arXiv CSopen access

Trigger Timing, Deadline Readiness, and Event-Aligned Accounting for Dynamic Ad Insertion

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Trigger Timing, Deadline Readiness, and Event-Aligned Accounting for Dynamic Ad Insertion

arXiv:2609.19899v1 [cs.MM] 17 Sep 2026

Prashant Chaudhary and Kapil Khandelwal

Abstract—Dynamic ad insertion comparisons can conflate trigger, reach, readiness, playback, billability and measurement even when the accounting is arithmetically correct. We separate these events with an observed-event ledger, a candidate-invariant reference deadline and pod-level contribution accounting. The deadline rule is fixed before candidate assignment and tests whether an admissible transition state remains valid, not whether preparation merely finished earlier. A restricted monotoneplayback representation states when media-position summaries suffice; a counterexample shows why they fail over a wider path class. An offline synthetic study exercises the definitions over nine short-lifetime conditions informative for the readiness comparison and nine long-lifetime conditions serving as analytic controls. Across 45,000 shared scripts, two trigger policies share imposed playback paths, latency draws and hypothetical value and cost coefficients. Playhead summaries substantially misclassify reach events in the nonmonotone mixtures, yet neither shortcut reverses the contribution contrast in this grid, because some errors cancel under the shared design. Replacing validity at the deadline with completion by the deadline reverses the comparison in three of the nine informative conditions, all at one of the three latency settings. Scoring readiness at actual viewer arrival rather than at the reference deadline shifts pause-path readiness but changes no contribution sign. These outcomes are consequences of the event definitions applied to established misclassification mechanisms. Conservative bounds retain uncertainty when records are missing, and the artifact records code, seeds, event histories and checking procedures. The evidence is synthetic, uses no commercial telemetry, and ranks neither server-side nor server-guided insertion. Index Terms—ad insertion, connected television, deadline readiness, measurement, simulation, trigger timing.

I. Introduction Dynamic ad insertion (DAI) creates an engineering deadline at a commercial opportunity. An ad opportunity, or avail, is a planned point where ads or fallback may replace or interrupt primary content. An ad pod groups the components considered at that opportunity. A manifest tells the player which media to request. A cue announces an opportunity, while a schedule records expected opportunities independently. We call an opportunity actionable when a permitted cue or schedule provides enough information to begin viewer-specific work. Preparation can include auction calls, selection, validation, transcoding, manifest processing and player-side loading. An earlier start gives these steps more time but exposes resources (Corresponding author: Prashant Chaudhary.) P. Chaudhary is an independent researcher, Santa Clara, CA 95051 USA (e-mail: [email protected]). K. Khandelwal is with the Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]).

to sessions that may never reach the opportunity. A later start reduces some exposure while shortening the preparation window. Server-side ad insertion (SSAI) and server-guided ad insertion (SGAI) describe presentation mechanisms, not fixed trigger times. Both stitched and guided workflows permit earlier or later session-specific resolution [1]–[10]. An analyst must also distinguish the resulting records. A resolver request establishes neither playback nor payment. A playback record needs reconciliation with the billable event, and a tracking declaration differs from an independent measurement result [11]–[14]. Media position and wall-clock time answer different timing questions. Treating these records as interchangeable can produce a plausible aggregate that counts the wrong events. Table I maps documented interfaces to the evidence that a comparison still needs. We contribute an event construction that makes probability and accounting tools usable for DAI. The model separates execution locus from resolution timing, records observed trigger and reach events, and fixes a deadline that a candidate cannot move through its own delays. Example 1 shows why terminal and maximum playhead positions cannot always recover those events. Pod-level accounting then combines reconciled value with event-attributable and shared costs. An application specification and an executed synthetic study show the distinct effects of playhead misclassification, completion-based readiness and incomplete capture, including cases where errors cancel in the policy comparison. Applied, the method produces one row per eligible planned pair with common opportunity/deadline identifiers and candidate-specific event, commercial and cost records; its principal comparison is the all-pair contribution difference in Equation (11), reached or not. Section II establishes the implementation context. Sections III and IV define the events and accounting, and Section V presents the study and application specification. Section VI states the scope and interpretation limits. II. Architecture, timing, and adjacent work A. Execution locus and resolution timing Execution locus identifies whether a server returns a personalized stitched presentation or the player resolves and switches to separately signaled content. Resolution timing identifies when personalized per-play work begins relative to the opportunity. These are separate design dimensions. Under a stitched presentation, per-play work can begin at actionability or be deferred while assembly remains serverside. A player-guided path can preload decision resources or media, or wait until nearer playback. Earlier resolution allows

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more preparation time and exposes more nonreaching pairs to work; later resolution shortens both that lead and some speculative exposure. The combination determines the relevant scheduling, cache and load behavior. Stitching refers to what the client receives as one presentation, whether or not the server concatenates files. Guided insertion does not require moving the auction into the client: selection may remain server-controlled while the player resolves, loads and executes a transition [1], [6], [7], [15]. Timing also changes load. Synchronized live triggers can create bursts that change latency, fill and value. AWS documents retrieval windows and transactions-per-second shaping for this reason [8]. Trigger and load-control policies must therefore be specified together. B. Standards and product mechanisms HTTP Live Streaming (HLS) interstitials signal asset or asset-list resources through EXT-X-DATERANGE metadata [2], [6]. The second-edition draft adds Preload Date Ranges for earlier resolution [2]. Dynamic Adaptive Streaming over HTTP (MPEG-DASH) edition 6 adds alternative Media Presentation Description (MPD) events for insertion or replacement [3]–[5]. DASH Industry Forum (DASH-IF) Livesim2 documents replacement events, personalized List MPDs and earliestResolutionTimeOffset [3], [16]. Earlier DASH-IF Part 5 supplies interoperability context [17]. AWS distinguishes stitched from guided prefetch [8]–[10], [18]. Schedule-based ad prefetching, listed as unsupported for SGAI in the compatibility matrix [10], is distinct from the guided-prefetch mechanism based on session-tagged manifest refreshes [9]. Guided prefetch depends on session-tagged HLS interstitial-media manifest refreshes reaching MediaTailor. Caching /v1/i-media prevents that trigger path, and the cited guide does not support DASH guided prefetch [9]. Equal schedule settings therefore need not imply equal request paths or costs. A cue-driven path needs cue receipt and interpretation unless a declared independent schedule supplies actionability. SCTE 35-1 and SCTE 224 locate signaling and scheduling in that chain [19], [20]. The Video Ad Serving Template (VAST) and Open Measurement supply metadata, tracking and measurement interfaces [11], [12], while OpenRTB’s tmax bounds a bid-response stage rather than end-to-end readiness [21]. Auction and billable events remain distinct [14]. Interactive Advertising Bureau (IAB) and Media Rating Council (MRC) guidance separates loading from rendering and addresses indirect SSAI signals [22], [23]. Table I maps these interfaces to the remaining evidence requirements. C. Relationship to adjacent work Predictive prefetching trades latency against speculative traffic [24], and timing can change queueing even at equal transfer volume [25]. Real-time scheduling formalizes deadlines [26]. Prior DAI work addresses URL-based pointer mapping [27] and sixth-edition server-guided insertion [1]; adjacent cloudcaching work supplies retention policies [28]. Streaming studies relate playback problems and session characteristics to

abandonment [29], [30], but supply no directly transferable DAI nonreach probability. The present construction combines the cue, session, pod, billability and reconciliation rules located in Section II-B with observed-event accounting. The measurement problem also involves post-assignment selection. Comparing only viewers who reach an opportunity can compare different populations when the policy affects reach; the general problem of posttreatment conditioning is established in causal inference [31]. The proposed deadline and eligible-pair denominator address this design issue but do not identify a causal effect without an appropriate assignment and observation scheme. Common random numbers and missingdata bounds likewise supply established tools rather than novel estimators [32], [33]. Misclassification can distort a comparison and, under suitable error mechanisms, reverse its direction; that phenomenon is established in measurement-error research, including Copeland et al. [34] and the qualifications summarized by Yland et al. [35]. This paper specifies the DAI events and time conventions whose substitution produces the concrete errors examined in Section V. III. Event-timed model A. The unit, the cohort, and the rules fixed in advance We analyze one planned pod–play pair: a named mid-roll opportunity and an initiated session, retaining slot outcomes within the pod. A candidate comprises its implementation, trigger policy, load controls and routing rule. Label the actual routed path in a hybrid fleet. Fix eligibility, consent, device support, pod construction and the analysis horizon before assignment, retaining pairs that never reach the opportunity. A qualifying crossing is the observed boundary crossing selected by a predeclared rule. Descriptive accounting may treat repeated crossings as separate pairs. A causal comparison instead retains candidate-induced seeks and restarts within the original pair or a separately declared session-level unit, without enlarging the cohort after assignment. Do not pool those denominators. For pre-roll, trigger and reach may both approach one while readiness still varies. Index 𝑎 identifies the candidate, 𝑖 the planned pair, and 𝑡 wall-clock time. A quantity indexed only by 𝑖 is common to candidates for that pair. We suppress indices only when their meaning is unambiguous. Declare one viewer-facing transition state that counts as ready for every candidate. A state is admissible only while it meets that criterion and remains valid and usable. A dependency graph comprises the processing steps and branches started under a trigger policy to create that state. Four further rules are fixed before the comparison begins: • the opportunity boundary is frozen, meaning that a candi-

date’s own delays cannot move it; rule says when a pair can no longer count as reached; • we record a qualifying initiation when a per-play resolution job starts and that start satisfies the declared trigger rule; and • the session-expiry

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• the primary-trigger rule selects which qualifying initiation

is used for timing and exposure accounting when more than one occurs. Two quantities are recorded once per pair and are common to every candidate: 𝐵𝑖 : the target opportunity position on the declared primarycontent timeline; • 𝐷 𝑖 : the common wall-clock deadline at which the viewerfacing transition state is required.

