Brazil’s runoff:
testing the turnaround

The statistical finding

The reported turnaround is far outside this model’s expectations

A transfer baseline assigning 30% of the eliminated-candidate vote to Lula, half the net turnout increase and half the net reduction in blank and invalid ballots predicts a Lula gain of 5.186 million votes. The reported gain was 10.377 million—approximately twice as large. Under the same assumptions, Flávio would finish ahead by 5.952 million votes. The reported result instead puts Lula ahead by 4.430 million: a 10.382-million difference in the final margin.

The benchmark is explicit: 70% of the votes previously cast for other candidates go to Flávio and 30% to Lula; the two net increases in valid voting divide evenly; and original finalist support remains stable. The probability analysis then allows uncertainty around these expected shares. The baselines and uncertainty ranges are scenario assumptions, informed by political orientation and compared with surveys, rather than parameters estimated from a complete voter-transition dataset.

Lula accounted for 73.60% of the two finalists’ combined net vote gain. Under the broadest of the three independent preference distributions examined here, the conditional probability of receiving at least 70% is 0.08209%—about one in 1,218. Matching or exceeding the reported 73.60% share has probability 0.01901%—about one in 5,260. The reported turnaround is highly improbable within this model. The statistical question is whether these expectations adequately describe the electorate and, if they do, what produced such an exceptional change.

In his post-election interview, Gustavo Segre connected the late voting extension with the increase in participation and the movement of votes from eliminated candidates. His claims that all additional participants supported Lula and that 84% of the other-candidate pool transferred to Lula are interpretations of aggregate counts. This analysis replaces those fixed allocations with explicit transfer scenarios and calculates the changes required to reproduce the result.^[4]

01 / Accounting

What actually changed between the rounds

For every 100 people who participated in the first round, 44.78 voted for Flávio, 43.01 for Lula, 7.44 for other candidates and 4.77 cast blank or invalid ballots. The runoff had 102.36 participants on that same scale. Holding the denominator constant prevents changing turnout from being mistaken for a transfer between candidates.^[1],[2]

Blank ballots fell by 870,302 and invalid ballots by 950,944, a combined decline of 1,821,246. These are net changes. The underlying movements can include invalid-to-valid voting, valid-to-invalid voting, new participation and abstention. Their balance is observed; the individual transitions are not.

Lula gained 10,376,694 votes; Flávio gained 3,721,538. Their combined gain equals the 9,316,747 votes previously cast for other candidates plus the 4,781,485 increase in valid votes. These are accounting identities. Treating the three related totals as three independent pieces of statistical evidence would count the same information repeatedly.

The 27 state/federal-district result sets plus overseas voting sum exactly to the retrieved national totals in both rounds. The arithmetic is therefore internally consistent at this level. The separate statistical question is whether the redistribution of votes fits a credible account of voter behavior.^[3]

02 / Political probabilities

Political orientation and measured voting intentions

A voter who chose a candidate advocating Milei-style reforms may be less likely to choose Lula. Renan Santos explicitly praised Milei’s model. But a Renan voter need not share every position, may reject both finalists, or may vote chiefly against Flávio. Renan declined to endorse either finalist. Cury, Caiado and Zema did endorse Flávio; together their first-round voters accounted for 68.48% of the eliminated-candidate pool, rather than all of it.^{[11],[12],[13],[14],[15]}

Quaest’s candidate-specific point estimates favor Flávio among Renan and Caiado supporters, while Cury’s group leaned toward Lula. The subgroup uncertainty is large. Palver’s October 18–20 survey instead reported Lula/Flávio intentions of 52/43 among Cury voters, 39/42 among Renan voters and 44/44 among Caiado voters. Palver used weighted, nonprobability digital recruitment. The disagreement cautions against treating a single transfer estimate as a precisely known population parameter.^{[16],[19],[20]}

Datafolha’s four-candidate pool contained 9,056,092 first-round votes. Its 26% Lula and 47% Flávio estimates become 35.62% Lula among respondents naming either finalist: 26 ÷ (26 + 47). This normalization excludes the other 27%; it does not predict how they ultimately voted. The subgroup sample size was not obtained, so the survey’s national sample of 2,520 and national margin cannot be applied to this subgroup.^[17],[18]

For the combined group that had abstained or voted blank/invalid, Quaest reported 40% Lula, 36% Flávio, 21% maintaining the earlier choice and 3% undecided, with a published ±6-point margin. Among finalist intenders, that is 52.63% Lula. Crucially, this is the whole prior abstention/blank/invalid group, not the people represented by the net turnout rise or net invalid-ballot decline.^[16]

The surveys support differentiated transfer expectations. They also show why a common rate for the whole eliminated-candidate pool is an approximation. The 30% Lula baseline represents a right-leaning aggregate scenario; its adequacy must be judged against subgroup preferences, participation changes and the uncertainty in those measurements.

