FRAME: narratives as causal graphs with an explicit residual

What this file is. A complete text version of the theoretical framework for the ERC Starting Grant 2026 proposal FRAME (PI: Benjamin Dybro Liengaard, Aarhus University). It reproduces everything in the interactive overview FRAME_residual_and_primitives.html: the visible text, the hover explanations, the worked examples and every cost breakdown. It adds development notes that are not in the HTML (section 12), a placement of the framework in the existing literature (section 13) and a list of the changes still needed in the proposal drafts (section 14). The equations alone are also kept in the Claude Doc “FRAME Model Equations”.

Status (13 October 2026). This framework is newer than the current proposal drafts (ERC_StG_2026_B1_v4.docx, ERC_StG_2026_B2_v4.docx). Where the drafts and this file disagree, this file is the newer version. All numbers come from a prototype model with illustrative parameters. The orderings (which holder pays more, which placement is cheaper, who adopts) held in all 27 parameter combinations tried; the magnitudes did not, so treat every number as illustrative. The utilities in section 7 are illustrative by construction: utility is exogenous to the model.

What changed on 12–13 October 2026. (1) Entrenchment is explained as resistance to changing a link, on a 0–1 scale. (2) Primitive 3 is now statement placement, formerly vertex mutation. (3) Primitive 4, narrative utility, now comes with the decision rule that binds the four primitives: the agent takes the narrative with the highest anticipated utility net of the revision cost, which acts as a switching cost. (4) The HTML overview has seven panes: The narrative (with strength and entrenchment as matrices in a folded section), Rigidity, Revision cost, Statement placement, Narrative utility, What it explains, Can it fail?

0. Terminology rules (read first)

The four primitives, always in this order:
1. Rigidity: a property of one graph.
2. Narrative revision cost: a property of a pair of graphs.
3. Statement placement: a choice.
4. Narrative utility: a payoff, and the decision rule that binds the four.
“Narrative revision cost” replaces “merge friction”. Reason: people do not merge narratives; they always revise their own. Use “merge friction” only when quoting the old drafts.
“Statement placement” replaces “vertex mutation”. Reason: the primitive is about where an agent places a new statement it accepts as true, when the message leaves its causal role open. Nothing about the statement mutates; only its links differ. Use “vertex mutation” only when quoting the old drafts.
A narrative is held by an individual agent, never by a household. The one exception is when quoting B1’s survey wording, e.g. “households holding open… narratives”.
Closed narratives never break; they only defend. Abrupt breaks belong to highly rigid semi-closed narratives, not to closed ones. If a draft says “closed narratives break abruptly”, correct it to “highly rigid (semi-closed) narratives”.
Explain quantities by what their scale means, not by their mechanics. Entrenchment is “the agent’s resistance to changing the causal link, on a scale from 0 (changes freely) to 1 (unchangeable)”, not “counter-evidence moves the link 20% of the way”. The mechanical reading is kept only where it is the argument, as in sticky expectations (section 9.1).
Utility is the destination, revision cost the route. Anticipated utility belongs to a narrative, wherever the agent starts from, and is exogenous. Revision cost belongs to a change and is computed from the held narrative’s entrenchment and rigidity. Never derive one from the other (section 12 records why).
Core vocabulary:
- statement (vertex)
- link (edge)
- strength (share attributed to a cause)
- entrenchment (resistance to changing a link, 0 to 1)
- residual (the unspecified causes; a vertex of its own)
- rigidity (the entrenchment of the residual)
- opening cost (the price of drawing a new cause out of the residual)
- revision cost (the cost of the cheapest revision that accommodates an incoming narrative; the switching cost in the decision rule)
- placement (where a new statement is linked into the graph)
- anticipated utility Û (what holding a narrative is worth to the agent)
- target statement B

1. A narrative

A narrative is one agent’s causal account of a target statement B:

Nᴮ : (Kᴮ ∨ Zᴮ) → B

Statements are vertices. A statement is held true within the narrative (e.g. “energy prices rose”).
Perceived causal links are edges.
Kᴮ, the specified causes: the causes of B that the agent names.
Zᴮ, the residual: every cause of B that the agent leaves unspecified. The residual is a vertex of its own, drawn like a statement box.

In full, a narrative is N = (S, W, R):

S is the set of statements.
W holds the strengths.
R holds the entrenchments.

Every statement v has its own residual vertex Z_v pointing into it, and the residual’s entrenchment is the rigidity ρ_v. What the narrative is worth to the agent, its anticipated utility, is not part of the object: it is a separate primitive (section 7).

The running example: “A simplified narrative of rising prices”

This is deliberately minimal: a cost-push narrative of inflation. In the HTML, every link carries two numbers, the strength above the link and the entrenchment below it.

Gas supply cut ──0.7 / 0.3──▶ Energy prices rose ──0.3 / 0.6──┐
                                                              ▼
Raw material prices rose ─────────0.5 / 0.8──────▶ Prices are rising   (target statement B)
                                                              ▲
Unspecified causes (residual Zᴮ) ·······0.2 / ρ = 0···········┘

Strength (the first number): the share of the effect that the agent attributes to the cause.
Entrenchment (the second number): the agent’s resistance to changing the link, from 0 (changes freely) to 1 (unchangeable). The residual’s entrenchment is the rigidity.

Statements:

Statement	Named causes	Residual share z
Gas supply cut	none	z = 1 (all its causes are unspecified)
Raw material prices rose	none	z = 1
Energy prices rose	gas supply cut (0.7)	z = 0.3
Prices are rising (target B)	raw materials (0.5), energy prices (0.3)	z = 0.2

Links:

Link	Strength w	Entrenchment r	Reading
Gas supply cut → Energy prices rose	0.7	0.3	weakly held: changes easily
Energy prices rose → Prices are rising	0.3	0.6	moderately held
Raw material prices rose → Prices are rising	0.5	0.8	firmly held
Residual Zᴮ → Prices are rising	z = 1 − 0.5 − 0.3 = 0.2	rigidity ρ = 0 (open)	opening a new path into prices is free

What the residual holds. Rising wages, loose monetary policy, strong consumer demand: anything the agent has not named.

