When Nine True Claims Carry One False One

1. The Missing Half of Claim-by-Claim Analysis

Traditional fact-checking assumes that claims can be separated, verified, and scored one at a time. That is indispensable for establishing what evidence supports each statement, but it does not describe how a reader receives the whole. Suppose an article carries ten material claims, of which nine are independently verified at ten out of ten and one scores two out of ten. A simple average reports

and the piece looks ninety-two percent credible. The arithmetic is correct and the interpretation can still be badly misleading, because the tenth claim may be the headline, the causal explanation, or the conclusion the author most wants believed, while the nine accurate claims form a scaffold around it. The weak claim has not become truer. It has become more believable, and the gap between those two things is the quantity this paper is about.

2. Truth Is Not Only Counted; It Is Arranged

Once several statements have been confirmed, the source accumulates trust, and that trust changes how later and weaker claims are received. A publication can therefore use accurate details to build an inaccurate implication, placing a poorly supported proposition at the centre of an otherwise impeccable factual structure. The surrounding truths do not make it objectively truer, but they can carry it across the threshold of belief. Nine true claims and one false claim do not always produce a ninety-percent-true article. Sometimes the nine exist principally to carry the tenth.

3. Intrinsic and Contextual Credibility, and Why One of Them Is Latent

Let the article contain \(n\) material claims, indexed \(i=1,\dots,n\). Give each an intrinsic evidence score

where zero means no credible independent support and one means exceptionally strong support. Let \(p_i \in [0,1]\) be the same claim's perceived credibility once it is read in context. The difference

is the credibility the claim has gained, or if negative lost, from its surroundings rather than from its own evidence.

One distinction has to be made before any equation, because the earlier drafts blurred it. Perceived credibility \(p_i\) is not something that can be read off the text. It is a latent quantity describing how a reader receives the claim, and in empirical work it would be estimated from reader judgements collected after exposure to the complete article, while \(c_i\) would be estimated from independent evidence assessed without the article's framing. The model computes a predicted \(p_i\) from assumptions about influence; it does not thereby measure the psychological effect of a real article, and the two must never be conflated. Everything that follows describes the behaviour of the proposed model, and stands or falls on whether measured perceptions later match it.

4. The Model's Assumptions, Stated Explicitly

Influence among claims is written as a matrix \(W=[w_{ij}]\), where \(w_{ij}\) is the weight with which claim \(j\) bears on the perceived credibility of claim \(i\). Three conditions are imposed.

The centralities introduced later satisfy \(v_i \in [0,1]\). These conditions are what make the propagation well behaved, as the next section shows.

5. Credibility Propagation, and Why It Is Well Posed

A first, one-step version blends a claim's own evidence with the evidence of its neighbours,

The stronger version lets influence propagate through perceived values rather than stopping at one step, which gives a fixed point,

and, solving when \(I-\alpha W\) is invertible,

Perceived credibility must never leave the zero-to-one scale, or the model would be meaningless. Write the operator in (5) as its convergent Neumann series,

which converges because the spectral radius of \(\alpha W\) is at most \(\alpha<1\). Since \(W\) is row-stochastic, \(W\mathbf{1}=\mathbf{1}\), and therefore

The entries of \(M\) are non-negative and each row sums to one, so \(M\) is itself row-stochastic, which means every \(p_i\) is a weighted average of the intrinsic scores \(c_j\) and thus lies in \([0,1]\). The propagation is well posed by construction.

With the formal behaviour of the propagation settled, two lineages should be named accurately, because an earlier draft overstated them. Equation (4) is not DeGroot's model, which repeatedly averages current opinions with no anchor. The continuing attachment to the intrinsic vector \(\mathbf{c}\) places it instead in the Friedkin and Johnsen family of opinion dynamics, in which agents combine social influence with a fixed attachment to their initial position, with DeGroot as the ancestor rather than the thing itself. The resolvent in (5) is algebraically analogous to a personalised or restart form of PageRank, though the interpretation differs, since \(\mathbf{c}\) carries credibility scores rather than a probability distribution. Both lineages are a strength, because the machinery has been studied for decades, and they mark the limit of what is new here: the propagation mathematics is borrowed and well understood. The proposed contribution lies in combining these established methods for the specific purpose of analysing credibility transfer within narratives; whether closely related models already exist for this purpose would require a fuller literature review than has been done here.