𝑜𝑖(0) ≥ 0 be that session’s validated live-edge delay, frozen from a measurement unaffected by candidate-specific work. Use the common reference deadline prog

𝐷𝑖 = 𝐷𝑖

+ 𝑜 𝑖(0) .

•

For each candidate and pair, we record: 𝐴 𝑎𝑖 : the earliest wall-clock time at which the required cue or schedule is usable by candidate 𝑎; • 𝐺 𝑎𝑖 : the scheduled media position at which candidate 𝑎 intends a per-play trigger to occur, when candidate 𝑎 defines such a position; • 𝑇𝑎𝑖 : one when at least one qualifying initiation is observed, zero otherwise under complete trigger-event capture; • 𝜏𝑎𝑖 : the wall-clock time selected by the primary-trigger rule, defined only when 𝑇𝑎𝑖 = 1; • 𝑅 𝑎𝑖 : one when the session reaches the frozen opportunity boundary before the common session-expiry rule applies, whatever is presented there, and zero otherwise; • 𝑍 𝑎𝑖 (𝑡): a binary state indicator equal to one exactly while the common admissible state is valid and usable at wallclock time 𝑡, and zero otherwise; • 𝐻 𝑎𝑖 : one if a dependency graph started by a qualifying initiation ever attains the common state, and zero otherwise; and • 𝐶 𝑎𝑖 : the first wall-clock time at which that attainment occurs, defined only when 𝐻 𝑎𝑖 = 1. •

The definitions imply 𝐻 𝑎𝑖 ≤ 𝑇𝑎𝑖 , which completely captured records must satisfy. An externally prepared state may be deadline-valid without policy attainment (Section III-D). A missing required start or terminal record leaves the corresponding quantity unobserved, not zero. ∗ for physical initiation and 𝑂 𝑇 for adequate Write 𝑇𝑎𝑖 𝑎𝑖 ∗ when 𝑂 𝑇 = 1, capture of that event class. Then 𝑇𝑎𝑖 = 𝑇𝑎𝑖 𝑎𝑖 and is unobserved otherwise; reach and validity follow the same construction. Sections III–IV assume complete capture; Section V-C4 relaxes it. The full notation and domains appear in Supplement S4. For multiple qualifying initiations indexed by 𝑗, record initiation times 𝜏𝑎𝑖 𝑗 . The default primary trigger is 𝜏𝑎𝑖 = min 𝑗 𝜏𝑎𝑖 𝑗 , with any alternative rule fixed before comparison. It opens the readiness window and trigger-cost exposure. Later retries and duplicates remain trigger-attributable costs without resetting 𝜏𝑎𝑖 . Record a successful graph’s own initiation separately when its processing time is needed. Apply service-tier and avail-mapping eligibility before constructing pairs, and common session-expiry and backgroundplayback rules afterward. We fix the rules that produce the opportunity boundary and deadline before candidate assignment. Their values for a particular pair need not be known before playback starts. For timeshifted live viewing, first map the authoritative opportunity prog presentation timestamp to a program-clock deadline 𝐷 𝑖 . Let

Share the frozen offset across candidates without reestimating it from later playback. Record its measurement time, clock authority, event identifier, uncertainty, discontinuities, presentation-timestamp (PTS) restamping, and schedule update or cancellation rule. Cue arrival and interpretation set 𝐴 𝑎𝑖 without moving 𝐷 𝑖 . An unknown offset leaves viewer-relative readiness unresolved unless bounds permit classification. An prog origin-relative analysis may use 𝐷 𝑖 but must name that different target and keep its rate separate. Three cases arise: an offset observed before assignment is frozen and shared; one observed after assignment is admissible only if the recorded ordering establishes that no candidatespecific work could have affected it; an offset obtainable only through candidate-dependent instrumentation cannot supply the common target. Earlier assignment at device, household or channel-day level helps only if it also establishes an unaffected common offset; moving assignment earlier alone does not. Otherwise use the separately named origin-relative target. For on-demand playback, derive 𝐷 𝑖 from a candidateindependent reference trajectory to 𝐵𝑖 . A descriptive ledger may use a predeclared pause or seek rule. A causal comparison excludes post-assignment actions from that reference or treats them as candidate-affected outcomes. Applying the same update rule does not itself make deadlines common. Candidate stalls, retries and transition delays move neither 𝐵𝑖 nor 𝐷 𝑖 . Reach is one whether the boundary presents paid ads, fallback, slate, house content or primary-content continuation, independently of ad playback. Readiness refers to the frozen reference schedule, including any validated candidate-unaffected offset. We report later drift, pauses, seeks and candidate-induced live-edge changes separately from readiness at that reference deadline. For a pair with 𝑇𝑎𝑖 = 1, the available wall-clock readiness window is 𝑊𝑎𝑖 = 𝐷 𝑖 − 𝜏𝑎𝑖 .

(1)

A primary trigger after the deadline gives 𝑊𝑎𝑖 < 0. On a cue-dependent or schedule-dependent path, 𝜏𝑎𝑖 ≥ 𝐴 𝑎𝑖 , so the actionability-based upper bound is max 𝑊𝑎𝑖 = 𝐷 𝑖 − 𝐴 𝑎𝑖 ,

max 𝑊𝑎𝑖 ≤ 𝑊𝑎𝑖 .

If 𝐴 𝑎𝑖 > 𝐷 𝑖 , the cue-dependent path cannot meet the deadline. A separately prepared admissible state may still be valid at that deadline. Where 𝐺 𝑎𝑖 is defined, the media distance 𝐵𝑖 − 𝐺 𝑎𝑖 is not a substitute for 𝑊𝑎𝑖 : the two coincide only under further assumptions about continuous 1× playback, buffering, clock alignment, and the definition of the deadline.

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B. Observed trigger and reach events For any event E, let 1{E} equal one when the event occurs and zero otherwise. The two events at the center of the exposure accounting are 𝑇𝑎𝑖 = 1{qualifying initiation observed}, 𝑅 𝑎𝑖 = 1{boundary reached before session expiry}

(2)

Both are observed, and neither is inferred from a single terminal playhead value. A live join after the scheduled trigger opportunity, a catch-up trigger fired at join, a forward seek, a backward seek, a restart, and a skipped pod each break any rule that depends only on final media position. Example 1 (non-identification from playhead summaries). Let the trigger position be 60 and the opportunity position be 100, and count a seek over the opportunity as a skip, not as reach. One session plays continuously from 0 to 110, restarts at 0, and stops at 10. A second plays from 0 to 40, seeks to 110, immediately restarts at 0, and stops at 10; assume no catch-up trigger. Both have final position 10 and maximum position 110. The first has (𝑇, 𝑅) = (1, 1) and the second (𝑇, 𝑅) = (0, 0). Thus neither the terminal position nor even the pair of terminal and maximum positions identifies these events over the allowed path class. The example is constructed under an explicit skip rule; it makes no claim about how often such paths occur. Write Pr for probability under the declared eligible-pair population or synthetic script distribution. For a finite cohort, the corresponding empirical rates use its stated weights. In probability subscripts, 𝑡 labels triggering and 𝑟 labels reach; the argument 𝑡 in 𝑍 𝑎𝑖 (𝑡) remains wall-clock time. The cost superscripts and readiness-conditioning labels are defined in Sections IV-A and III-D, respectively; Supplement S4 collects them. Define the trigger probability 𝑝 𝑡 ,𝑎 = Pr(𝑇𝑎𝑖 = 1) and the reach probability 𝑝 𝑟 ,𝑎 = Pr(𝑅 𝑎𝑖 = 1). Where the conditioning event has positive probability, the reached-pair trigger rate 𝑔 𝑎 and the nonreached-pair trigger rate ℎ 𝑎 are

can cut short the work actually performed, and any reusable output has to be allocated to whichever pairs benefit from it. Because 𝑇𝑎𝑖 admits a late trigger as well as an early one, pre define the pre-deadline subset 𝑝 10,𝑎 = Pr(𝑇𝑎𝑖 = 1, 𝑅 𝑎𝑖 = 0, 𝜏𝑎𝑖 < 𝐷 𝑖 ) and report it wherever the timing distinction matters. The comparison 𝜏𝑎𝑖 < 𝐷 𝑖 is evaluated only on the 𝑇𝑎𝑖 = 1 subpopulation on which 𝜏𝑎𝑖 is defined. An initiation exactly at the deadline has 𝑊𝑎𝑖 = 0 and belongs to total triggering but not to this strictly pre-deadline subset. It does not replace the total trigger probability in the cost calculation of pre Section IV. Reported together, 𝑝 10,𝑎 and 𝑝 10,𝑎 distinguish total trigger-without-reach exposure from its pre-deadline subset. Revenue earned by a reached pod stays attached to that pod and is not credited to a nonreaching pair merely because work was reused. Any contractually retained value, credit, or charge on an 𝑅 𝑎𝑖 = 0 pair is handled as a separate nonreach branch in Section IV-A. The complementary cell 𝑝 01,𝑎 counts reached opportunities that lack the designated trigger, which includes missed-trigger cases and cases where some other permitted path had already prepared an admissible state. An implementation that guarantees a designated per-play trigger on every reached pair satisfies 𝑅 𝑎𝑖 ≤ 𝑇𝑎𝑖 , which we call the event-containment condition. Under it, 𝑝 01,𝑎 = 0, and 𝑔 𝑎 = 1 whenever 𝑝 𝑟 ,𝑎 > 0. For 0 < 𝑝 𝑟 ,𝑎 < 1 this reduces Equation (3) to

𝑝 𝑡 ,𝑎 = 𝑝 𝑟 ,𝑎 + ℎ 𝑎 (1 − 𝑝 𝑟 ,𝑎 ).