2. The two matrices: strength W and entrenchment R

In the HTML these sit in a folded section under the figure, “Strength and entrenchment as matrices”. Each has a row for each cause and a column for each effect. Every statement has a residual pointing into it, though the figure draws only the target’s: the residual’s row in W is the share left to unspecified causes, and its row in R is the rigidity.

Strength W

W_uv is the share of statement v that the agent attributes to cause u. The named shares plus the residual add up to 1. This is Kelley’s discounting principle: naming a big new cause discounts the others.

cause effect	Energy prices rose	Prices are rising
Gas supply cut	0.7	·
Raw material prices rose	·	0.5
Energy prices rose	·	0.3
residual z	0.3	0.2
column sum	1	1

The residual is the slack of W:

z_v = 1 − Σ_u W_uv

Whatever share of v its named causes do not take is left to unspecified causes. Every statement has a residual. A statement with no named causes, like the gas supply cut, has z = 1.

Entrenchment R

R_uv is the agent’s resistance to changing the causal link u → v, on a scale from 0 (changes freely) to 1 (unchangeable). Behaviourally, r is the share of an evidence-implied revision that the agent refuses (see “From entrenchment to a price”).

cause effect	Energy prices rose	Prices are rising
Gas supply cut	0.3	·
Raw material prices rose	·	0.8
Energy prices rose	·	0.6
rigidity ρ (residual row)	0	0

In this example ρ = 0 for both statements, so new paths open for free. An agent who insists “the only reasons are raw materials and energy” would have ρ near 1 for prices (section 4).

From entrenchment to a price

φ(r) = r / (1 − r)

φ(r) is 0 at r = 0 and grows without bound as r → 1.
Evidence of precision Π moves a link a share G = Π / (Π + φ(r)) of the way. At Π = 1 this is exactly 1 − r: counter-evidence of unit precision moves the gas → energy link 70% of the way, energy → prices 40%, raw materials → prices 20%.
G is what you get if the agent trades off the revision cost ½ φ(r)(w − w₀)² against the misfit to the evidence ½ Π(w − w*)². This is the decision rule of section 7 at the level of one link, with the value of fitting the evidence as the utility.

Statement entrenchment e

Each statement also has its own entrenchment e ∈ [0, 1]: the agent’s resistance to giving up the statement itself, from 0 (given up freely) to 1 (cannot be given up). In the prototype, e ranges from 0.3 to 0.6; “energy prices rose” has e = 0.5.

3. The price list: what each change to a narrative costs

Every change to the narrative has a cost, and a revision costs the sum of its changes:

C(N → N″) = Σ c(change)

Change	Cost	Example in the running narrative	Cost
Add a statement	κ⁺	(plus its links, priced separately)	0.10
Add a statement and its link	κ⁺ + κ + ½ φ(r₀) w²	new: wages rose → prices, at 0.2	0.21
Give up a statement and its links	κ⁻ + ½ φ(e) + Σ (κ + ½ φ(r) w²)	give up “energy prices rose”	0.97
Add a link	κ + ½ φ(r₀) w²	new: gas supply cut → prices, at 0.1	0.10
Remove a link	κ + ½ φ(r) w²	remove raw materials → prices (0.5)	0.60
Change a link’s strength	½ φ(r) (Δw)²	raw materials → prices, 0.5 → 0.4	0.02
Open new paths into a statement v	½ φ(ρ_v), once per revision	semi-closed holder (ρ = 0.9): any new cause of prices	4.50

What each term means

κ⁺, the fixed cost of a new statement. The price of accepting a statement the agent did not hold before, such as “wages rose”. Its links are priced on top. Here κ⁺ = 0.1.
κ⁻, the fixed cost of giving up a statement. The price of dropping a statement the agent held true. Belief perseverance suggests κ⁻ may exceed κ⁺; here both are 0.1.
κ, the fixed cost of a link. Paid for every link formed or removed, whatever its strength. The number of links that must be re-formed predicts which revision people choose and how long it takes (Krumnack et al., 2011). Here κ = 0.1.
½ φ(r₀) w², the strength cost of a new link. A new link is priced as a move from 0 to w at r₀ = 0.3, the entrenchment that new links start with. Stronger new links cost more.
½ φ(r) w², the strength cost of a removed link. A removed link is priced as a move from w to 0 at the link’s own entrenchment. Strong, entrenched links are the costliest to remove; at r = 1, removal is impossible.
½ φ(e), the cost of giving up a statement. Giving a statement up is a full change, from held to not held, so the variable cost is ½ φ(e). The Σ in the formula runs over every link into or out of the statement; each goes with it at κ + ½ φ(r) w².
½ φ(r)(Δw)², the entrenchment term. Every link whose share changes costs this much. The term is convex, so it produces partial adjustment: narratives bend.
½ φ(ρ) = ½ ρ / (1 − ρ), the opening cost. The price of getting anything out of the residual: either a new cause, or, when the residual is empty, the admission that other causes exist. It is paid once per revision, however many paths open. It is a fixed cost, which is why rigid narratives break rather than bend. At ρ = 1 it is infinite.

Rules for the residual

An open residual moves for free. z = 1 − Σ w takes up or gives up share at no cost.
If a new cause needs more share than the residual holds, the other links shrink, and each shrink is priced as a change of strength.
Opening means drawing on the residual. It happens when a revision names a new direct cause of v, or when z_v = 0 and the revision moves share into the residual (admitting that other causes exist).
New paths are direct links into v only. An upstream cause, such as shipping costs → raw material prices, is priced by the residual of raw material prices, not by the residual of the target.
Once the opening cost is paid, the residual stays open (ρ is set to 0). How a residual re-entrenches afterwards is an open question (section 12).