03 / Testing the transfer baseline

How much must change to reproduce the result?

The 30/50/50 benchmark

The model separates three components: class 1, 9,316,747 votes for eliminated candidates; class 2, the 1,821,246 net reduction in blank and invalid ballots; and class 3, the 2,960,239 net increase in participants. The latter two are accounting quantities. Treating them as simple pools of converters and entrants is a simplifying assumption.

Let p be Lula’s share of the eliminated-candidate pool, q his share of the net turnout increase and r his share of the net blank/invalid decline. Assume all eliminated-candidate voters remain valid voters; all original finalist supporters stay with their finalist except a net shift S from Flávio to Lula; and there is no offsetting participation or invalid-ballot churn. The model’s accounting equation is:

Under a 70% Flávio / 30% Lula transfer baseline for the other-candidate pool and q = r = 50%, the model accounts for 5.186 million additional Lula votes. The remaining 5.191 million is the size of the departure requiring explanation. It could be supplied by different transfers, original-supporter switching, participation exchanges omitted from the model, error in the initial expectations or an intervention affecting the reported totals. The 30% Lula rate is the stipulated scenario baseline, rather than a fitted estimate. For comparison, substituting the Datafolha conditional 35.62% share leaves a 4.668-million residual. That comparison extrapolates from four candidates to the full pool and assumes those not naming a finalist eventually divide in the same ratio; the survey does not establish either assumption.

Even if every net additional participant and every net reduction in invalid ballots went to Lula, a no-switch/no-churn model would require at least 60.06% of the other-candidate pool to choose Lula to reproduce the reported total. With p ≤ 50%, it cannot reproduce that total without relaxing those restrictions. Keeping both other pools at 50% and allowing no switching would instead require 85.72% of the other-candidate pool to choose Lula. Reproducing the reported result therefore requires a major change in at least one component of the baseline.

The result does not require all three pools to favor Lula. For example, the restricted equation can be satisfied with p = 30%, q = r = 50% and S = 5.191 million. If attributed entirely to original-finalist switching, that residual is equivalent to a net movement of 9.25% of Flávio’s first-round support to Lula. This identifies the required scale of change; the aggregate data do not observe that movement directly. More broadly, some former other-candidate voters could become invalid voters while an even larger number of formerly invalid voters become valid voters. The net decline would conceal that exchange. Aggregate totals cannot identify the full transition matrix.^[28]

Joint scenarios with the 70/30 baseline

Separate groups of voters can respond to the same political events. The calculations initially assume independent group preferences, then show a dependent comparison. Independence is a modeling choice; it does not follow merely from the groups containing different people.

Uncertainty about each group’s underlying preference is represented by P₁ ∼ Beta(0.3κ, 0.7κ) and P₂, P₃ ∼ Beta(0.5κ, 0.5κ). The expected Lula shares are 30%, 50% and 50%. The concentration κ controls the spread around those expectations. Higher κ expresses greater confidence in the baseline and makes large upward departures less probable. This models uncertainty about group preferences, rather than treating millions of voters as independent fair coins with precisely known probabilities.

For κ = 10, the probability that all three groups exceed 50% Lula is 2.2461%; for κ = 40 it is 0.1184%; for κ = 100 it is 0.000576%. The same baseline thus produces radically different tails depending on how confidently it is imposed. The concentrations are illustrative choices, not estimated from the polling data.

For threshold t, independence gives Pr(P₁ > t) × Pr(P₂ > t) × Pr(P₃ > t). If the groups instead respond perfectly together to a common political influence while preserving these marginal distributions, their joint tail can equal the smallest individual tail. At the 50% threshold, that is the “Class 1 >50%” column—four times the independent result. Neither dependence structure has been established here.

These probabilities describe joint group-share events. The actual totals do not establish that all three groups exceeded any listed threshold. A result model must use their unequal sizes, the finalists’ original votes and all transitions. The weighted calculation below evaluates the reported result under the same transfer model.

How improbable is a 70% share of the additional votes?

The relevant denominator is 14,098,232: the sum of the two finalists’ net gains, comprising 9,316,747 votes previously cast for other candidates and 4,781,485 additional valid votes. Lula’s gain of 10,376,694 is 73.6028% of that total. This is an observable accounting share. Interpreting it as the share received from those three pools additionally uses the model’s assumption of stable original-finalist support and no offsetting participation exchanges.

