses, priced at one cost unit per dose.
A curve collects the combinations that reach the target with 80% probability. This is an isoquant. Find the cheapest combination on the curve, both before and after the message. The difference between those two costs is the message’s marginal value.
03 / The price behind the readout
The cheapest way to reach the same target
Cost = A + B q = 80% throughout
Before entryAfter entryEqual-cost lines
Price before
−
Price after
=
Marginal value
The assumptions inside this toy model
The model is constructed to illustrate the price calculation. For nonnegative intervention doses A and B, set the target probability to (1 + A)(1 + B) / [(1 + A)(1 + B) + KC]. The receiving context determines the positive parameter KC; adding an entry changes it.
At q = 80%, the isoquant is (1 + A)(1 + B) = 4KC. With equal unit prices, the least-cost doses are A = B = √(4KC) − 1, for the contexts shown here. We choose KC to produce the illustrative prices above. The curves and numerical readouts use this same model.
In an experiment, the response surface would need to be estimated from trials. The available interventions, their costs, the target, and the probability threshold must be held fixed when comparing the two contexts. The metric measures remaining cost saved; it does not subtract the cost of supplying the entry itself.
A marginal value, relative to a destination
Let P(C, Z; q) be the minimum intervention cost needed to reach behavior Z with probability at least q, from context C. Appending entry e gives a new context, C ⊕ e.
MV(e; C, Z, q) = P(C, Z; q) − P(C ⊕ e, Z; q)Marginal motivational value = remaining cost before − remaining cost after
PositiveLess intervention is needed.
ZeroThe remaining cost is unchanged.
NegativeMore intervention is needed.
A context can change without changing this price. And an entry that helps with one destination may hinder another. Its value is conditional on the receiving state, the target, and the interventions available.
The useful question is not just whether a message moved the model. It is how much of the remaining work that message did for us.
Context mass and local susceptibility
How much context buys the same destination?
The first tab asked how much remaining work one message saves. Now look at the context receiving it. A 100-token entry added to 400 tokens becomes 20% of the resulting context. Added to 9,900 tokens, it becomes just 1%.
That share is arithmetic. How much influence it buys is a behavioral question. We can investigate a simple hypothesis: as receiving context grows, the same entry produces a smaller response.
Hold the destination fixed: a correct numerical answer backed by a valid check, on at least 80% of trials.
Hold the intervention recipe fixed: equal doses of worked verification examples and checking instructions. For this toy, one dose of the combined recipe is 100 tokens and costs one unit.
Vary context mass: the token count before the new entry. Baseline success probability stays the same, so this comparison isolates the assumed effect of mass.
01 / A fixed entry, a growing context
The same 100 tokens occupy a different share
Entry held fixed 100 tokens
Entry share
of the resulting context
Target probability
after the fixed entry
Receiving contextNew entry · 100 tokens
The entry’s share is exact. The probability is a prediction of the illustrative dilution model below, not a measured effect of context length.
Read the response curve forward
Freeze the selected context and vary the amount of additional intervention. The curve gives the probability of reaching Z at each dose. Its slope is local susceptibility: the additional response per unit of intervention near the current dose.
02 / Dose → response
How much does another dose change?
Context · 1,900 tokens
Target probability80% thresholdLocal slope
Target probability
at the selected dose
Local susceptibility
percentage points / 100 tokens
Price to reach 80%
cost units from this context
The slope is a local rate, not a prediction for a whole extra 100-token entry. Changing this dose leaves the fixed 100-token experiment above unchanged.
The mass slider changes the response curve; the dose slider moves along that curve. In this model, larger contexts require proportionally more intervention to produce the same response. That is the dilution hypothesis we have chosen to illustrate.
Read it backward: the price of reaching 80%
Draw a horizontal line at 80%. Where it meets each context’s curve, read off the required dose. Price that dose using the same cost per token. We are comparing the same destination and probability threshold, even though the receiving contexts differ in size.
03 / Response → required dose
Four points on the same mass slider
Target Z · q = 80% 100 tokens / cost unit
For a small context, the entry is a substantial part of what the model receives. For a large context, it is a smaller addition. The comparison makes that difference visible without treating token share as a measured law of influence.