Parameters in the prototype

κ = κ⁺ = κ⁻ = 0.1
r₀ = 0.3
statement entrenchment e from 0.3 to 0.6
φ(r) = r/(1 − r), chosen so that unit evidence moves a link 1 − r of the way
anticipated utility Û: exogenous; the values in section 7 are illustrative

4. Primitive 1 of 4: Rigidity (a property of one graph)

Definition. Rigidity ρ is the entrenchment of the residual Zᴮ, the vertex that points into B and holds every cause not yet named. Opening any new path into B draws on it and costs ½ φ(ρ) once, however many paths open. Rigidity runs from open (ρ = 0), where new paths are free, to closed (ρ = 1), where they are impossible.

Rigidity is a property of the residual, not a summary of how firmly the named links are held: the two come apart (section 12). It is measured, not derived from utility; what a high ρ reveals about the agent is discussed in section 12.

The typology

Type	ρ	Residual	Opening cost	Description	Typical text marker
Open	0	Explicitly non-empty and revisable (in the example z = 0.2)	0.00	The scientific stance	A hedge: “other factors may play a role”
Semi-closed	0 < ρ < 1 (default 0.3; 0.9 for the late-break examples)	Treated as empty (z = 0) for fast, practical reasoning, but can be opened at a finite cost	½ φ(ρ): 0.21 at ρ = 0.3, 4.50 at ρ = 0.9	Most everyday reasoning. The higher ρ, the later and more abruptly it breaks	—
Closed	1	Denied: “the only reason” (z = 0)	∞	No evidence can open a new path, so the narrative never breaks; it can only defend. The question “what else could have caused B?” is out of reach, which links closure to dogmatism and to sacred values	A completeness claim

In the figures, the closed residual is drawn as a crossed-out Zᴮ.

The model holders

The holders differ only in how closed “prices are rising” is:

Holder	Raw materials → prices	Energy prices → prices	z for prices	ρ for prices
Open	0.5	0.3	0.2	0
Semi-closed	0.6	0.4	0	0.3
Semi-closed, highly rigid	0.6	0.4	0	0.9
Closed	0.6	0.4	0	1

Everything else is the same for all of them:

gas supply → energy prices at 0.7, with z = 0.3 for energy prices
all link entrenchments as in section 2
the residuals of energy prices and of raw materials are open (ρ = 0)

Worked example: “Energy prices did not rise”

The same message gets three different responses:

Open holder (C = 0.97). The share of energy prices moves to other factors: z 0.2 → 0.5.
- give up “energy prices rose” (e = 0.5), κ⁻ + ½φ(e): 0.60
- remove gas supply → energy prices (0.7, r = 0.3): 0.21
- remove energy prices → prices (0.3, r = 0.6): 0.17
- residual of prices 0.2 → 0.5: free
Semi-closed holder, ρ = 0.3 (C = 1.24). The holder opens the residual: z 0 → 0.4.
- give up the statement: 0.60
- remove gas supply → energy prices: 0.21
- remove energy prices → prices (0.4, r = 0.6): 0.22
- open new paths into prices, ½φ(0.3): 0.21
Closed holder (C = 1.35). The holder doubles down: raw materials 0.6 → 1.0, “the only reason”.
- give up the statement: 0.60
- remove gas supply → energy prices: 0.21
- remove energy prices → prices (0.4): 0.22
- raw materials → prices 0.6 → 1.0 (r = 0.8), ½φ(r)Δw²: 0.32

For the semi-closed holder, opening (0.21) is cheaper than doubling down (0.32). For the closed holder, opening is impossible, so the share that is freed must go to the remaining named cause.

5. Primitive 2 of 4: Narrative revision cost (a property of a pair of graphs)

Definition. People do not merge narratives; they revise their own. The revision cost of an incoming narrative N′ is the cost of the cheapest revision of the held narrative N that accommodates it:

C(N′ | N) = min over N″ consistent with N′ of C(N → N″), where C(N → N″) = Σ c(change)

The changes are priced as in section 3. Rigidity enters only when the incoming narrative opens a new path into a statement. In the decision rule (section 7) the revision cost is the switching cost: it is subtracted from the anticipated utility of every narrative the agent does not already hold.

Worked example: “Demand surged, not costs”

The incoming narrative is closed: demand takes the whole of “prices are rising” (a new link at 1.0), and the cost links must go.

Open holder (C = 1.18):
- new statement “demand surged”, κ⁺: 0.10
- remove raw materials → prices (0.5, r = 0.8): 0.60
- remove energy prices → prices (0.3, r = 0.6): 0.17
- new link demand → prices at 1.0 (r₀ = 0.3): 0.31
- residual of prices 0.2 → 0: free
Semi-closed holder, ρ = 0.3 (C = 1.67): 0.10 + 0.82 (remove raw materials at 0.6) + 0.22 (remove energy at 0.4) + 0.31 + 0.21 (opening).
Semi-closed holder, ρ = 0.9 (C = 5.95): the same items, with the opening cost at 4.50.
Closed holder: ∞.

The holder × message table

Incoming narrative	What it does	Open (ρ = 0)	Semi-closed (ρ = 0.3)	Closed (ρ = 1)
Shipping costs rose → raw material prices	Adds an upstream cause: one statement and one link at the periphery	0.22	0.22	0.22
Wages also rose → prices	Adds a cause of prices without contradicting anything	0.21	0.44 (opens)	∞ (impossible)
Demand, not costs, drove prices	Replaces the causes of prices and claims completeness	1.18	1.67 (opens)	∞ (impossible)
Energy prices did not rise	Denies a statement: its links go with it, and its share of prices must go somewhere	0.97	1.24 (opens)	1.35 (doubles down)
Rising prices pushed energy prices up	Reverses a link: energy prices become an effect of prices, not a cause	0.29	0.55 (opens)	0.66 (doubles down)

What the table shows:

Shipping costs are priced the same for every holder. The new path goes into raw material prices, whose residual is open for everyone (z 1.0 → 0.7). Rigidity of the target does not matter for upstream additions.
“Wages also” is cheap for the open holder and impossible for the closed one. Nothing is contradicted, but a new path into prices must open.
Denials and reversals are always possible, even for the closed holder, because they remove links instead of adding new paths into B. The closed holder pays by pushing the freed share onto raw materials (0.6 → 1.0).