(5)

The endpoint cases follow from Equation (4); event containment additionally gives 𝑝 𝑡 ,𝑎 = 1 when 𝑝 𝑟 ,𝑎 = 1. Containment is a system property that complete event capture must confirm; placing a scheduled trigger before the opportunity does not establish it.

C. Restricted monotone-playback representation 𝑔 𝑎 = Pr(𝑇𝑎𝑖 = 1 | 𝑅 𝑎𝑖 = 1),

ℎ 𝑎 = Pr(𝑇𝑎𝑖 = 1 | 𝑅 𝑎𝑖 = 0).

For 0 < 𝑝 𝑟 ,𝑎 < 1, the law of total probability gives 𝑝 𝑡 ,𝑎 = 𝑔 𝑎 𝑝 𝑟 ,𝑎 + ℎ 𝑎 (1 − 𝑝 𝑟 ,𝑎 ).

(3)

The same relationship can be written so that no conditional probability is ever taken over an empty cell. Let the first digit of a cell label record 𝑇𝑎𝑖 and the second record 𝑅 𝑎𝑖 , and define 𝑝 10,𝑎 = Pr(𝑇𝑎𝑖 = 1, 𝑅 𝑎𝑖 = 0) and 𝑝 01,𝑎 = Pr(𝑇𝑎𝑖 = 0, 𝑅 𝑎𝑖 = 1). Then 𝑝 𝑡 ,𝑎 = 𝑝 𝑟 ,𝑎 + 𝑝 10,𝑎 − 𝑝 01,𝑎 .

(4)

At 𝑝 𝑟 ,𝑎 = 0, Equation (4) collapses to 𝑝 𝑡 ,𝑎 = 𝑝 10,𝑎 = ℎ 𝑎 , and the empty reached-pair rate 𝑔 𝑎 drops out. At 𝑝 𝑟 ,𝑎 = 1 it collapses to 𝑝 𝑡 ,𝑎 = 1 − 𝑝 01,𝑎 = 𝑔 𝑎 , and the empty nonreachedpair rate ℎ 𝑎 drops out. The 𝑝 10,𝑎 cell is the measure of trigger-without-reach exposure. It should not be read as pure waste, since cancellation

Media position can stand in for the event ledger, but only inside a stratum where the playback assumptions hold. A stratum is a subset of pairs sharing the same trigger position, opportunity position, entry condition, and playback assumptions, indexed by 𝑠; a conditioning suffix | 𝑠 means the rate is computed within that stratum alone. Let 𝐺 𝑠 be the common trigger position and 𝐵𝑠 the common opportunity position, with 𝐺 𝑠 ≤ 𝐵𝑠 . The representation additionally requires session entry at or before 𝐺 𝑠 , a qualifying initiation when 𝐺 𝑠 is crossed, continuous forward playback, and no seeks, restarts, discontinuity jumps, or catch-up triggers. The observation stops at the common session-expiry boundary; a crossing recorded only after expiry is excluded. Expiry after a valid crossing does not revoke that earlier reach. Let 𝑆 𝑎𝑖 be the final media position reached before session expiry for pair 𝑖 under candidate 𝑎, and for any mediaposition threshold 𝑥 define the stratum-specific survival function 𝐹 𝑆,𝑎|𝑠 (𝑥) = Pr(𝑆 𝑎𝑖 ≥ 𝑥 | 𝑠). When Pr(𝑆 𝑎𝑖 < 𝐵𝑠 | 𝑠) > 0,

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𝑇𝑎𝑖 = 1{𝑆 𝑎𝑖 ≥ 𝐺 𝑠 },

𝑝 𝑡 ,𝑎|𝑠 = 𝐹 𝑆,𝑎|𝑠 (𝐺 𝑠 ),

𝑅 𝑎𝑖 = 1{𝑆 𝑎𝑖 ≥ 𝐵𝑠 },

𝑝 𝑟 ,𝑎|𝑠 = 𝐹 𝑆,𝑎|𝑠 (𝐵𝑠 ),

ℎ 𝑎|𝑠 =

𝐹 𝑆,𝑎|𝑠 (𝐺 𝑠 ) − 𝐹 𝑆,𝑎|𝑠 (𝐵𝑠 )

(6)

. 1 − 𝐹 𝑆,𝑎|𝑠 (𝐵𝑠 )

Equation (6) assumes no particular abandonment distribution. Its positive-denominator condition is required because the nonreached-pair trigger rate has no meaning when every session reaches 𝐵𝑠 . In that endpoint case, 𝑝 𝑟 ,𝑎|𝑠 = 𝑝 𝑡 ,𝑎|𝑠 = 1 and 𝑝 10,𝑎|𝑠 = 0; ℎ 𝑎|𝑠 is not needed. Strata do not pool freely. Combining strata with different opportunity positions, trigger rules, or included sessions does not generally return the average of the stratum-specific rates. When aggregating, weight the event-aligned value and variable-cost components, and apply shared horizon cost once to the pooled cohort unless a declared nonoverlapping rule allocates that cost across strata instead. For live and seekable services, estimate the event probabilities directly from joined logs, using Equation (4) when a conditioning cell is empty. Reach stays candidate indexed throughout. Player scheduling, blocking work, incompatibility, and candidate-induced stalls can all affect pre-boundary behavior and abandonment. The boundary definition is common and frozen, but equality of reach across candidates is a hypothesis to be tested rather than a premise, and nothing in the framework imposes an unconditional reach penalty on SSAI or on SGAI. D. Deadline readiness and downstream states Before comparison, declare a common viewer-facing criterion: required playable prefix, transition-timing tolerance and no remaining critical-path network action. Guided instructions with media still unfetched fail that criterion; server-validated stitched segments alone also do not prove client readiness. Each path needs evidence of the whole criterion. Supplement S7 instantiates it on two constructed paths, testing the mapping rather than deployment collectability. Using the state variable 𝑍 𝑎𝑖 (𝑡) from Section III-A, the deadline-readiness indicator is 𝑄 𝑎𝑖 = 𝑍 𝑎𝑖 (𝐷 𝑖 ).

(7)

A fallback that meets the common criterion at 𝐷 𝑖 has 𝑄 𝑎𝑖 = 1, and lower monetary value alone does not turn it into a miss. If no admissible state is valid at 𝐷 𝑖 , then 𝑄 𝑎𝑖 = 0, and that holds whether the state arrives late or arrived early and has since expired, been evicted, or been revoked. Changing the readiness criterion changes the event definition, which invalidates an otherwise unchanged comparison. 𝑄 𝑎𝑖 is readiness at the reference deadline, not at actual arrival. A post-assignment pause can delay arrival beyond 𝐷 𝑖 , allowing a state valid at 𝐷 𝑖 to expire before it is needed. Arrival-relative readiness depends on a potentially candidateaffected time and cannot replace this candidate-invariant target. Section V-C3 reports it alongside reference readiness for the imposed, shared paths of the synthetic study (SIM1) in Section V-C.