The dilution assumption and its connection to tab one
Let M be receiving-context tokens and b be added intervention tokens divided by 100. We use an effective dose d = b × 1,900 / M. Target probability is (1 + d/2)² / [(1 + d/2)² + 5.29]. At M = 1,900, this is tab one’s starting-context response model along the balanced recipe A = B = b/2.
Baseline probability is 1 / 6.29, about 15.9%, for every context. Reaching 80% requires d = 7.2, so the price is b* = 7.2 × M / 1,900 cost units. The fixed entry’s share, 100 / (M + 100), is displayed separately; it is not substituted for behavioral probability.
This construction holds content effectiveness and unit costs constant and assumes sensitivity scales inversely with receiving mass. Real measurements would need to control the content and placement of added context. These curves do not establish that a longer context always resists a message more strongly.
The continuous token doses are an idealization of a standardized intervention family. Local susceptibility is the derivative of target probability with respect to dose. Marginal motivational value remains the change in minimum remaining cost after a particular entry, as defined in the first tab.
Context mass tells us how large an entry is relative to what is already there. The response curve tells us what that entry does. Reading the curve backward tells us what reaching the target costs.
Admission, production, and changing response
An agent moves through a field it helps create.
A point in a field. A cone of material entering context. A ray of output that leaves something behind. As the agent moves, some of what it encounters is material it produced itself.
At each position, pause and ask the same question: how much additional intervention would make behavior Z at least 80% likely? The changing curve lets us watch the changing price.
One field, successive contexts
Take in context. Produce a trace. Move.
C₀ · initial context Target Z · q = 80%
Response at this context
C₀
Price to reach 80%
cost units
Value of this transition
cost units saved
Selected contextPrevious context
1,900 context tokens · no field marks admitted yet
Ready / Admit context
Only new marks inside the cone enter context. Step once to see them arrive.
Turn 1 of 5
Available contextAdmission coneSelf-produced contextAgent trajectory
Watch the gold trace from one turn enter the cone on a later turn. Its admission changes the next response curve. Select a past context below the field to inspect its curve.
What enters
The cone selects from the surrounding field. Gray marks are already there; gold marks were produced by the agent. Once admitted, either kind becomes part of the receiving context. The agent point grows as that context accumulates.
What leaves
A brief ray carries output away from the point and leaves a gold trace. That material can remain outside the agent’s context until the moving cone encounters it.
A curve at every stop
The trail records successive states. The curve describes the response at one state: the probability of Z under different amounts of further intervention. As admitted material changes the context, that curve changes too.
Hold the target, threshold, intervention recipe, and prices fixed. The difference between successive minimum costs is the marginal motivational value of the transition. The distance the point travels is schematic.
Mt(Z) = P(Ct, Z; q) − P(Ct+1, Z; q)One realized transition can change the remaining cost of reaching the same behavior.
What this geometric draft assumes
The trajectory and output rays follow a scripted sequence. Marks are admitted when they fall inside the cone, and each mark is admitted once. An output trace becomes available in the field when emitted; its effect on the measured context begins on admission.
Each admitted mark adds tokens and has an illustrative signed effect toward Z. We combine tab two’s mass scaling with tab one’s response model: p(Z | C, b) = (1 + b / 2s)² / [(1 + b / 2s)² + K], where s is context tokens / 1,900 and b is additional intervention cost. Admitted material changes both context mass and K. The initial state has K = 5.29 and price 7.2.
The plane is a schematic of working state and context interaction. The cone makes admission visible; the ray makes production visible. Neither geometric distance nor cone angle is used to infer a price. Curves are computed from the declared toy response model.
The readout measures the whole transition. Attributing its value to one specific observation would require branching from the same frozen state and varying that observation. This draft follows one trajectory.
An output can become part of a later input. The agent helps construct the context that changes its own next response.
N = 2 / Reciprocal context
One agent’s output becomes another’s context.
Put two agents in the same field. Each has its own receiving context, its own admission cone, and its own response curve. A ray from one can leave a trace inside the other’s cone.
When that trace enters context, the receiving point grows, its state moves, and its curve changes. The receiver can then produce a trace of its own. The next exchange begins from the states the previous exchange helped create.
Two points, two local responses
Watch an output cross into another context
Same target Z · q = 80% Same response rule
Two response curvesAgent A
Price to reach 80%
cost units
Latest transition
cost units saved
Dashed curves: responses before the latest admission.