First new path pays

The opening cost is paid once per revision, however many paths open. Take a semi-closed holder with ρ = 0.9 (opening cost 4.50):

Revision	C
Wages alone	4.73
Demand alone	4.73
Wages and demand at once	5.00
Demand after wages has been absorbed (residual now open)	0.21

A closed holder cannot absorb either at any price. Once the first alternative cause has been let in, the rest come almost free. This route dependence is also what separates the revision cost from utility in data (section 7).

6. Primitive 3 of 4: Statement placement (a choice)

Definition. Where an agent places a new statement it accepts as true, when the message leaves its causal role open. The agent picks the placement with the lowest revision cost on the existing narrative, so rigid narratives place new facts where they leave the core intact.

N*(s) = argmin over N″ ∈ P(s, N) of C(N → N″), and C(s | N) = C(N → N*(s))

P(s, N) holds the revisions that link s to the narrative: as a cause of a statement already held, as an effect of one, or between two.
Leaving s unconnected is not a placement. It would cost only κ⁺ and always win. Whether to take s on board at all is decided by utility, as for any incoming narrative.
Revision cost is the minimum; placement is where it is attained. Placement is the decision rule of section 7 restricted to the placements of s, with every placement worth the same. If one placement is worth more to the agent (it keeps the blame where the agent wants it), the general rule applies: the placement with the highest utility net of cost.
Reinterpretation in the strict sense (reading “energy prices did not rise” as “rose less than feared”) changes the statement itself and is not part of this primitive. It would need a distance between statements, which the model does not have.

Worked example: the incoming statement “wages rose”

The agent accepts it as true, but the message does not say whether wages are a cause or an effect of rising prices.

Placed as a cause (wages rose → prices are rising). This opens a new path into B, so the agent draws on Zᴮ and pays the opening cost ½ φ(ρ): nothing for an open narrative, infinite for a closed one.
Placed as an effect (prices are rising → wages rose). B gains a consequence, not a cause, so no new path into it opens and its residual is untouched. The new link takes share from the residual of wages (1.0 → 0.8), which is open.

Revision cost of each placement:

Holder	as a cause	as an effect	placement
Open (ρ = 0)	0.21	0.21	either: the cost does not decide
Semi-closed (ρ = 0.3)	0.44	0.21	as an effect
Semi-closed (ρ = 0.9)	4.73	0.21	as an effect
Closed (ρ = 1)	∞	0.21	as an effect

Where the numbers come from:

The effect placement costs 0.21 for every holder: new statement κ⁺ 0.10, plus a new link prices → wages at 0.2, 0.11.
The cause placement for the semi-closed holder (ρ = 0.3) costs 0.44:
- new statement: 0.10
- new link: 0.11
- shrinking raw materials 0.6 → 0.55: 0.01
- shrinking energy 0.4 → 0.25: 0.02
- opening: 0.21

Two readers of the same article. The open reader can take rising wages as a cause, a sign that inflation is broadening. The closed reader places them as an effect of rising prices, and costs stay “the only reason”. Both graphs survive. It is also the second-round-effects debate: a wage–price spiral for one reader, catch-up for the other.

Prediction: placing a fact as an effect becomes more common as rigidity rises.

Measurement. Placement can be elicited as a forced choice (cause, effect or unrelated) or by asking people to draw the fact into their graph. This replaces the free-text coding of reinterpretations in B2’s H4 (section 14).

7. Primitive 4 of 4: Narrative utility (a payoff) and the decision rule

Definition. Narrative utility is the anticipated utility of holding a narrative: what it is worth to the agent, from comfort, moral order, belonging and predictive grip. It is exogenous to the model and belongs to the narrative, wherever the agent starts from.

The agent takes the narrative whose anticipated utility, net of the revision cost of getting there, is highest. Keeping the held narrative costs nothing:

N* = argmax over N″ of [ Û(N″) − C(N → N″) ], with C(N → N) = 0

The revision cost is a switching cost: a cognitive burden subtracted from every narrative the agent does not already hold. Utility is the destination, revision cost the route. The two are built from different things: utility is exogenous, the cost is computed from the entrenchment and rigidity of the narrative the agent holds now.

Adoption and diffusion

An incoming narrative N′ is adopted when its utility gain exceeds its revision cost, with C(N′ | N) the cheapest consistent revision (every revision consistent with N′ is taken to be worth Û(N′)):

adopt N′ when Û(N′) − Û(N) > C(N′ | N)

Diffusion is the share of agents for whom this holds. The same narrative spreads among open holders and stalls among rigid ones, with no difference in how it is valued.

Where utility comes from

Comfort: reassurance that the world makes sense and things will be fine.
Moral order: the story assigns responsibility and blame.
Belonging: the story is shared with the agent’s group.
Predictive grip on a domain: the story helps forecast what happens next. A larger residual means less grip. Fitting the evidence is this value: under accumulating evidence the rule becomes bend or break (section 8).

Worked example (HTML figure): the same narrative, two holders

The incoming narrative is “demand, not costs”. Both holders value it alike; only the revision cost differs. Utilities are illustrative, costs come from the model.

