The truce in the numbers

Part 1 of 4| By Aarez Osman, Research Fellow

Key takeaways

At Acme, a hypothetical $300 million B2B software company, the CMO reports $42 million of marketing-sourced pipeline in the third-quarter review. The CRO's CRM says $27 million. One quarter later, the board deck shows a single figure: $27 million, labeled "sourced pipeline (sales definition)."

The gap is gone. Nothing in the marketing automation platform changed. Nothing in the CRM changed. The attribution model was not revisited, and marketing still plans spend internally on the 180-day window that produced $42 million. So what, exactly, was reconciled?

A piece of applied mathematics answers that question more precisely than any operating review we know of. The company is invented; the mechanism is real and has a formal description.

Consensus lives on the edges

The paper is "Opinion Dynamics on Discourse Sheaves" by Jakob Hansen and Robert Ghrist, published in the SIAM Journal on Applied Mathematics in 2021 (Hansen and Ghrist). The structure is simple once the notation is stripped away.

Start with a network: people as nodes, relationships as edges. Each person holds a private opinion in a space that belongs only to them. Each relationship has a shared space, the topics those two people actually discuss. Connecting the two is a restriction map: the translation a person applies when turning a private opinion into what they say to that particular neighbor. Each person can have a different translation for every relationship. Withholding an opinion is a map that sends it to zero. Exaggerating it stretches it. Softening, selective emphasis, and reversal are all just different maps (summary in The Cartan-Topos Protocol).

Consensus has a precise meaning in this model. It is a global section: an assignment where every connected pair says matching things across every shared topic.

A global section constrains what is said on the edges. It places no constraint on what is believed at the nodes. Two people can agree perfectly in public while holding private opinions that are nothing alike, as long as their translations land in the same place. Ghrist described the result plainly when the paper came out: the system can evolve "towards a public agreement while people can maintain their private opinions" (Penn / Phys.org).

In a later workshop paper he gave it a better name. A network in a harmonic state, he writes, is "not a strict consensus – what does it mean for two agents with entirely different internal ideologies to agree? – but rather an expressed external consensus: a truce" (Ghrist, Harmonicity in Networked Social Information Dynamics).

Translations move faster than beliefs

Most opinion-dynamics models assume that when people disagree, their opinions drift toward each other. Hansen and Ghrist also let the translations drift. In a section titled "Learning to Lie," they hold private opinions fixed and let the restriction maps evolve to reduce disagreement. In their worked example, two people start far apart and the system reaches concord when one person's map flips sign: they learn to express the opposite of what they believe.

When both opinions and maps are allowed to change, the maps adapt more readily, because, in the authors' words, "the discrepancy between opinions is so great" (Hansen and Ghrist).

There is a second mechanism. A nonlinear variant lets agents ignore communications where expressed disagreement is too large (n-Category Café). The network reaches a kind of peace by no longer talking across its widest gaps.

The same dynamics, inside one company

Now replace the people with functions.

Acme's CMO holds a private representation in the marketing automation platform. Her translation to the CRO counts any opportunity created within 180 days of any marketing touch: $42 million. Her translation to the CFO, for the budget conversation, counts only opportunities where marketing was the first touch: $31 million. The CRO's representation is the CRM, where pipeline is "marketing-sourced" only if the opportunity originated from an inbound form, with SDR-sourced deals excluded: $27 million. Three honest maps, three numbers, one underlying set of deals.

Figure 1. Acme (hypothetical) across two quarters. In the Q3 review, two handoffs carry visible disagreement. By the Q4 board deck, the expressed gap is zero, but neither underlying system has changed. The pipeline edge was resolved by adopting another function's translation; the bookings edge was resolved by no longer discussing it.

Both of the paper's mechanisms appear in Figure 1.

The pipeline edge is learning to lie, performed by people who would never describe it that way. The marketing-to-sales disagreement falls from $15 million to zero because one translation moved to match another. The gap left the room without leaving the company.

The bookings edge is the nonlinear version. The COO has raised the $1.4 million gap between CRM closed-won and ERP bookings for two quarters running. Each time it is explained as "timing." By the third quarter she stops raising it. Six months later, operations finds that a recurring set of deals marked won in the CRM had been waiting on customer procurement approvals that sometimes never came. The onboarding team had been staffed for customers who did not arrive.

The pattern shows up in real companies too. When McKinsey asked CEOs and CMOs of the same companies to name their top three marketing metrics, the pairs agreed only half the time. CEOs tended to name revenue growth and margin; CMOs tended to name brand awareness and recognition (McKinsey). Two leaders can nod at the same slide while translating it through different definitions of success.

What silence tells you

Most organizations measure alignment by the absence of argument. The discourse sheaf shows why that instrument is unreliable: agreement in expression is compatible with arbitrarily large disagreement in the underlying representations. This has two practical consequences for anyone building or buying a decision system.

Disagreement has a location. In a sheaf model, the useful question is which edge carries the disagreement, and through which map. At Acme, the pipeline gap and the bookings gap are different problems with different owners. The first belongs jointly to the CMO and CRO. The second belongs to the CRO and COO. A system that reports a single "alignment" score cannot tell them apart.

Reconciliation has a direction. Every time a disagreement is resolved, something absorbed it: either the evidence changed or a translation did. A decision system that records only the final agreed number discards the most important fact, which map moved to produce it. Had Acme's board deck carried one extra line, "pipeline restated to sales definition; marketing's internal planning figure remains $42 million," the CFO would have known that next year's marketing budget was being built on a number the board had never seen.

A note on the examples

Acme is a hypothetical company, and its figures are illustrative. We use it to show how the discourse-sheaf mechanism maps onto a familiar operating problem. The next three posts build on it: Part 2 traces where the mathematics comes from and what it can compute, Part 3 looks at documented public cases, and Part 4 maps the published work closest to this question.

Next in the series: Part 2, "The math came first"

References

Hansen, J. and Ghrist, R. Opinion Dynamics on Discourse Sheaves. SIAM Journal on Applied Mathematics, 2021. www2.math.upenn.edu

Ghrist, R. Harmonicity in Networked Social Information Dynamics. ICWSM Workshop Proceedings, 2022. workshop-proceedings.icwsm.org

McKinsey & Company. The power of partnership: How the CEO–CMO relationship can drive outsize growth, 2023. mckinsey.com

Penn / Phys.org. New mathematical tools to study opinion dynamics, 2021. phys.org

Learning to Lie with Sheaves. The n-Category Café, 2022. golem.ph.utexas.edu

The Cartan-Topos Protocol (summary of discourse sheaves). alphaxiv.org