A validity spell includes its attainment instant and excludes its expiry or invalidation instant. For one spell attaining the state at 𝐶𝑎𝑖 and ending at 𝐸 𝑎𝑖 > 𝐶𝑎𝑖 , where 𝐸 𝑎𝑖 denotes that spell’s invalidation time, the convention is 𝐶𝑎𝑖 ≤ 𝑡 < 𝐸 𝑎𝑖 . Thus attainment exactly at 𝐷 𝑖 counts; expiry exactly at 𝐷 𝑖 does not. With multiple spells or prepared paths, 𝑍 𝑎𝑖 (𝑡) is one when any admissible spell is active. A cancellation, failure or unresolved job is not ready unless another admissible state is active. Bounded validity is documented in MediaTailor: prefetch schedules define a consumption window, and recurring schedules can limit retrieved-ad availability by an expiration setting [8]. This establishes a finite-usability mechanism, not the value of SIM1’s stress lifetime. A deadline probe observes one instant; clock uncertainty requires transitions across the uncertainty interval. When 𝑇𝑎𝑖 = 𝐻 𝑎𝑖 = 1 and 𝜏𝑎𝑖 ≤ 𝐶𝑎𝑖 , define policy triggerto-attainment time 𝐿 𝑎𝑖 = 𝐶𝑎𝑖 − 𝜏𝑎𝑖 ≥ 0. It includes parallel or sequential work, retries, failed branches and speculative attempts. Record the successful graph’s own initiation when its processing time is needed. A state predating 𝜏𝑎𝑖 or prepared outside the policy receives no 𝐿 𝑎𝑖 , and its source must be recorded. Policy attainment at 𝐶𝑎𝑖 ≥ 𝜏𝑎𝑖 still defines policy latency as 𝐶𝑎𝑖 − 𝜏𝑎𝑖 , otherwise it remains undefined. The inequality 𝐿 𝑎𝑖 ≤ 𝑊𝑎𝑖 establishes first attainment by 𝐷 𝑖 , equivalently 𝐶𝑎𝑖 ≤ 𝐷 𝑖 , not continued validity. Expiry, eviction or revocation may still make 𝑄 𝑎𝑖 = 0. Logging can miss first attainment while directly observing readiness at 𝐷 𝑖 . A stage timeout such as tmax affects attainment, fill or value without replacing either elapsed-time measure. Timestamps must share a clock. Map 𝐴 𝑎𝑖 , 𝜏𝑎𝑖 , 𝐶𝑎𝑖 , the state-validity transitions, and 𝐷 𝑖 onto one, and let 𝜀 ≥ 0 be a documented bound, in wall-clock units, on relative deadlineversus-state-transition alignment error. This is a bound on relative alignment, not a bound on each timestamp. Uncertainty intervals include both endpoints; they are not half-open validity spells. When the error bound is zero, the interval reduces to the single deadline. When only separate timestamp bounds are available, propagate both, as below. A pair is definitely ready only when 𝑍 𝑎𝑖 (𝑡) = 1 throughout [𝐷 𝑖 −𝜀, 𝐷 𝑖 +𝜀], and definitely not ready only when the state is zero throughout that interval. Anything else is marked timing-indeterminate and analyzed for sensitivity rather than forced into a binary value. Directional bounds are handled in Supplement S5. Any reported difference built from these timestamps must propagate the bound of every timestamp inside it. Two symmetric per-timestamp bounds of size 𝑒 give a worst-case difference bound of 2𝑒, rather than 𝑒; Supplement S5 gives the directional enclosure. An ordering whose difference interval crosses zero remains indeterminate. Do not drop indeterminate pairs: report lower and upper readiness rates by assigning those pairs zero and one respectively, or use a calibrated measurement-error model defined in advance. Fig. 1 collects the event ledger, the restricted position representation, and the wall-clock readiness relation defined above. Three readiness rates distinguish conditioning and scoring time. Superscripts 𝐽 and 𝑅 condition reference-deadline readi-

6

Observed events first; position survival only in a restricted stratum GENERAL SOURCE-JOINED EVENT LEDGER

Actionable

Trigger

cue or schedule Aai

observed job Tai, τai

event order shown is illustrative and not to scale

Validity

job Hai, Cai state Zai(Di), Qai

Reach

Playback

Commercial

frozen opportunity Rai

start / progress / fallback

billability and measurement

Live joins, seeks, restarts, catch-up triggers, and missed triggers are represented by these observed events — not by a final playhead scalar.

RESTRICTED MONOTONE-PLAYBACK SPECIAL CASE Trigger position

Gs

entry at/before Gs; fires at crossing; forward only

Opportunity position

Sai < Bs

Sai ≥ Bs

Bs

Gs ≤ Bs and Pr(Sai < Bs ∣ s) > 0 are required; otherwise estimate trigger and reach directly.

TRIGGERED-JOB WALL CLOCK W max = D − A i ai ai

independent from media position

Wai = Di − τai Actionable Observed trigger

Aai

Valid state begins

Frozen deadline

Cai

Di

τai

VALID AT DEADLINE SPELL ENDS EARLY

Zai(Di) = 1 ⇒ Qai = 1 expiry/eviction ⇒ Qai = 0

Fig. 1. The general model logs actionability, an observed trigger, state validity, frozen opportunity reach, playback, billability, and measurement. The wall-clock panel distinguishes a state valid at the deadline from one that expires or is evicted first. The media-position panel applies only to the continuous-forward-playback special case; live joins, seeks, restarts, and catch-up triggers require the observed event ledger.

arr be ness on a triggered job and a reached opportunity. Let 𝑡 𝑎𝑖 the first qualifying arrival time, defined only when 𝑅 𝑎𝑖 = 1; it is distinct from actionability 𝐴 𝑎𝑖 . Superscript 𝐴 denotes scoring at arrival. On positive-probability conditioning events, define

𝑞 𝑎𝐽 = Pr(𝑄 𝑎𝑖 = 1 | 𝑇𝑎𝑖 = 1),

𝑞 𝑎𝑅 = Pr(𝑄 𝑎𝑖 = 1 | 𝑅 𝑎𝑖 = 1),
arr 𝑞 𝑎𝐴 = Pr 𝑍 𝑎𝑖 (𝑡 𝑎𝑖 ) = 1 | 𝑅 𝑎𝑖 = 1 . (8) The job-conditioned rate includes cancellation and terminal errors, and can include readiness from shared or prepared paths outside the policy. Joint (𝐻 𝑎𝑖 , 𝑄 𝑎𝑖 ) cells distinguish policy attainment. If several paths supply 𝑍 𝑎𝑖 (𝐷 𝑖 ), record which supplied the deadline-valid state. The reach-conditioned rate includes missed triggers and describes schedule-relative readiness among reached boundaries. All three rates are diagnostics, not extra multipliers in the event-weighted accounting of Section IV. Record actual arrival and post-departure cancellation separately.

A hard readiness gate rejects candidates below a specified readiness threshold. Use it only when a contract, service-level objective or nonnegotiable playback requirement supplies the threshold and target population. Otherwise incorporate misses and fallback into pod value. Estimate or bound Equation (8) directly without a parametric latency model, preserving viewer, session, stream, device, region and incident dependence in cluster or block resampling. Illustrative path-specific readiness boundaries appear in Supplement S6. IV. Pod-level economic accounting A. Net value and event-aligned cost net be the complete pod’s realized monetary value after Let 𝑉𝑎𝑖 reconciliation. The superscript denotes reconciliation, and the value may be positive, zero or negative. Record no-fill, partial fill, readiness miss, fallback, playback, billability, invalidtraffic filtering, discrepancy settlement, makegoods, credits, clawbacks and payment-dependent measurement conditions. Write E for expectation.

7

net | 𝑅 Where 𝑝 𝑟 ,𝑎 > 0, define 𝑣 𝑎 = E[𝑉𝑎𝑖 𝑎𝑖 = 1]. Where the corresponding readiness cells also have positive probability, ready net | 𝑅 = 1, 𝑄 miss = E[𝑉 net | define 𝑣 𝑎 = E[𝑉𝑎𝑖 𝑎𝑖 𝑎𝑖 = 1] and 𝑣 𝑎 𝑎𝑖 𝑅 𝑎𝑖 = 1, 𝑄 𝑎𝑖 = 0]. Readiness is then already inside 𝑣 𝑎 : ready

𝑣 𝑎 = 𝑞 𝑎𝑅 𝑣 𝑎

+ (1 − 𝑞 𝑎𝑅 )𝑣 miss 𝑎 .

(9)

Equation (9) uses occupied cells only. If 𝑞 𝑎𝑅 = 1 it reduces ready to 𝑣 𝑎 = 𝑣 𝑎 and the empty miss mean is omitted; if 𝑞 𝑎𝑅 = 0 it reduces to 𝑣 𝑎 = 𝑣 miss and the empty ready mean is omitted. 𝑎 The miss term is reconciled monetary pod proceeds, possibly zero or documented paid value usable after 𝐷 𝑖 . Slate and primary-content continuation earn no pod revenue merely by preserving playback. Report engagement separately. An analysis starting from ready-cell value applies readiness once with explicit miss value, never again to the already averaged 𝑣𝑎. Conditional values belong to the candidate policy. Synchronized load and request timeouts can affect readiness, readystate value or miss-state value, with the channel determined by measurement. Let 𝑁 > 0 be the number of planned pod–play pairs in the analysis cohort over the delivery horizon, and let 𝑧 be the workload vector, whose components encode workload quantities or mix: channel count, opportunity count, deviceclass mix, distinct-creative count, creative-minutes, or reserved capacity. Costs attach to events. Let 𝜅 𝑡𝑎𝑖 ≥ 0 be the realized variable cost attributable to triggered work for candidate 𝑎 and pair 𝑖, covering downstream requests started by the trigger, redirects, retries, duplicate work that was not eliminated, and cleanup through the declared terminal state, cancellation point, or timeout cutoff. Let 𝜅 𝑟𝑎𝑖 ≥ 0 be the realized downstream delivery, playback, reporting, and reconciliation cost attributable to a reached opportunity. Superscript 𝑡 marks trigger-attributable cost and superscript 𝑟 marks reach-attributable cost. Where the corresponding event has positive probability, the conditional cost means are 𝑐 𝑡 ,𝑎 = E[𝜅 𝑡𝑎𝑖 | 𝑇𝑎𝑖 = 1],

𝑐𝑟 ,𝑎 = E[𝜅 𝑟𝑎𝑖 | 𝑅 𝑎𝑖 = 1].