	Held narrative	Incoming, open holder (ρ = 0)	Incoming, rigid holder (ρ = 0.9)
Anticipated utility Û	4.5	6.5 (a gain of 2.0)	6.5 (the same gain)
Revision cost C	0 (keeping it costs nothing)	1.18	5.95, of which 4.50 is the opening cost
Net, Û − C	4.5	5.32	0.55
Decision	the value to beat	adopts (5.32 > 4.5)	keeps the held narrative (0.55 < 4.5)

A closed holder (ρ = 1) faces an infinite cost and never adopts.

Identification: utility differences from choices

The cost is computed from measured entrenchment and rigidity, so every adoption reveals that Û(N′) − Û(N) > C(N′ | N), and every rejection the reverse. Utility differences are therefore identified from choices at known prices, as switching costs are separated from preferences in markets (Dubé, Hitsch and Rossi, 2010). Two signatures tell the cost from the utility:

Route dependence. The cost depends on the order in which claims arrive: for a holder with ρ = 0.9, demand costs 4.73 on its own and 0.21 after wages have been absorbed. Utility depends only on the narrative held at the end.
Physiology (B2). Aversion on contradiction measures the cost side; reward on confirmation measures the utility side.

Û is not read off R. The value of a held explanation inferred from its entrenchment would be the cost of giving it up, counted a second time (section 12).

Open question. Should a larger residual lower utility through lost predictive grip? In the revision cost, moving share into an open residual is free. It should arguably cost predictive grip instead, so that utility falls as the residual grows. How to price this is not yet decided.

8. The decision rule binds the four primitives

Formerly “the acceptance-and-defence mechanism”. One rule covers acceptance, defence, placement and the response to evidence: take the narrative with the highest anticipated utility net of the revision cost; keeping the held narrative costs nothing.

Acceptance. A narrative is adopted when the utility it promises exceeds the revision cost of getting there (section 7).
Defence. When the rule says keep, the held narrative is defended. Defence takes three forms: rejecting the incoming narrative outright; placing the incoming statement where it leaves the core intact (section 6); and doubling down, pushing freed share onto the named causes when the residual cannot open (section 4).
Placement. The rule restricted to the placements of a new statement; with every placement worth the same it is the cheapest one (section 6).
Evidence: bend or break. With the value of fitting the evidence as the utility, Û(N″) = −½ Π · misfit(N″, N′)², the rule becomes: minimize ½ Π · misfit² + C(N → N″). The convex strength costs ½ φ(r)Δw² give partial adjustment, bending; the fixed opening cost ½ φ(ρ) gives thresholds, breaking (section 9.1).

9. What it explains

Measured structure turns the rule into explanations of economic and cognitive phenomena.

9.1 Sticky expectations

Entrenchment is stickiness. One unit of evidence moves a link only 1 − r of the way, so expectations that rest on entrenched links adjust slowly. In the workhorse models of expectation formation (Coibion and Gorodnichenko, 2015), this pins down the stickiness parameters: they become measured properties of the narrative instead of free parameters. The mapping is an observational equivalence: the same regression coefficients arise from resistance to revision rather than from missing information. The model output below is the decision rule of section 7 with fitting the evidence as the only utility, Û(N″) = −½ Π · misfit²; comfort, moral order or belonging attached to the held narrative would enter its utility and raise the bar for switching further.

Evidence of precision Π moves a link a share G = Π / (Π + φ(r)) of the way.
Noisy information: the coefficient of forecast errors on forecast revisions is β = (1 − G)/G = φ(r)/Π. So β = r/(1 − r) at Π = 1.
Sticky information: the share that does not update is λ = 1 − G. So λ = r at Π = 1.

Rigidity adds thresholds. Opening a new path has a fixed cost, so a rigid narrative ignores the evidence until it outweighs that cost, and then jumps. Opening pays once ½ Π w² exceeds ½ φ(ρ) plus the other costs, so the evidence needed grows with the odds of rigidity and with 1/w². Expectations de-anchor abruptly instead of drifting.

Model output. Evidence accumulates that wages carry a 0.4 share of rising prices. The wage share each holder accepts, as a function of accumulated evidence Π (log scale):

Holder	When it moves	How it moves
Open (ρ = 0)	at Π ≈ 3.0, once the evidence covers the small fixed costs κ⁺ + κ of a new statement and its link	adjusts gradually towards 0.4: it bends
Semi-closed (ρ = 0.3)	at Π ≈ 6.4: it must open a path first, which is cheap	then adjusts gradually
Semi-closed (ρ = 0.9)	ignores the evidence until Π ≈ 60, when the gain finally exceeds the opening cost ½φ(0.9) = 4.5	jumps almost all the way at once: it breaks
Closed (ρ = 1)	never: the opening cost is infinite	it can only defend

Prediction: survey expectations adjust more slowly where the prevailing narrative’s links are more entrenched, and where it is more rigid they ignore the evidence and then de-anchor abruptly. Entrenchment sets the speed of adjustment; rigidity sets the threshold and the jump. (B1’s draft prediction says stickiness rises with rigidity; that conflates the two, see section 14.)

9.2 Paradigm shifts (Kuhn, 1962)

In Kuhn’s account, a paradigm is a dominant narrative about a field’s causal structure, held with a nearly closed residual. Kuhn’s cycle maps onto the framework:

Normal science, where rigidity rises. Work inside the paradigm entrenches it, and its residual closes. (How entrenchment and rigidity build up after adoption is an open question, section 12.)
Anomalies, which are placed to fit. They are absorbed by statement placement: each is placed where it leaves the core intact, at the lowest revision cost.
Crisis, where costs pile up. Each anomaly lowers the paradigm’s predictive value, but nothing moves while the gain from switching stays below the revision cost, dominated by the opening cost ½ φ(ρ).
Revolution: the break. Once the gain outweighs it, the new paradigm is adopted under the decision rule: the residual opens and the core is rewired at once. This is the same jump as the ρ = 0.9 holder in 9.1.

9.3 Thinking fast and slow (Kahneman, 2011)

Under time pressure, people reason with folded narratives, in which one link stands for a whole chain. Given time, they unfold a link into its subnarrative: more statements, more links, and a residual at each step.