The test asks for the probability of a share at least 70%, rather than an arbitrarily narrow interval around 70%. It then evaluates the more demanding threshold of matching or exceeding 73.6028%.

Condition on the observed pool sizes and set S = 0. The modeled Lula gain is Y = 9,316,747P₁ + 2,960,239P₃ + 1,821,246P₂. A Lula victory requires Y > 8,161,598.5; a 70% share requires Y ≥ 9,868,762.4; reproducing or exceeding his reported gain requires Y ≥ 10,376,694. These are the relevant weighted tail events under these particular restrictions.

The broadest illustrated distribution gives a 2.4552% probability of a Lula win, a 0.08209% probability of at least 70% of the combined gain, and a 0.01901% probability of at least the reported 73.6028%. A narrow victory and a turnaround of the reported magnitude are therefore very different statistical events. Under all three independent scenarios, the reported gain lies far outside the expected range.

The broad scenario is already permissive about individual group preferences: the standard deviation is 13.82 percentage points around the class-1 mean of 30%, and 15.08 points around each 50% mean. Its small weighted tail arises because a very large combined gain requires substantial favorable movement across the components.

Dependence remains consequential. Keeping the same broad marginal distributions but letting one common influence place all three preferences at the same percentile raises those probabilities to 1.3326% and 0.6439%, respectively. This is an illustrative alternative to independence, not an estimate of actual political coordination. It demonstrates that the size of the anomaly depends on how common political influences are represented. The event remains uncommon in that particular alternative, but considerably less so.

Method: independent Beta propensities; integrate the class-1 survival probability over classes 2 and 3 using Gauss–Jacobi quadrature. Results agree at 64, 128 and 256 nodes per dimension. The dependent comparison uses a common uniform percentile with each group’s inverse Beta distribution. κ is an assumed concentration, not a survey sample size. No ballot-level coin-flip variance is added.

The calculation conditions on the observed net turnout and invalid-ballot changes and holds original finalist support fixed. It therefore tests a specific prediction about vote allocation. Shared campaign effects, offsetting participation changes and switching between the original finalists belong among the candidate explanations for its failure. Their contribution must be quantified against the 5.191-million residual.

04 / Geography and turnout

Weather and the geography of participation

There was real disruption: Jundiaí reported 80 mm of rain in 12 hours overnight into election day. INMET had forecast substantial rain in parts of the South and Southeast while much of the Northeast remained dry. A forecast is not a completed rainfall dataset, and one city cannot represent the electorate.^[21],[22]

A serious test would match observed hourly rainfall, flooding, road closures, transport availability and polling-site changes to precincts, then compare turnout changes while accounting for earlier turnout, demographics and local contests. That geographic comparison would determine whether participation increased where the weather most strongly predicted a decline.

Higher runoff participation is also not unprecedented: the TSE’s 2022 summaries show an increase of 570,424 participants. The 2026 increase is substantially larger. Its exceptional character should be measured against comparable elections and local conditions.^[23],[24]

05 / Timing and authority

The late extension as a possible explanatory mechanism

Justice Cristiano Zanin’s former role as Lula’s lawyer creates an obvious question about impartiality and public confidence. The published order arose from a coalition complaint concerning transport in three Bahia municipalities and invoked enforcement of an earlier STF transport ruling. The order’s stated procedural basis was enforcement of that earlier ruling. Its national scope, timing and unequal practical availability are the features relevant to evaluating its electoral effect.^[5],[6],[8]

Timeline sources: original order, contemporary reconstruction and the TSE’s own notice. All times Brasília. This is a documentary timeline, not a forensic reconstruction from system logs.^[5],[7],[8]

Flávio’s campaign also requested an extension through the TSE for rain-affected areas in five states. That targeted request differs from the STF intervention. Flávio later refused to recognize the result and announced a challenge. His objection focused attention on the intervention and its consequences.^[9],[10]

The extension creates two different causal questions. It could have affected which people were able to vote, or the interval before publication could have been used to change already-cast results. The second hypothesis concerns alteration after voting and must be traced across machine closure, local tally creation, transmission, receipt, validation, totalization and publication. Each stage creates a different record against which the reported outcome can be compared.

Segre’s suggested comparison of voting times could help identify unusual machine activity, provided definitions, clock accuracy, queues and ballot complexity are controlled. His seven-second/four-second example is hypothetical. Such timing analysis cannot, by itself, detect a substitution occurring later in a central database.^[4]

The timing supplies a concrete intervention to examine alongside explanations based on voter behavior. Its explanatory value depends on the number and direction of votes it affected, or on any changes between original local records and published totals. The model residual provides a scale against which such effects can be assessed.