Let 𝐾 𝑎 (𝑁, 𝑧) be the total cost over the delivery horizon that neither variable-cost measure captures. It can include integration work, platform licenses, minimum commitments, capacity tiers, shared creative preparation, and effects on the primary manifest or content path. Deduplication and allocation make the three cost buckets mutually exclusive. Keep retries, duplicate requests and cancellation cleanup in 𝜅 𝑡𝑎𝑖 unless explicitly allocated elsewhere, without also charging 𝜅 𝑟𝑎𝑖 or 𝐾 𝑎 (𝑁, 𝑧). Use unconditional event-weighted terms, assuming finite first moments and assigning zero to inactive branches. They remain defined at empty cells. When the relevant event probability is positive, the value, reach-cost and trigger-cost terms in Equation (10) factor as 𝑝 𝑟 ,𝑎 𝑣 𝑎 , 𝑝 𝑟 ,𝑎 𝑐𝑟 ,𝑎 and 𝑝 𝑡 ,𝑎 𝑐 𝑡 ,𝑎 , respectively. The first of the three expectations gives the reached-pair value contribution. Using it as the entire value contribution

assumes that reconciliation leaves no retained monetary value when 𝑅 𝑎𝑖 = 0. Where the governing contract permits retained revenue, credits, or charges on a nonreached pair, replace it net ] and report the with the unconditional contribution E[𝑉𝑎𝑖 𝑅 𝑎𝑖 = 0 branch separately. Expected monetary contribution per planned pair is then net 𝑚 𝑎 (𝑁, 𝑧) = E[𝑅 𝑎𝑖 𝑉𝑎𝑖 ] − E[𝑅 𝑎𝑖 𝜅 𝑟𝑎𝑖 ] (10) 𝐾 𝑎 (𝑁, 𝑧) . − E[𝑇𝑎𝑖 𝜅 𝑡𝑎𝑖 ] − 𝑁 All terms are currency per planned pair. The delivery horizon fixes 𝑁 and cost allocation. Attribute later makegoods and clawbacks to the original cohort within a declared maturation window, without adding later traffic to 𝑁. Report any settlement cutoff and remaining unsettled exposure. Book reuse once through its realized consequence. Documented avoided processing reduces incremental cost. Laternet and enters only a horipod revenue stays in that pod’s 𝑉𝑎𝑖 zon containing that pair. Cross-cohort reuse needs common starting inventory and terminal adjustment, wash-in/wash-out, or another declared boundary rule for carried savings and contractually measurable terminal monetary value. Otherwise one cohort bears creation cost while another collects its benefit. Keep nonmonetary terminal value outside 𝑚 𝑎 (𝑁, 𝑧) and 𝑐 𝑡 ,𝑎 . Use incremental avoidable cost for architecture selection and report fully allocated cost separately for budgeting. Allocate shared costs by a causal driver, such as requests, creatives, creative-minutes, channels or capacity. Test disputed denominators that could reverse the comparison. Let subscript 0 denote the current implementation, and compare Δ𝑚 𝑎,0 (𝑁, 𝑧) = 𝑚 𝑎 (𝑁, 𝑧) − 𝑚 0 (𝑁, 𝑧) with zero. The baseline does not itself have zero absolute contribution. Compare candidates 𝑎 and 𝑏 at the same declared cohort size 𝑁 and workload 𝑧, and define the contribution difference as

Δ𝑚 𝑎,𝑏 (𝑁, 𝑧) = 𝑚 𝑎 (𝑁, 𝑧) − 𝑚 𝑏 (𝑁, 𝑧).

(11)

B. Cost regimes and pod expansion A fixed-cost assumption applies only within one implementation scope and capacity regime. Allocate commitments and tiers to 𝐾 𝑎 (𝑁, 𝑧), and request-linked charges to 𝜅 𝑡𝑎𝑖 or 𝜅 𝑟𝑎𝑖 by their driving event. Evaluate tiers and shared costs at each candidate workload. Let 𝑀𝑎𝑖 count realized required ad or fallback components. A count determined after triggering is an outcome. Compare within 𝑀𝑎𝑖 groups only if component count was fixed before assignment. Retain the count, component presence, pod outcome and fallback. A zero-slot or no-fill result is not automatically ready; the readiness, value and fallback rules decide. Estimate readiness of all required components jointly rather than multiplying separate slot-level rates, and where one decision call resolves the whole pod, charge its shared decision cost once. For sequential resolution, 𝑇𝑎𝑖 remains pair-level and 𝜅 𝑡𝑎𝑖 includes every initiation and attributable operation under the primary-trigger rule. When readiness requires all components,

8

𝐶𝑎𝑖 is their first joint attainment and 𝐿 𝑎𝑖 the chain latency from primary initiation. Record component processing times separately. Sum slot values or costs only when interactions are immaterial, otherwise use the pod record. Equation (10) gives the general comparison. Supplement S3 gives a local fixed-plus-per-pair break-even form, valid only at a fixed workload and inside the applicable cost regime for each candidate. V. Measurement and application A. Minimum event joins We join server and client evidence whenever the player owns part of the transition. Connected television (CTV) guidance distinguishes server records from observable client events [13]. Table II lists the evidence needed for each event class. We retain a status, timestamp, source and join key for every event. An ephemeral session–opportunity identifier is sufficient. We check rendering through client evidence rather than server receipt, and account for cached or player-guided actions when a request is absent. We keep VAST tracking resources, Open Measurement results and contractual billable events distinct. MediaTailor also documents manifest, session-initialization and tracking-response log types, some requiring debug mode [36]. Common-state readiness still requires the joined evidence in Table II. B. Comparison design and data minimization Equation (10) is accounting. Causal identification additionally needs the same target population and a defensible assignment rule. Prefer randomized session routing or cluster assignment. Where candidates share a constrained queue or service, the assignment cluster must include units that can affect one another through that load. A non-serving shadow path compares only the outputs and load conditions it faithfully reproduces. An observational comparison must justify comparable groups, covariate overlap, adjustment for competing explanations and the target population. Disjoint groups support descriptive comparison; pairing them needs a justified construction [37]. Assign candidates before candidate-specific actionability or triggering. If the candidate affects 𝑅 𝑎𝑖 , reach-conditioned diagnostics such as 𝑞 𝑎𝑅 compare post-assignment selected groups. They remain descriptive unless the analysis defends assumptions about pairs that would reach under both candidates [31]. The target is the average contribution difference over all eligible planned pairs, reached or not, Δ𝑚 𝑎,𝑏 (𝑁, 𝑧) in Equation (11), at the declared cohort size, workload, assignment unit and allocation regime. Under valid random assignment, correct measurement and control of shared-load effects, the unconditional event-weighted contrast, including 𝐾 𝑎 (𝑁, 𝑧)/𝑁, estimates the policy effect on contribution. Its reach, readiness, value and cost components remain diagnostics rather than separately identified causal effects. Individual routing on constrained shared infrastructure estimates a mixed-load policy, which may differ from an all-sessions deployment in latency, queueing, and the tier and reserved-capacity terms of 𝐾 𝑎 (𝑁, 𝑧).

Specify trigger and concurrency controls together. Examine time, device, region, load and incident-window variation. A paired analysis needs both quantities on the same pair or another justified paired construction; shared aggregates alone do not establish pairing. Collect only fields required by the event dictionary and strata. Where consent, technical, contractual or legal constraints prevent row-level joins, use compatible modular estimates or bounds. Propagate uncertainty and name unobserved quantities and coverage changes instead of substituting server proxies for client truth. C. Synthetic verification and stress demonstration (SIM1) A stipulated worked example using the same coefficients appears in Supplements S1–S2; it is pedagogical and separate from this executed study. 1) Design and what it can test: SIM1 checks that the analyzer computes the declared quantities and demonstrates the failure modes those definitions distinguish. The generator assigns a scripted path. The ledger and its bounds use only the declared event classes and capture metadata. The export also contains playhead records for the position comparators, but the ledger ignores them when an event class is missing. A separately implemented geometric oracle and event extractor check the computations. We use a program-aligned synthetic 1× schedule in which position 100 corresponds to wall-clock deadline 100. EARLY and LATE initiate at positions 60 and 90. The clock-translation control adds 40 seconds to deadlines, session starts, actionability, session expiry and observation end. A late join at media position 𝑒 starts at wall-clock 𝑒 plus this translation, rather than at zero. A continuous-play crossing counts as reach, whereas a seek over the opportunity counts as a skip. Restarts remain within one pair. Both policies use catch-up on late-join scripts: entry at or beyond the policy’s trigger position, including equality, starts the job. Skip scripts disable catch-up. Each pair permits one job and no retriggering. Jobs continue after departure, and their trigger cost includes that work. Session expiry occurs at 180 seconds and observation ends at 400 seconds, with the same clock translation. Neither the expiry gate nor the actionability gate suppresses an event in the random grid. Constructed fixtures test those suppressive cases, retries, cancellation, deduplication, failure and prepared states. The export omits invalidations after the observation end, so the loss procedure cannot delete them. This horizon-specific convention determines which validity records can become unobserved. The five path types are monotone playback, restart, skip, late join and pause. Restarts and skips require a seekable timeline (on-demand or live timeshift); late joins are liveshaped. M1/M2 are stress geometries, not service profiles. The completion-versus-validity reversal also occurs in purely monotone M0, so it does not depend on seekability. M0 selects only monotone paths. M1 assigns probability 0.60 to monotone paths and 0.10 to each other type. M2 assigns 0.20 to every type. Latency follows a lognormal distribution with median

9

TABLE I Examples of documented interfaces and their measurement limits (interfaces checked on the access dates given in the references). This is a source-based interface map, not an instrumented vendor comparison.