Fast (folded): gas supply cut → prices are rising. Quick to use, but it can only be kept or dropped as a whole.
Slow (unfolded): gas supply cut → energy prices rose → prices are rising. The intermediate statement is explicit, with its own links and entrenchment.

Unfolding takes time and working memory. The same message can then cost differently, because a contradiction can land on one step instead of on the whole link. And since every statement has a residual, unfolding opens new places where other causes can enter.

Not yet formalized: folding will be treated as a quotient of the graph, a chain collapsed into one link (B2 plans quotient graphs). Folded and unfolded narratives stay under “What it explains” until that definition exists; they are not a fifth primitive.

10. Falsifiability: the theory can fail

The framework forbids three patterns. Observing any one of them falsifies it:

Contradictions with a low revision cost routinely trigger defence responses. A low revision cost is computed with the agent’s own entrenchment and rigidity: “wages also matter” costs an open holder 0.21 but a closed holder ∞.
Households with open, low-rigidity inflation narratives de-anchor as abruptly as those with highly rigid ones. This keeps B1’s survey wording “households”.
Prevalence-only indices forecast expectations as well as structure-augmented models do.

11. Main results of the prototype, in one list

Closed holders double down. When a named cause is discredited, its share goes to the remaining named causes (“the only reason”), never to the residual.
The first alternative cause pays for the rest. The opening cost is fixed and paid once per revision; after it is paid, further new causes are cheap (4.73 vs 0.21 at ρ = 0.9).
Placement protects the core. For rigid holders the cheapest placement of an incoming fact is the one that opens no new path: wages placed as an effect of prices, not as a cause. For open holders the cost does not decide.
Bend versus break follows from convex versus fixed costs.
- Open narratives bend (gradual adjustment).
- Highly rigid semi-closed narratives break (late, abrupt jumps).
- Closed narratives never break.
Upstream additions are priced the same for every holder, because they draw on the upstream statement’s residual, not on the target’s.
Same gain, different outcome. Two agents who value an incoming narrative alike can split on adopting it: the revision cost, not the utility, decides (1.18 vs 5.95 for “demand, not costs”).

12. Open questions and development notes (not shown in the HTML)

These come from the development discussion and are not part of the overview page.

Open questions

Whether a larger residual should cost predictive grip in utility (section 7).
Whether closing a residual should cost anything.
How entrenchment and rigidity form after adoption, and how narratives re-close after a break. In the model, ρ is set to 0 once opened. Conjecture, testable: they rise with the anticipated utility the agent placed on the narrative when adopting it.
How to formalize folding and unfolding (quotient graphs).
The choice set of placements. “Between” placements (energy → wages → prices) and the unconnected case need a reason to connect at all; the natural candidate is a utility penalty for leaving a relevant fact unexplained, the same open question as predictive grip. An unconnected fact, accepted but shelved, matches Kuhn’s anomalies and should stay a possible outcome.

Utility and rigidity: three proposals, and what was adopted (12–13 October 2026)

Rigidity derived from utility (ρ_v = f(u_v), with u_v the perceived value of the paths into v). Not adopted. It needs predictive grip kept out of u_v, since an agent who values an explanation for its forecasts has a reason to open it, not to close it; and it must leave link entrenchment alone, or a firmly held but open explanation becomes impossible.
Utility derived from entrenchment (an explanation is worth what it would cost to give up: Σ (κ + ½ φ(r) w²) + ½ φ(ρ), where the opening-cost term is a “bonus” for exclusivity). Not adopted as a utility term. It is the revision cost of abandoning the narrative under another name, so using it beside the revision cost would count it twice. It also cannot value a narrative not yet held, whose links would all start at r₀. The intuition survives as an interpretation: a high ρ reveals that the agent values exactly that set of causes and no others, and that value enters the decision once, as the opening cost.
Utility exogenous, revision cost subtracted as a switching cost. Adopted. Utility is the destination, revision cost the route; the rule in section 7.

Under discussion, not adopted: an individual opening cost

Φ_i(ρ_v) = c_i + ½ φ(ρ_v)

It would be paid once whenever v’s cause set is opened. So naming a cause from the residual would cost c_i even at ρ = 0.
c_i is a trait of the agent (the counterpart of need for closure); ρ_v is a state of the narrative.
In the prototype:
- all orderings held for c up to 0.4
- from c ≈ 0.11, semi-closed holders double down instead of reopening
- an open narrative held by someone with c = 0.3 resists more than a semi-closed one held by someone with c = 0
Identification needs at least two narratives per respondent, or a trait scale.

Rejected alternatives (do not reintroduce)

Rigidity as a summary of the entrenchment of a statement’s in-links. Rejected because the two come apart: a loosely held but closed narrative rejects new causes, doubles down and places wages as an effect, while a firmly held but open one does the opposite.
Pricing the residual exactly like a link (½ φ(ρ)Δz², no fixed cost). Rejected because it kills the core results:
- a closed holder would pay 0.23 instead of 4.73 for “wages also”
- doubling down would be only partial
- “first cause pays” would be lost
- there would be no break (Π ≈ 3.6 instead of 60), which contradicts B1’s claims about abrupt de-anchoring
“Vertex mutation” as reinterpretation of the statement’s content. Dropped from primitive 3; it needs a distance between statements.

13. Where the framework fits in the existing literature

Each strand is listed with what FRAME takes from it and what FRAME adds.

Narrative economics. Shiller (2017, 2019) treats narratives as contagious stories that drive economic fluctuations, and measures them by prevalence (how often a story appears). Akerlof and Snower (2016) argue that narratives are how people make sense of experience and that they shape motivation and decisions. FRAME takes the premise that narratives are economic objects worth measuring, and adds structure: a narrative is a causal graph with a residual, and its primitives (rigidity, revision cost) are what prevalence misses. Falsification test 3 is the direct comparison.