6. Transferred Credibility and Aggregate Contextual Shift

Subtracting \(\mathbf{c}\) from both sides of the fixed point (4) gives the transfer directly,

This is the grouping hypothesis in closed form. A weak claim, low \(c_i\), sitting among claims the reader already perceives as strong, high \((W\mathbf{p})_i\), receives a large positive transfer. It is a specific modelled quantity rather than loose language about proximity.

The arrangement also has an aggregate signature. Summing the transfer over the article defines the aggregate contextual shift,

If \(W\) is doubly stochastic, meaning its columns as well as its rows sum to one, then \(\mathbf{1}^{\top}W=\mathbf{1}^{\top}\), and by the same series argument \(\mathbf{1}^{\top}M=\mathbf{1}^{\top}\), so \(A=0\) for every \(\mathbf{c}\): the arrangement only redistributes credibility, taking from some claims what it lends to others. Double stochasticity is sufficient for this conservation but not necessary for a given article, since a particular score vector can still cancel to zero even when the columns are unbalanced. When \(W\) is not column-stochastic, the total need not be preserved, and a positive \(A\) marks an arrangement that raises perceived credibility across the article as a whole. The word chosen here is deliberately modest: \(A\) is an aggregate contextual shift, not manufactured credibility, because a sum of perceived-credibility scores is not a conserved physical quantity and calling its increase manufacture would be interpretation dressed as a theorem.

7. Narrative Centrality

Claim-counting makes a second error: it treats a trivial background fact and a load-bearing conclusion as equally important. Define the narrative centrality of a claim, \(v_i\in[0,1]\). This is candidly the model's most provisional quantity, a single number standing in for several distinct cues, and the research agenda proposes decomposing it into separately measurable components rather than leaving it to intuition. Centrality rises when the claim appears in the headline, is repeated, supplies the causal explanation, carries the conclusion, is reinforced by images and captions, or would change the article's meaning if removed. A weak claim with low centrality is an ordinary error. A weak claim with high centrality can be the article's entire persuasive payload.

8. Propaganda Leverage: a Candidate Index

Combining transfer, centrality, and weakness gives a per-claim leverage. It is a proposed diagnostic, not a consequence of the propagation model, and its particular form is one defensible choice among several. Because transfer can be negative, the sign has to be handled deliberately. The positive-leverage index is

and the opposite case, where context discounts rather than elevates a claim, is recorded separately as contextual suppression,

The behaviour of \(L_i^{+}\) is sensible: a well-supported claim contributes little because \((1-c_i)\) is small, an isolated weak claim contributes little because the transfer is small, and a weak but trivial claim contributes little because \(v_i\) is small. The dangerous case, a weak claim that inherits credibility and carries the conclusion, is the one that scores high. Two honest caveats belong with the formula. The transfer and \((1-c_i)\) both grow as intrinsic credibility falls, so weakness is weighted twice; this is defensible read as transfer multiplied by the stakes of that transfer, but it is a modelling choice, and simpler forms such as \((p_i-c_i)v_i\) are equally admissible. And \(L^{+}\) is unnormalised, so comparing two articles requires dividing through, for instance by total centrality, \(\bar L = L^{+}/\sum_i v_i\). Which functional form best separates real cases is an empirical question, not one the algebra settles.

9. The Group as a Cooperative Game: Shapley Attribution

The intuition that a set of claims can jointly produce a credibility that none of them holds alone is real, and it has a rigorous home in cooperative game theory, where the question is not what each player does in isolation but how the value created by a coalition should be attributed to its members. Treat the claims as the players and the target conclusion \(t\) as the quantity to be explained. Because \(v_i\) already denotes centrality, the coalition worth is written \(g\), not \(v\).