Implementation surface

Documented request, identifier, or event

What still needs measurement

MediaTailor stitched workflow

Session initialization returns manifest and tracking URLs; client tracking retrieves avail/ad metadata [38]. Client-mode asset lists contain a TRACKING section rather than using GetTracking; guided prefetch uses session-tagged manifest refreshes [9], [38]. Registration yields a stream ID and metadata/verification URLs; the client requests a pod manifest and sends activity pings [15]. The implementation documents session IDs, an ad-decision endpoint, impression/quartile beacon collection, and session-status endpoints [16].

Player boundary crossing and the full declared playable-prefix criterion are not established by a successful metadata response. Heartbeat arrival is not proof that a particular viewer has reached the break or begun rendering.

MediaTailor guided workflow

Google server-guided DAI

DASH-IF Livesim2 SGAI

A pod response alone does not establish the common deadline state or contractual payment. A documented testbed interface is not a completed player test, production trace, or calibrated viewer-behavior model.

TABLE II Minimum observability by event class.

Event class

Server-side evidence

Client, joined, or external evidence

Scope, cue, and trigger

Schedule, cue receipt, manifest, session, ad decision service (ADS), resolver, retry, duplicate, and cancellation timestamps. Planned opportunity position and request proxies.

Player receipt, actionability, primary-trigger rule, catch-up policy, session entry, shared event identifier, and clock alignment. Frozen boundary event, actual presentation or continuation, playhead path, seeks, restarts, background state, skip, discontinuity, and session expiry. Buffer, asset-list or MPD resolution, media loading, decode and validity transitions, common-clock error bound, transition policy, player/device version, and end-to-end trace. Ad start, quartiles, completion, stall, skip, slate, content continuation, and deduplication rules. Contract definition, qualifying playback event, settlement cutoff, and cohort attribution. Supported measurement runtime, vendor evidence, coverage, and reconciliation; tracking and measurement remain distinct. Delivery horizon, causal driver, allocation convention, tier, and sensitivity to disputed denominators.

Opportunity reach and media path

Deadline and readiness

Configured target; ADS, validation, transcode, packaging, manifest, segment, expiry, eviction, and revocation timestamps.

Playback and fallback

Beacon receipt and server-side segment requests, with duplicate and cache caveats. Ad-server, order, invalid-traffic, makegood, credit, and finance records. Declared verification resources.

Billability and reconciliation Independent measurement

Cost

Cloud, content delivery network (CDN), platform, ADS, operations, and reserved-capacity records.

2, 12 or 35 seconds, log-scale standard deviation 0.7 and a cap at 120 seconds. State lifetime is 15 or 300 seconds. Validity follows the half-open rule in Section III-D. With 𝑈 [𝑙, ℎ] denoting a uniform draw, durations and positions are in seconds at unit playback rate. Monotone paths play 𝑈 [20, 140]. Restart paths play 𝑈 [105, 120], restart at zero, then play 𝑈 [0, 40]. Skip paths play 𝑈 [20, 50], seek to 𝑈 [105, 120], then play 𝑈 [0, 20]. Late joins enter at 𝑈 [65, 95] and play 𝑈 [0, 50]. Pause paths play 𝑈 [20, 50], pause for 𝑈 [10, 60], then play 𝑈 [20, 70]. Each then quits; successive draws use the recorded pseudorandom stream.

represented conditions and 180,000 policy realizations. Trigger placement, opportunity position and coefficients remain fixed across this verification grid.

The 18 base conditions each contain five seeded replicates of 500 independent scripts, giving 45,000 scripts. Within each script, the policies share the path and latency draw through common random numbers [32]. The clock copies give 36

The incomplete-capture views delete 1% or 5% of susceptible initiation, reach and validity records under nested masks. We draw masks separately for each policy and reuse each policy’s mask in its translated copy. Common random numbers

Terminal- and maximum-position comparators replace trigger and reach indicators while retaining actual deadline readiness. The completion comparator replaces only readiness, counting any attainment by the deadline. Every method uses hypothetical ready value 0.030, miss value 0.010, reached cost 0.003, trigger cost 0.002 and horizon cost per pair 0.002. We deliberately reuse Supplement S1’s coefficients. Miss value represents a stipulated later paid outcome, not slate revenue.

10

therefore couple paths and latencies, not capture loss. Capture metadata identify the affected event classes, and costs remain completely observed. The deleted records correspond to 𝑂 = 0 in Section III-A; missing indicators remain unknown. Conservative bounds enumerate admissible reach/readiness states while retaining the observed costs. These are finite-sample feasible intervals rather than statistical confidence intervals [33]. We use Wilson score intervals for proportions with positive denominators [39]. Contribution standard errors use paired script differences, and the artifact supplies all five replicate means. Neither clock copies nor alternative capture views enter the independent sample count. We report every condition without a significance filter. 2) Event misclassification and cancellation in the policy contrast: Both position shortcuts exactly reproduce trigger and reach in the monotone positive control. The nonmonotone mixtures expose two different errors: terminal position can forget an earlier crossing after a restart, while both terminal and maximum position can interpret a seek over an opportunity as a reach. In M1, terminal-position reach misclassification ranges from 18.52% to 20.40% across the six latency/lifetime settings; maximum-position misclassification ranges from 9.24% to 10.44%. The corresponding M2 ranges are 38.60–40.84% and 18.48–21.16%. For M2 with median latency 12 seconds and lifetime 15 seconds, terminal position misclassifies 994 of 2,500 reach indicators: 39.76%, with a 95% Wilson score interval (reported only for a positive denominator) of 37.86–41.69%. Nevertheless, its aggregate reach estimate is 41.64%, compared with 42.68% under the event ledger. False positives and false negatives partly offset one another. Maximum position misclassifies 484 indicators, or 19.36% (17.86–20.96%), and reports 62.04% reach. The EARLY nonreached-trigger rate ℎ 𝑎 changes from 722/1,433 = 0.50384 under event accounting to 722/949 = 0.76080 under maximum position because the nonreached denominator is distorted. Neither position shortcut reverses the EARLY-minus-LATE contribution contrast in any of the 18 base conditions. Maximum-position errors cancel exactly: on skip records both policies have 𝑇 = 𝑅 = 𝑄 = 0, while the shortcut assigns 𝑇 = 𝑅 = 1 and leaves 𝑄 = 0, so with policy-specific constant coefficients each false assignment adds 𝑣 miss − 𝑐𝑟 ,𝑎 − 𝑐 𝑡 ,𝑎 , and 𝑎 the shared coefficients make that error equal on both sides in each realization: common random numbers give both policies the same skip records. Under the same marginal path law without paired draws, equality would hold only in expectation. Unequal policy-specific terms break the cancellation unless their net errors coincide; changing ready-cell value alone leaves these zero-readiness records untouched. Terminal position changes magnitudes without reversing signs. A correct-looking policy difference is compatible with incorrect event levels. 3) Completion and deadline validity: Here the contribution contrast is Δ𝑚 EARLY,LATE from Equation (11), and readiness among reached pairs is 𝑞 𝑎𝑅 . We selected the short lifetime to expose expiry as a stress mechanism. For a single state with lifetime 𝜆 𝑎𝑖 > 0, attainment time 𝐶𝑎𝑖 = 𝜏𝑎𝑖 + 𝐿 𝑎𝑖