Narratives as causal models in economic theory. Spiegler (2016) models agents who reason with a misspecified causal model (a directed acyclic graph) and fit the data to it. Eliaz and Spiegler (2020) make narratives such causal models and let agents adopt the narrative that promises the highest anticipated utility, with competing narratives in equilibrium. FRAME takes the graph object and the anticipated-utility adoption from this work. It adds (i) the residual vertex, which makes the unspecified causes explicit, (ii) entrenchment and rigidity as measured properties of a held graph, and (iii) the revision cost as a switching cost, so that adoption is net of the cost of getting there. The same narrative can then spread in one population and stall in another at equal utility, which Eliaz and Spiegler’s model cannot produce. Bénabou and Tirole (2016) give the utility sources their foundation: people derive utility from beliefs (comfort, self-image) and manage them. Bénabou, Falk and Tirole (2018) treat narratives as instruments of moral reasoning that assign responsibility, which is FRAME’s “moral order”.

Expectation formation and information rigidity. In sticky information (Mankiw and Reis, 2002), rational inattention (Sims, 2003) and the noisy-information framework that Coibion and Gorodnichenko (2015) test, the degree of stickiness is a parameter. FRAME maps it onto measured entrenchment (β = r/(1 − r), λ = r at unit precision) and adds the thresholds that rigidity creates, so that de-anchoring can be abrupt (section 9.1). Gabaix (2014) models agents who attend to a few variables and set the rest to defaults; the residual is the explicit counterpart of the causes an agent leaves out, with its own price for being opened. Andre, Haaland, Roth and Wohlfart (2022) elicit households’ and experts’ causal narratives about inflation from open-ended survey text and show that they are heterogeneous and tied to expectations; this is the closest measurement precedent. FRAME adds strength, entrenchment and the residual to what is elicited, and the cost calculus that turns the elicited structure into predictions.

Switching costs and state dependence. The decision rule is a switching-cost model: the value of the destination minus the cost of moving (Klemperer, 1987, 1995; Farrell and Klemperer, 2007). Separating the switching cost from preferences is the identification problem Dubé, Hitsch and Rossi (2010) address for consumer inertia, and FRAME’s two signatures (route dependence, and the physiological split between aversion on contradiction and reward on confirmation) are its answer for narratives. Status quo bias (Samuelson and Zeckhauser, 1988) is the behavioural face of the same rule.

Belief revision and the web of belief. Quine (1951) pictured beliefs as a web in which the periphery is revised first and the centre is protected; Harman (1986) made conservatism a principle of belief change. The AGM theory of belief revision (Alchourrón, Gärdenfors and Makinson, 1985) formalizes minimal change, and epistemic entrenchment (Gärdenfors and Makinson, 1988; Gärdenfors, 1988) is an ordering that decides which beliefs are given up first. FRAME’s entrenchment matrix is the quantitative, link-level counterpart, and the revision cost is a weighted minimal-change metric with a price for every edit. Thagard (1989) evaluates belief systems by explanatory coherence over a network of propositions; FRAME prices moves in such a network instead of scoring states. Lakatos (1970) distinguished a research programme’s hard core from its protective belt; statement placement puts anomalies in the belt, and Kuhn (1962) supplies the cycle that ends in a break (section 9.2).

Psychology of belief and its defence. Cognitive dissonance (Festinger, 1957) and motivated reasoning (Kunda, 1990) are the reasons a held narrative is defended; belief perseverance (Ross, Lepper and Hubbard, 1975) is why giving up a statement may cost more than accepting one (κ⁻ ≥ κ⁺). Need for cognitive closure (Webster and Kruglanski, 1994) is the trait behind the individual opening cost under discussion. Sacred values (Baron and Spranca, 1997; Tetlock, 2003) are closed narratives, ρ = 1, and the finding that affective response scales with value centrality (Hamid et al., 2019) is extended from importance to structural revision cost in B2’s laboratory work, with Larsen, Norris and Cacioppo (2003) supplying the physiological measures. Kelley’s discounting principle (Kelley, 1973) is why the shares into a statement sum to one. Causal-model accounts of cognition (Sloman, 2005) justify the graph object, and the number of links that must be re-formed predicts revision choices and times (Krumnack et al., 2011), the basis for κ.

Depth, folding and the limits of explanation. People build and use compressed causal models (Kinney and Lombrozo, 2024) and reason differently under time pressure (Kahneman, 2011): the folded and unfolded narratives of section 9.3. The illusion of explanatory depth (Rozenblit and Keil, 2002) and its link to extremism (Fernbach et al., 2013) suggest the prediction that unfolding exposes residuals and lowers rigidity. Local thinking (Gennaioli and Shleifer, 2010), in which only some scenarios come to mind, is the counterpart of the residual on the side of recall.

Measuring narratives from text. Ash, Gauthier and Widmer (2024) extract narratives from text as graphs of entities and relations. FRAME’s text pipelines (B2) build on this kind of extraction and add the markers of rigidity: hedges for open residuals and completeness claims for closed ones.