The worth of a coalition has to be defined by an explicit construction, or the Shapley value cannot be computed. For a coalition \(S \subseteq N\setminus\{t\}\) of supporting claims, build the induced network on \(S\cup\{t\}\): delete every edge incident to an excluded claim, renormalise each remaining non-empty row so that it again sums to one, and give any claim left with no neighbours zero contextual influence, so that its perceived credibility falls back to its own evidence. Let \(p_t^{S}\) be the target's equilibrium credibility in that induced network, and define

The Shapley value then attributes the conclusion's modelled contextual uplift to each supporting claim by averaging its marginal contribution across every order of assembly,

It is the unique attribution satisfying efficiency, so that the parts sum to the whole; symmetry, so that equal contributors receive equal credit; the null property, so that a claim contributing nothing receives nothing; and additivity. Its interpretation is only ever as sound as the coalition rule in (12), which is a modelling decision rather than a fact. Exact evaluation sums over all \(2^{n}\) subsets and is infeasible beyond small articles, so in practice the value is estimated by sampling random orderings and averaging marginal contributions, a method with known error behaviour. This is not an idle borrowing: the Shapley value already has an established use in attributing centrality in networks, and specially structured network games admit exact polynomial-time formulae, so exponential cost is not always unavoidable. The cooperative framing also keeps the intuition that first motivated this work, familiar from the dramatisation of Nash's life, that a group's outcome is not the sum of its members' isolated moves, while assigning it to the right instrument, the cooperative value rather than the equilibrium concept with which the popular version is often confused.

10. From an Intuitive Reading to a Reproducible Statistical Signal

The wish is for propaganda to declare itself rather than rest on a reader's impression. A null model turns that into a reproducible statistic, but only once the randomisation is fully specified, so it is specified here. Hold fixed the claims, their intrinsic scores \(\mathbf{c}\), the number of influence links, each claim's row-sum normalisation, and the structural roles such as which node is the headline. Randomise what remains: the placement of the influence links, that is which claims are connected to which, together with the assignment of the observed centralities to positions, drawing many arrangements from this restricted set. Recomputing the leverage for each draw produces a null distribution \(L^{(1)},\dots,L^{(B)}\) under the hypothesis of no deliberate credibility shaping. The primary result is the empirical tail probability,

the reproducible chance that a neutral arrangement of the same facts would match the observed leverage. A standardised effect size can accompany it,

but the tail probability is primary, because the null distribution is not assumed normal. A small \(\hat p\) means the real arrangement concentrates credibility on weak, central claims far more than a neutral arrangement of the same facts would. This is reproducible once the annotations and the randomisation are fixed, but it is not fully objective, because the inputs \(c_i\), \(v_i\), and \(W\) still rest on measurement choices and annotation judgement. It replaces an impression with a statistic; it does not replace judgement with certainty.

11. Spin as Optimisation: an Analogy to Bayesian Persuasion

The Bayesian persuasion framework of Kamenica and Gentzkow provides an analogy and a possible future extension, not the present model's formal foundation. In that literature an informed sender chooses a signal structure, subject to Bayes plausibility, so that a rational receiver who updates correctly is moved towards the sender's preferred action; it requires states of the world, a prior, signals, posterior beliefs, receiver actions, and utilities, none of which this model yet contains. The corresponding conjecture here is narrower: that an author may choose the arrangement, emphasis, and linkage of true claims so as to raise the contextual credibility of a target proposition. Written as a narrative-arrangement optimisation over the admissible arrangements \(\mathfrak{A}\),

where an arrangement \(\mathcal{A}\) fixes the influence matrix and the centralities. Spin, defined this way, is not an insult but a proximity, how close a real article sits to the arrangement that would be optimal for its apparent conclusion, and it is offered as inspired by Bayesian persuasion rather than as an instance of it. The ethical question stays where it belongs. Structure can be measured; intent cannot be read off structure, because carelessness, house style, and honest emphasis can imitate optimisation. Intent therefore remains a separate axis, rated cautiously and never inferred from a single arrangement.