and no other prepared path, readiness is equivalent to 𝑊𝑎𝑖 − 𝜆 𝑎𝑖 < 𝐿 𝑎𝑖 ≤ 𝑊𝑎𝑖 . A completion proxy checks only the upper inequality. This equivalence applies to the declared onespell model; the general ledger admits multiple spells. For a reached monotone script, the EARLY and LATE windows are 40 and 10 seconds. At a 15-second lifetime, their validlatency bands are (25, 40] and [0, 10], respectively, while the completion proxy uses [0, 40] and [0, 10]. The possibility of a discrepancy is therefore built into the design, not discovered by the simulation; the simulation measures its consequences under the fixed design. These readiness bands alone do not determine the all-pair contribution contrast, which also includes triggerwithout-reach cost, path mixtures and finite-sample variation. The completion-by-deadline ablation reverses the comparison in three of the nine informative short-lifetime conditions. All have a 12-second median latency and 15-second lifetime, with one reversal at each path-mixture level. The clocktranslated copies reproduce these outcomes exactly; they do not constitute additional independent reversals. Table III gives the three contrasts and their Monte Carlo standard errors. The fixed lognormal explains the pattern analytically for monotone scripts. At a 2-second median almost every job completes within LATE’s 10-second window; completionbased readiness slightly favors EARLY, but in M0 that value gain is smaller than EARLY’s extra trigger cost. At 12 seconds EARLY completes with probability 0.9573 yet remains valid with probability only 0.1045, against 0.3973 for LATE under both definitions. At 35 seconds both definitions favor EARLY. Path-specific exposure and the shared coefficients determine the mixed-path contrasts. The grid demonstrates reversals without locating their boundaries. Table IV in the Appendix reports every base condition, and unrounded values accompany the code artifact. In the M2 condition, EARLY readiness among reached opportunities is 216/1,067 = 20.24% when current validity is required, but 848/1,067 = 79.48% when any completion before the deadline is counted. LATE readiness is 376/1,067 = 35.24% under both definitions in that condition. An earlier trigger can create more time to complete while also allowing more time for the resulting state to expire. A prospectively selected restart trace illustrates the distinction. The viewer crosses the opportunity at time 100, continues to media position 112.3198, restarts and ends at 28.2250. The EARLY job becomes valid at 63.7355 and expires at 78.7355. The LATE job becomes valid at 93.7355 and expires at 108.7355. At the frozen deadline of 100, both policies have reached the opportunity, but only LATE has a usable state. Their configured contributions are 0.003 and 0.023. Terminal position marks both pairs unreached, while completion-bydeadline marks both ready. Each shortcut erases the pair’s −0.020 contribution contrast, but for a different reason. The nine 300-second-lifetime conditions are analytic consistency controls: any attained state starts no earlier than the translated time origin, and the deadline is 100 seconds after that origin. A state completed by the deadline therefore cannot have expired there. Completion-proxy disagreement, and hence a reversal caused by that proxy, is impossible in these nine

11

TABLE III Completion versus continued validity: the three sign-reversal conditions.

Path mixture M0 M1 M2

Event-ledger contrast

MC standard error

Completion-proxy contrast

MC standard error

−0.0025296 −0.0023328 −0.0016744

0.0001555 0.0001647 0.0001673

+0.0031104 +0.0034592 +0.0033816

0.0001600 0.0001637 0.0001611

All entries are hypothetical currency units per planned pair; MC denotes Monte Carlo. Each row uses median latency 12 seconds and usable-state lifetime 15 seconds.

conditions. M0 playhead agreement is likewise guaranteed by its monotone construction. Scoring the same reached pairs at actual arrival rather than at the reference deadline changes EARLY readiness by −1.36 to +4.51 percentage points and LATE readiness by +0.18 to +5.54 across the 12 M1/M2 conditions. Every changed indicator lies on a pause path: all 11,314 nonpause reached scripts are invariant under both scoring times. The mechanism is forced by the design. Each pause precedes both trigger positions and exceeds LATE’s 10-second reference window, so LATE is never ready at the reference deadline on a pause path and recovers its full window at arrival, while EARLY gains readiness through later attainment or loses it through expiry. Rescoring the all-pair contributions at arrival with unchanged coefficients preserves all 12 signs (Supplement S9). Two readings must be excluded. This is not a policy ranking: arrival time can itself be candidate-affected, so it cannot replace the reference target, and the paired comparison is available here only because SIM1’s paths and arrivals are imposed and shared. Nor do the moving 300-second rows weaken their control role. Those controls compare completion against validity at one fixed deadline; work unfinished at that deadline can finish by a later arrival, which is a different comparison. Fig. 2 shows all 18 comparisons rather than selecting only reversals. 4) Incomplete telemetry and near-zero comparisons: Bounds use event-class records and capture metadata, not event occurrence inferred from retained costs, even where the cost schedule implies it. They are therefore valid conservative feasible intervals, not the sharp identification region given all retained variables. At 1% susceptible-event loss, unknown deadline readiness ranges from 0.32% to 1.84% across policy-condition combinations; at 5% loss it ranges from 1.92% to 8.00%. The ledger preserves this uncertainty rather than filling missing event classes from exported playhead records or generator internals. Conservative contribution-contrast bounds include zero in one of the 18 base conditions at 1% loss and six at 5% loss. For M2, median latency 12 seconds and lifetime 15 seconds, the 5% bounds are [−0.0025116, −0.0004684], preserving a negative contrast in this finite generated sample. For M2, median latency 2 seconds and lifetime 300 seconds, they are [−0.0013172, +0.0011412]. Even with complete capture, the latter condition is close to zero: its mean contrast is −0.0000720, Monte Carlo standard error 0.0000549, and one replicate mean is positive. It should not be presented as a re-

liable policy preference. No missing-as-zero point comparison reverses sign in this grid. Correlated record loss could move these bounds in either direction relative to the independent-loss realization reported here. The guaranteed ordering is narrower: for nested information sets under the same constraints, removing information cannot shrink the feasible set. These results are not a lower bound on field fragility, nor proof that 5% telemetry loss is universally fatal. 5) Verification and scope: Regeneration from the recorded seeds matched all 180,000 realizations and 540,000 observation views: complete capture, 1% loss and 5% loss for each realization. A separate event extractor and integer-milliunit checker reproduced primary quantities and paired summaries. Monotone agreement, occupied-cell identities, conservative bounds and clock-translation invariance passed. Latencies use correctly rounded exponential and logarithm functions, so regeneration gives identical records on Linux x86-64 and macOS arm64 (Supplement S10). The post-study arrival diagnostic reads retained records without new draws or generator execution; its separately implemented checker matched all 120,000 policy/clock classifications and all 72 aggregate rows. The geometric oracle uses the generated script and parameters, while the separate event extractor uses the exported events and an integer monetary calculation. Their trigger, reach and readiness computations are distinct from the analyzer’s; verification orchestration reuses the generator and masking utilities. The arrival checker shares recorded inputs and declared semantics with its scorer, but reconstructs crossings and validity separately and imports neither the scorer nor simulator. None independently validates the common path/latency sampler: a stable generation error can reproduce itself. D. Application-protocol specification The following six steps specify how an analyst would apply the method. 1. Define the comparison before assignment. Name the eligible cohort, actual routed implementation, trigger/load policy, opportunity, delivery horizon and settlement lag. For deployment selection, include the current implementation; SIM1 instead compares two hypothetical candidates. 2. Freeze the event dictionary. Fix crossing, trigger, readiness, deadline, expiry, fallback and reconciliation rules. Retain playback, billability and measurement as separate records.

12

Valid at deadline

Completed by deadline

M0 2 s 15 s M0 2 s 300 s M0 12 s 15 s M0 12 s 300 s M0 35 s 15 s M0 35 s 300 s M1 2 s 15 s M1 2 s 300 s M1 12 s 15 s M1 12 s 300 s M1 35 s 15 s M1 35 s 300 s M2 2 s 15 s M2 2 s 300 s M2 12 s 15 s M2 12 s 300 s M2 35 s 15 s M2 35 s 300 s −6

−4

−2

0

2

4

EARLY − LATE contribution (10⁻³ hypothetical units per planned pair) Fig. 2. EARLY-minus-LATE contribution in every base condition, using the event ledger and completion proxy. Labels identify mixture, median latency and state lifetime in seconds. The horizontal scale is in 10 −3 hypothetical currency units per planned pair. Horizontal error bars show one Monte Carlo standard error of each paired-script contrast, not a 95% confidence interval. Table III and the artifact give the underlying uncertainty and replicate results. Neither the translated copies nor masks add independent samples.

3. Join the evidence. Use per-pair keys and source/clock metadata. Report 𝑊𝑎𝑖 only for triggered pairs; omit emptycell conditionals and bound unobserved states rather than replacing them with zero. 4. Apply justified requirements. Protocol, privacy and device requirements may constrain admission. A readiness threshold requires an external operational or contractual justification. 5. Compute contribution. Attribute realized value and mutually exclusive costs with Equation (10); allocate reuse and horizon costs once. Keep unsettled exposure explicit. 6. Interpret with the design. Use joint uncertainty only where pairing or assignment supports it, and separate conditional diagnostics from causal effects. Report indeterminate when supported comparison intervals cross zero.

The event-dictionary template supplies editable fields for these steps and a constructed example row.