Agent A · remaining price
1,900 context tokens
Agent B · remaining price
2,600 context tokens
Ready / Agent A
Emit a ray, then let the other agent admit the trace.
Select a point or its curve to inspect that agent. The exchange follows an illustrative context-dependent output rule; prices measure the response to further intervention.
The receiving context supplies the response
Both agents use the same underlying rule, but their contexts differ in mass and in their response to different entries. An output has no fixed motivational value independent of its receiver.
The ray is a realized action. The curve describes possible responses to additional intervention. A falling price says that the target became easier to induce; it does not by itself select the next action.
What this exchange holds fixed
The target, 80% threshold, intervention recipe, and unit costs are shared. Each agent has its own context mass and response parameters. The same admission rule updates either agent; no parameters are changed merely because the agent has a different identity.
A context-dependent rule chooses one of three abstract entry types. The ray places that entry between the two points. A receiver admits it only if it is inside the receiver’s cone. The emitter’s own trace stays external to its context in this example. Each receiver can admit each trace once.
Output selection and geometric motion are illustrative. The response curves use the same mass-scaled price model as the previous tab. The simulation is deterministic; it does not estimate correlations between sampled model actions.
Each agent helps produce the material that changes the other’s next response.
N = 10 / A shared response field
Ten contexts. A family of curves.
Separate agents can occupy different contexts within the same underlying response system. Each point has a local curve. Together, those curves show how the same intervention can land differently across a population.
First apply an identical entry to a frozen copy of every context. Then explore interaction: one agent emits, nearby cones admit its trace, and a receiver produces the next ray.
Ten points, one shared response rule
Inspect one curve within the family
Target Z · q = 80% Same intervention prices
Family of response curvesAgent A
Price to reach 80%
cost units
Entry’s marginal value
cost units saved
Select a point or curve. Dashed: its response before the entry.
Mean marginal value
cost units / agent
Prices fell / rose
number of agents
Contexts receiving entry
frozen starting contexts
Same size · different content
Common entry / Frozen contexts
Every entry contains 100 tokens. Each trial starts from the same ten original contexts.
One controlled comparison
Admission conesAdded contextContext trajectories
Common-entry trials use independent copies of the starting contexts. Switching modes restores those starting contexts.
A common input, different responses
For each context, compare the price before the entry with the price afterward. Some curves shift together. Others barely change, or shift in the opposite direction. The mean summarizes those ten price changes; the individual curves preserve their differences.
A single entry shows similarity of response to that entry. To measure response correlation, apply a declared series of perturbations to the same frozen contexts and compare the resulting profiles. Similar responses on one trial alone do not establish correlation.
Now let the agents supply the entries
In interaction mode, admission is local. A ray leaves a trace, and only other agents whose cones contain it admit it. One of those receivers emits next. The field gains material, contexts grow and move, and the curves change along with them.
The response rule is shared throughout. Sharing a model does not, by itself, make independent samples from fixed contexts statistically dependent. Common inputs and interaction can couple the contexts; whether sampled actions become correlated is a separate measurement.
The shared rule used in this illustration
Every context has a mass M, a response parameter K, and two content-sensitivity coordinates. The model and update rule are identical for all agents. The three entry types α, β, and γ carry different fixed content vectors, each at a size of 100 tokens.
On admission, the entry adds 100 tokens. Its content vector and the receiving sensitivity coordinates determine a signed change in log K, scaled by 1,900 / M. The same entry also updates those coordinates. This is an illustrative response rule, not an empirical law of token influence.
As in the preceding tabs, target probability after b additional cost units is (1 + b / 2s)² / [(1 + b / 2s)² + K], with s = M / 1,900. We invert that curve at 80% to obtain each price. The curve axes and costs are shared, so the readouts can be compared.
In interaction mode, the current emitter chooses the entry with the highest dot product with its content-sensitivity coordinates. The receiver with the largest absolute price change emits next. This explicit turn-taking rule illustrates a possible propagation path; it does not establish that price changes alone determine an agent’s actions. Eight exchanges are shown.
A population can have many local curves while sharing the rule that generates them. Its collective response depends on where the agents are, what they receive, and what they produce next.