12. A Worked Illustration

Take ten claims, nine verified and one weak and central.

A simple average again reports \((9(1.00)+0.20)/10 = 0.92\). Suppose the weak claim is connected chiefly to the nine strong ones, so its neighbourhood average is \(1.00\), and take susceptibility \(\alpha=0.65\). From the one-step relation (3),

so its transfer is \(T_{10}=0.52\) and, since the transfer is positive, its positive leverage is

These values are illustrative, chosen for clarity rather than measured. The point is structural: one weak claim can dominate the meaning of an otherwise accurate article when it is central, highly connected, and contextually elevated, and each quantity above, the transfer, the leverage, the Shapley share, and the tail probability against a neutral arrangement, is in principle computable rather than merely felt, once the inputs have been measured.

13. The Semiotic Layer

None of this is produced by words alone. A meme, video, report, or infographic is a package of signs: photographs and footage, uniforms, flags, logos, maps, charts, captions, quotation marks, official terminology, typography, colour, music, editing pace, and institutional branding. These need not make explicit factual claims, yet they raise the perceived credibility of nearby propositions. A typeface, though, does not possess evidential credibility in the way a proposition does, so it cannot simply be added as another claim node carrying its own \(c_i\). The faithful place for these signs in the present model is not as claims but as modifiers: they raise the influence weights \(w_{ij}\) flowing into a proposition, or they raise a reader's susceptibility \(\alpha\), lending authority, salience, or apparent authenticity without themselves being evidence. Treating signs as a second class of nodes with their own scores, rather than as claims, is a natural extension and is set out in the agenda.

14. Error, Bias, Persuasion, and Propaganda

A high leverage score does not prove malicious intent, because the same structure can arise from carelessness, confirmation bias, advocacy, sensationalism, thin sourcing, or editorial compression as readily as from design. The model measures propaganda function or propaganda risk, not proven purpose, and the categories should be kept distinct.

15. The Same Mechanism Builds Honest Meaning

Everything above can read as an indictment, as though credibility transfer were only a fault to be policed. It is not. The movement of credibility through a narrative is also the ordinary machinery of understanding, teaching, and original thought. A teacher places a new idea among established facts so a learner can see how it connects to what they already accept; a scientist situates a novel hypothesis within verified literature; a good essay earns confidence in its conclusion by grounding it in claims the reader can check. In each the perceived credibility of the new proposition exceeds its isolated evidential credibility, and that surplus is not a trick but the way a mind travels from the known to the unknown.

Originality depends on the same effect even more sharply, because a genuinely new synthesis begins with little independent track record, low not because it is weak but because it is unfamiliar. What lets it be taken seriously at all is the verified material around it and the integrity of its reasoning. The line between this constructive use and propaganda is therefore not the presence of transfer, which is present in both, but narrative integrity: whether the conclusion actually follows from the verified premises, whether uncertainty is signposted, and whether the surrounding facts bear on the conclusion or merely decorate it. Honest synthesis uses grouping to help a true or testable conclusion be understood; propaganda uses the identical grouping to move an unearned one past scrutiny. The mathematics cannot separate them on its own. Only the relationship between premises and conclusion can, which is why narrative integrity, not leverage alone, is treated as the decisive measure.

16. What Would Make This Real: a Research Agenda

The framework is at present a hypothesised conjecture, and the equations above are analytical proposals rather than measured facts. Making it real means several distinct pieces of work, named here so that the next steps are clear and none is pretended to be finished.