VI. Implications and limitations A. Scope This framework combines DAI-specific event construction, a common eligible-pair denominator and a candidate-invariant deadline with synthetic verification. Its probability, misclassification and accounting methods are established. SIM1 verifies computations and distinguishes failure modes under stipulated inputs, not their field frequencies. Supplements S1–S2 are pedagogical. The arrival diagnostic quantifies reference-time sensitivity on imposed paths, not policy effects on arrival, abandonment or revenue. SIM1 is an offline verifier with no media player, decoder, commercial ad server, queue or observed audience; its policies are not SSAI and SGAI. Path mixtures, latency laws, lifetimes and monetary coefficients are uncalibrated stress inputs. Common imposed paths hold abandonment and primary-playback effects fixed, while shared latency excludes load feedback. Trigger placement and opportunity position are fixed; mixtures are not profiles of one delivery mode. The grid excludes recurrent viewing, long-run retention, cross-session caps and bidding equilibrium. Clock copies test coordinate invariance,

13

not independent viewers or the live-edge-offset construction. Loss masks model neither correlated outages, invalid-traffic filtering nor uncertainty about captured event classes. Deployment requires a dated, rights-cleared client–server join, capture-coverage evidence, defensible cost allocation and uncertainty treatment. Client validity transitions cannot be replaced by nominal server timestamps; each path must demonstrably satisfy the same readiness criterion. Table I’s documentation does not establish a complete collection path. Collectability on target runtimes with bounded relative alignment error is a precondition, not a demonstrated capability. The join must reconcile player event definitions, timestamp origins and reporting cadences, with instrumentation and clock mapping supporting 𝜀 in Section III-D. Incomplete coverage leaves unresolved states or bounds. Pod value requires contractual reconciliation; disputed shared-cost allocation requires sensitivity analysis. The local extension is regime-specific at fixed workload. Hybrid deployments need path labels, and changing sources need dated checks. The next experiment is an instrumented server-guided testbed with a scripted client, such as DASH-IF Livesim2 [16]. It would collect the client transitions behind 𝑍 𝑎𝑖 (𝑡) and measure 𝜀 rather than stipulate it. This successor work would test collectability on one path, not across commercial CTV runtimes or as calibration of SIM1’s stress inputs; it has not been executed here. B. Implications for architecture analysis An earlier trigger can enlarge the readiness window and expose more nonreaching pairs to work. It can also allow a prepared state to expire before the deadline. The resulting contribution depends on actual events, value, cancellation/reuse and shared costs, not the architecture label. Guided insertion can change cacheability, scheduling and observability; stitched insertion can also preload. The documented guided-prefetch heartbeat is one implementation-specific example [9]. In synchronized live delivery, many sessions can reach the same ad break within seconds of one another. Concentrated requests can overload the ad-decision service; AWS documents retrieval-window and transactions-per-second shaping to limit timeouts and low fill [8]. SIM1’s shared latency draws exclude this feedback. Measured field effects enter candidate-specific ready latency and readiness, conditional pod values 𝑣 𝑎 and 𝑣 miss 𝑎 , variable costs, and reserved-capacity cost 𝐾 𝑎 (𝑁, 𝑧). Concurrency, timeout and load-control settings therefore belong in the candidate definition and comparison record. Request-time or late-binding assembly is another candidate. With fixed job-duration and lifetime laws, delaying resolution shortens 𝑊𝑎𝑖 , reducing expiry exposure while increasing deadline-miss exposure; synchronized request-time work can also concentrate load. Which effect dominates requires measurement and the accounting in Equation (10), not an inference from the mechanism’s name. Common reach and equal coefficients are assumptions to test. When ad work can affect primary playback, retain candidate-specific reach. Keep pod-horizon contribution separate from long-run engagement or retention. The framework

identifies which records a comparison needs; it does not supply records a deployment cannot observe. C. Joint uncertainty and decisions A point estimate should not decide an architecture when the difference it reports is smaller than its own uncertainty. Use the common-cohort contribution contrast defined in Equation (11). Estimate or bound Δ𝑚 𝑎,𝑏 (𝑁, 𝑧) jointly, retaining covariance only where the assignment and the data construction actually supply it. Genuine pairing can come from randomized paired observations, or from a non-serving shadow calculation on the same opportunity. Alternatives observed only on separate pairs are not paired. For deterministic sensitivity analysis, let 𝜃 be the vector of uncertain inputs, let U be a specified nonempty set of jointly admissible values, and write Δ𝑚 𝑎,𝑏 (𝑁, 𝑧; 𝜃) for the difference produced by one allowed input vector. Each evaluated input must satisfy the accounting assumptions and finite-moment conditions. Compactness is not required for the infimum/supremum definition; a bounded, explicitly specified scenario set makes numerical evaluation and its limitations transparent. Holding 𝑁 and 𝑧 fixed, candidate 𝑎 is robustly dominant only if inf 𝜃 ∈ U Δ𝑚 𝑎,𝑏 (𝑁, 𝑧; 𝜃) > 0, and candidate 𝑏 is robustly dominant only if sup 𝜃 ∈ U Δ𝑚 𝑎,𝑏 (𝑁, 𝑧; 𝜃) < 0. For statistical inference, construct a confidence interval directly for Δ𝑚 𝑎,𝑏 (𝑁, 𝑧), or map a simultaneous joint confidence region through Equation (11). Two marginal intervals with the same individual coverage do not automatically deliver that coverage jointly. A positive lower confidence limit supports a positive difference at the stated confidence level without establishing deterministic robust dominance. Under either analysis, a range containing zero is indeterminate. The same discipline governs a contract-supplied or otherwise externally justified readiness threshold. A point estimate above the threshold is not sufficient when the relevant onesided confidence bound falls below it. Incident clustering and device heterogeneity should be preserved rather than averaged away. VII. Conclusion DAI comparisons need separate trigger, reach, state-validity and commercial records on one eligible-pair denominator. SIM1 shows substantial playhead misclassification without a contrast reversal in this grid, while substituting completion for deadline validity reverses the comparison in three of nine informative conditions, all at one latency setting; rescoring readiness at actual arrival changes no contribution sign. Nine long-lifetime conditions provide analytic controls for the completion comparison. The event ledger, accounting and inspectable artifact make these consequences explicit, while the application specification identifies the joins, cost records and uncertainty treatment needed for deployment.

14

Appendix Complete synthetic condition results TABLE IV EARLY-minus-LATE contribution across all 18 base conditions.

Mix

Median (s)

Lifetime (s)

Ledger

Terminal

Maximum

Completion

M0 M0 M0 M0 M0 M0 M1 M1 M1 M1 M1 M1 M2 M2 M2 M2 M2 M2

2 2 12 12 35 35 2 2 12 12 35 35 2 2 12 12 35 35

15 300 15 300 15 300 15 300 15 300 15 300 15 300 15 300 15 300

−0.0070848 −0.0004584 −0.0025296 +0.0032784 +0.0009408 +0.0031048 −0.0066256 −0.0002400 −0.0023328 +0.0029048 +0.0010704 +0.0031776 −0.0067976 −0.0000720 −0.0016744 +0.0033720 +0.0012344 +0.0029248

−0.0070848 −0.0004584 −0.0025296 +0.0032784 +0.0009408 +0.0031048 −0.0047216 −0.0002720 −0.0017088 +0.0018568 +0.0005824 +0.0021936 −0.0028936 −0.0001040 −0.0004344 +0.0011240 +0.0003624 +0.0007968

−0.0070848 −0.0004584 −0.0025296 +0.0032784 +0.0009408 +0.0031048 −0.0066256 −0.0002400 −0.0023328 +0.0029048 +0.0010704 +0.0031776 −0.0067976 −0.0000720 −0.0016744 +0.0033720 +0.0012344 +0.0029248

−0.0004208 −0.0004584 +0.0031104 +0.0032784 +0.0029888 +0.0031048 −0.0002176 −0.0002400 +0.0034592 +0.0029048 +0.0028944 +0.0031776 −0.0001256 −0.0000720 +0.0033816 +0.0033720 +0.0030024 +0.0029248

Seven decimal places are shown for comparison with the artifact outputs rounded to the same precision; the artifact retains additional digits. This is display precision: Monte Carlo standard errors for the contrasts in this table range from 0.0000252 to 0.0001868 (Fig. 2, Table III and the artifact).

Acknowledgment The authors used ChatGPT (OpenAI) and Claude (Anthropic) for drafting and LaTeX preparation throughout the article and supplement, and to check derivations in Sections III– IV and Supplement S1–S5, S7 and S9. ChatGPT was also used for figure, simulation and verification code. The mathematical development, event definitions, analyses and conclusions are the authors’ own. The authors reviewed all AI-assisted text and code and take full responsibility for the work. Code and data availability. Repository: https://github.com /methodtrace/2026-dai-timing. Release tag: v1.0.0. Zenodo version DOI: 10.5281/zenodo.22774143. Code: Apache-2.0; data, tables, fixtures and the event-dictionary template: CC BY 4.0. The artifact includes source, configuration, 90 seeds and 90 event shards, masks, all tables, 19 fixtures, separate verifiers, file/content hashes, a transport-equivalence audit, the correctly rounded math module with its tests, and an execution history. The arrival diagnostic adds two scoring/checking scripts, its frozen protocol and input manifest, 72 condition rows, 360 replicate rows, 60,000 pair scores, selected event traces and eleven separate known-answer checks; its 60 input shards reuse the same stored data. A separate exact revaluation check preserves both scoring conventions. Section V-C5 reports the verification. References [1] A. Giladi, N. Levy, and T. Griebel, “Server-guided ad insertion in 6th edition of MPEG DASH,” in Proceedings of the 5th Mile-High Video Conference, pp. 59–65, 2026, doi: 10.1145/3789239.3793274. [2] R. Pantos, “HTTP Live Streaming 2nd edition,” Internet-Draft draftpantos-hls-rfc8216bis-22, May 2026, work in progress. Accessed: Sep. 8, 2026. [Online]. Available: https://datatracker.ietf.org/doc/html/draft-pan tos-hls-rfc8216bis-22

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