14. Changes still needed in B1/B2 (v4)

The drafts predate the renames and the decision rule. Suggested replacements:

Everywhere: “merge friction” → “narrative revision cost”; “vertex mutation” → “statement placement”; “closed narratives break abruptly” → “highly rigid narratives break abruptly”.
B1 abstract, the list of primitives: “… narrative revision cost (the cost of the cheapest revision that accommodates new information), statement placement (where a new fact is placed: the revision that minimizes that cost), and narrative utility (the anticipated utility of holding a narrative, net of which the revision cost acts as a switching cost) …”
B1, the sentences after Kinney and Lombrozo (2024): “Depth also widens statement placement. When a message asserts a fact but not its causal role, the holder places it by the cheapest revision, and an unfolded narrative offers more places where a fact can attach without touching its core. It is how two readers of the same article each find confirmation of opposite beliefs: both accept that wages rose, one as a cause of inflation, the other as its effect. This yields a testable prediction: placements that protect the core become more common as rigidity rises.”
B1 and B2, the acceptance-and-defence mechanism: “a narrative is adopted when the utility it promises exceeds the revision cost of getting there; the revision cost is a switching cost, and once adopted a narrative is defended”.
B2, H4: “placements of an open-role fact that protect the core, elicited as a forced choice (cause, effect or unrelated), become more common as the revision cost of the alternative placement rises”. In the fallback and risk lines, “mutation coding” → “placement choice”.
B1, falsification: “households holding open narratives de-anchor as abruptly as those holding closed ones” → “… as those holding highly rigid ones”.
B1, WP prediction (i): “the stickiness of survey expectations … increases with the rigidity of the prevailing narrative” → “survey expectations adjust more slowly where the prevailing narrative’s links are more entrenched, and de-anchor abruptly where it is more rigid”. In the model, stickiness (partial adjustment, β = r/(1 − r)) comes from entrenchment; rigidity produces thresholds and jumps, not slower adjustment. Prediction (ii) is consistent as it stands.

15. References

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Appendix: full cost breakdowns

The notation is “item: cost”. “Free” means the share moves through an open residual at no cost. Unless stated otherwise, the semi-closed holder has ρ = 0.3.

Add “wages rose” → prices at 0.2 (open holder): 0.21 - new statement, κ⁺: 0.10 - new link wages → prices at 0.20 (r₀ = 0.3), κ + ½φ(r₀)w²: 0.11 - residual of prices 0.20 → 0.00: free

Add gas supply cut → prices at 0.1 (open holder): 0.10 - new link at 0.10: 0.10 - residual of prices 0.20 → 0.10: free

Remove raw materials → prices (open holder): 0.60 - remove link (0.50, r = 0.8), κ + ½φ(r)w² = 0.1 + 0.5: 0.60 - residual of prices 0.20 → 0.70: free

Weaken raw materials → prices 0.5 → 0.4 (open holder): 0.02 - ½φ(0.8)(0.1)²: 0.02 - residual 0.20 → 0.30: free

Shipping costs rose → raw material prices (all holders): 0.22 - new statement “shipping costs rose”: 0.10 - new link shipping → raw materials at 0.30: 0.12 - residual of raw materials 1.00 → 0.70: free

Wages also rose → prices - Open: 0.21 (as above) - Semi-closed: 0.44 - new statement: 0.10 - new link: 0.11 - raw materials 0.60 → 0.55: 0.01 - energy 0.40 → 0.25: 0.02 - opening ½φ(0.3): 0.21 - Semi-closed, ρ = 0.9: 4.73 (the same items with opening ½φ(0.9) = 4.50) - Closed: the same items, but opening ½φ(1) = ∞, so the total is ∞

Demand, not costs, drove prices - Open: 1.18 - new statement: 0.10 - remove raw materials (0.50): 0.60 - remove energy (0.30): 0.17 - new link demand → prices at 1.00: 0.31 - residual 0.20 → 0.00: free - Semi-closed: 1.67 - new statement: 0.10 - remove raw materials (0.60): 0.82 - remove energy (0.40): 0.22 - new link at 1.00: 0.31 - opening: 0.21 - Semi-closed, ρ = 0.9: 5.95 (the same items with opening 4.50) - Closed: the same items, but opening ∞, so the total is ∞

Energy prices did not rise - Open: 0.97 - give up statement (e = 0.5), κ⁻ + ½φ(e): 0.60 - remove gas → energy (0.70, r = 0.3): 0.21 - remove energy → prices (0.30, r = 0.6): 0.17 - residual of prices 0.20 → 0.50: free - Semi-closed: 1.24 - give up statement: 0.60 - remove gas → energy: 0.21 - remove energy → prices (0.40): 0.22 - residual of prices 0.00 → 0.40: free - opening: 0.21 - Closed: 1.35 - give up statement: 0.60 - remove gas → energy: 0.21 - remove energy → prices (0.40): 0.22 - raw materials → prices 0.60 → 1.00 (r = 0.8): 0.32

Rising prices pushed energy prices up - Open: 0.29 - remove energy → prices (0.30): 0.17 - residual of prices 0.20 → 0.50: free - new link prices → energy prices at 0.30: 0.12 - residual of energy prices 0.30 → 0.00: free - Semi-closed: 0.55 - remove energy → prices (0.40): 0.22 - residual of prices 0.00 → 0.40: free - opening: 0.21 - new link prices → energy at 0.30: 0.12 - residual of energy prices 0.30 → 0.00: free - Closed: 0.66 - remove energy → prices (0.40): 0.22 - raw materials 0.60 → 1.00: 0.32 - new link prices → energy at 0.30: 0.12 - residual of energy prices 0.30 → 0.00: free

Wages placed as an effect (all holders): 0.21 - new statement: 0.10 - new link prices → wages at 0.20: 0.11 - residual of wages 1.00 → 0.80: free

First new path pays (semi-closed holder, ρ = 0.9) - Wages alone: 4.73 - new statement: 0.10 - new link at 0.20: 0.11 - raw materials 0.60 → 0.55: 0.01 - energy 0.40 → 0.25: 0.02 - opening ½φ(0.9): 4.50 - Demand alone: 4.73 (the same items with “demand surged” in place of wages) - Wages and demand at once: 5.00 - two new statements: 0.20 - two new links at 0.20: 0.22 - raw materials 0.60 → 0.49: 0.02 - energy 0.40 → 0.11: 0.06 - opening once: 4.50 - Demand after wages has been absorbed: 0.21 - new statement: 0.10 - new link at 0.20: 0.11 - raw materials 0.55 → 0.53, energy 0.25 → 0.21 and wages 0.20 → 0.06: each about 0.00 - no opening cost: the residual is already open