The first is measurement of the inputs. Intrinsic credibility \(c_i\) needs a scoring protocol precise enough that independent annotators, given the same claim and the same evidence, arrive at similar scores, and the reliability of that agreement has to be measured rather than assumed. Perceived credibility \(p_i\), being latent, has to be elicited from readers after exposure, not inferred from the model. The influence matrix \(W\) has to be defined operationally, whether annotated by hand from cues such as sequence, causal language, repetition, shared sourcing, and headline-to-body linkage, or learned from the text, with the row-stochastic and hollow structure of Section 4 imposed. Every such score would carry an uncertainty interval, because neither evidential credibility nor reader perception can be measured without error.

Several extensions are named but deliberately not built into the core, because each would complicate the clean boundedness result and belongs in its own study. Readers differ, so a single \(\alpha\) should give way to audience-specific propagation, with a class \(a\) carrying its own susceptibility,

so that an article has not one contextual-credibility vector but a distribution of effects across audiences. Narrative centrality should be decomposed into annotatable components rather than intuited as a single number, for example

for headline prominence, repetition, causal role, multimodal reinforcement, and the dependence of the conclusion on the claim. Rebuttals and disclaimers require signed weights and a bounded non-linear update, which must be introduced carefully because signed weights invalidate the simple row-stochastic proof. And the semiotic layer of Section 13 is properly handled by a second class of context nodes carrying authority or salience scores rather than evidential ones, with the state split into claim and context blocks,

so that signs influence claims through \(W_{CS}\) without pretending to be claims themselves.

The last piece, and the one that decides whether any of this is worth pursuing, is validation against ground truth. The natural design is a labelled corpus, pieces widely regarded as propaganda alongside pieces regarded as honest reporting, drawn at both scales, the small and carefully annotated and the large and coarsely labelled, and including both known and held-out unknown cases. The question is simply whether the leverage index, or the Shapley concentration on weak central claims, separates the labelled groups better than a claim-count accuracy percentage does. Only if it separates them does calibration become meaningful, and only then could thresholds be proposed. Until that separation is demonstrated, the model should be described as theoretically grounded, mathematically explicit, psychologically plausible, and testable, but not yet validated. The next phase therefore requires both formal refinement, of the coalition rule, the audience model, and the semiotic block above, and empirical measurement; further algebra alone will not establish whether the model works.

17. The Central Principle

Truth is not only counted. It is arranged.

The reliability of an article cannot be reduced to the fraction of its atomic claims that pass verification. A complete method needs two levels: atomic analysis, asking whether each proposition is independently supported, and grouped analysis, asking what happens to that proposition once it is embedded in the whole. The tools assembled here, the propagation and its transfer identity, the aggregate contextual shift, the Shapley attribution, the leverage index and its null model, and the persuasion analogy, are a first language for that second level. The final question is not how many statements in an article are true, but how much credibility those true statements transferred to the one that had not earned it.

Conclusion

Claim-by-claim checking remains indispensable, and nothing here replaces it. What it adds is a way to describe the architecture of persuasion rather than only the strength of the bricks. The model is deliberately built from established machinery, the Friedkin and Johnsen and PageRank propagation, the Shapley value, and by analogy Bayesian persuasion, so that its claim to seriousness rests on mathematics others have tested, and its own novelty is confined to the application. It remains a conjecture. Its quantities are now defined precisely enough to be argued with and eventually measured, but they have not been measured, and the model is not yet a validated instrument. The next move is both refinement and measurement together.

References

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Status Note

The Nash reference is retained as a motivating metaphor for interacting values and joint outcomes; the load-bearing cooperative game theory is Shapley's value and its network-centrality use by Michalak and colleagues, with permutation estimation after Castro and colleagues. The propagation belongs to the Friedkin and Johnsen family, with DeGroot as ancestor and personalised PageRank as an algebraic analogue. The Bayesian persuasion connection is an analogy and a proposed extension, not a formal foundation. The illusory-truth references support the psychological premise that repetition and presentation can raise perceived truth independently of evidence. The Narrative Credibility Transfer Model itself is a proposed analytical framework: not yet empirically calibrated, experimentally validated, or peer reviewed, with numerical values illustrative rather than measured.