06 / Distinguishing the explanations

Compare evidence created before publication with the published totals

The most direct test of alteration after local counting is to compare authenticated original Boletins de Urna—the polling-machine result reports—with published section totals, and then recompute the national result. Originals independently retained by observers before publication are especially valuable because a second download from the same central system is not an independent reference.^[26]

1. Preserve originals and logs immediately. Obtain legally authorized, independently witnessed copies of original media, signed BU files, digital vote records, machine logs, transmission and receiver logs, totalization logs, configurations and backups. Record provenance, hashes, custody and subsequent access.
2. Reconcile the complete section inventory. Account for operating, aggregated, uninstalled, substituted, pending, rejected, manually processed and resubmitted sections. A clean comparison of selected files can miss an omitted or duplicated section.
3. Recompute and quantify discrepancies. Match authenticated original BUs to published results, then independently sum all included sections. Reconcile candidate, blank, invalid and participation totals. Publish every discrepancy and its resolution.
4. Reconstruct the interval. Separate closure, printing, transmission attempts, receipt, validation, processing and disclosure. Correct time zones and clock drift. Compare independently retained timestamps with central logs; look for unexplained revisions, not merely late arrivals.
5. Trace authorization and coordination. Match privileged access, deployments, overrides and corrections to approvals. Through lawful processes, obtain communications relevant to requesting, drafting and implementing the extension. Link any alleged instructions to actual unauthorized changes.
6. Use probability sampling where originals cannot all be checked. Pre-specify representative selection, maximum-error bounds and escalation rules. Treat missing records conservatively. Add targeted anomaly checks, while keeping them separate from the representative sample. Release methods, code and non-sensitive evidence for both campaigns and independent analysts.

The compiled TSE Resolution 23.673/2021 already provides routes for inspection: Article 46 lists relevant logs, BUs, digital vote records, machine records and section data; Articles 47–49 address requests and preservation; Article 51 provides for post-election verification supported by facts and a work plan. Eligible oversight bodies should promptly request preservation and disclosure through those procedures.^[25]

The audit must measure margin error

Summing e_i measures the published margin’s overstatement relative to the original reports. Investigators must establish whether the total could reach the reported 4,430,191-vote margin. Reversing the winner would require, for example, at least 2,215,096 votes transferred from Flávio to Lula, or 4,430,192 extra Lula votes, if either were the sole mechanism.

The statistical guarantees of an audit depend on a complete, trustworthy evidence trail and a valid escalation procedure. A BU is a machine-generated tally, not a voter-verified paper record of every voter’s intention. Agreement between independently preserved BUs and published totals would strongly constrain the proposed later alteration mechanism, but would not by itself exclude misrecording before the original BU was created. Digital signatures likewise depend on the integrity of their keys and signing process.^[27]

07 / Interpretation

A substantial departure demands a substantial explanation

The result fails the 30/50/50 baseline by 5.191 million Lula votes and 10.382 million votes in the final margin. Under the broadest independent uncertainty distributions examined, the probability of at least a 70% share of the combined gain is 0.08209%; matching or exceeding the reported 73.60% has probability 0.01901%. Within this model, the departure is extraordinary.

1. First explanation: the baseline substantially misread the electorate. First-round choices may be much weaker predictors of runoff behavior than the baseline assumes. Ideological labels, endorsements, turnout selection or the survey estimates may have misrepresented the relevant preferences. In this explanation, the apparently anomalous outcome exposes a failure in the initial model of the electorate.
2. Second explanation: extraordinary voter movement away from sound expectations. If the baseline reasonably captured the starting preferences, the observed gain requires a large shift in support, mobilization or participation. The model shows the size and conditional improbability of that shift. An explanation based on ordinary politics must account for its magnitude, including movements concealed by net totals.
3. Third explanation: an omitted influence or intervention affected the election or its reported result. This includes mechanisms absent from the voter-transfer model, such as a material effect of the late procedural intervention, administrative error or deliberate alteration. A specific explanation must produce effects of the appropriate direction and scale. The processing interval and independently preserved local tallies provide concrete places to test it.

These categories can operate together. The strength of the statistical inference depends on how well the expected transfers, uncertainty ranges and dependence assumptions represent the electorate. A 70/30 expectation alone does not determine a tail probability; confidence in that expectation is also part of the model.

The clear conclusion is that the reported outcome falls far outside the range of routine variation described by the baseline and uncertainty distributions examined here. Explaining it requires substantially different initial preferences, exceptional subsequent voter movement, or another material mechanism affecting the result. Establishing which account can explain the full discrepancy is the central analytical task.