Temporary Transitional Aid Guarantee

The hold-harmless. When the Fair School Funding Plan computes less for a district than a prior-year baseline, the guarantee tops the district up to that baseline instead. verified Line [I] of the School Finance Payment Report, R.C. 3317.019.

The mechanism, and the clawback inside it # ↑ Contents

The department’s formula is

[I] = max( [H2] Funding Base  -  [I1] Open Enrolment Adjustment  -  [H] Foundation Funding,
           0 )

[H2] is the guarantee funding base — the FY2020 anchor, published per district and never re-based. [I1] is the open enrollment clawback, a reduction for a guaranteed district that has shed entering open enrollment beyond a threshold. Each is a mechanism in its own right that this line composes rather than contains. verified

Because the guarantee is a difference rather than a rate, any increase in computed funding reduces it one-for-one until it reaches zero and the district returns to formula.

It is a nominal floor, so it erodes by construction # ↑ Contents

A guarantee written in dollars and anchored to a fixed historical year cannot hold its value. A district it protects loses ground every year through no decision by anyone, and nothing in the formula records that it happened. “Held at FY2020” is a statement about nominal dollars only, and across FY2020-FY2025 prices rose 25.1%. verified

So “temporary transitional” describes the intent and not the mechanism. A guarantee anchored to a fixed historical year does not expire; it is escaped, district by district, only as the formula grows past a level set in 2020. On the FY2027 evidence, for half of these districts that is not close. inference

Whether the statute or subsequent budget acts contemplate re-basing is answered in the findings: the statute does not, and the acts did — every biennium for a decade, and not once since FY2022.

What this repository computed # ↑ Contents

Not what Ohio publishes. Figures derived here from committed fixtures, each one citing the test that reproduces it.

The limit case, and the panel cannot see it # ↑ Contents

North Bass Local receives $19,672.64, and every cent of it is this guarantee. Zero base cost, zero categoricals, zero transportation, zero of everything the formula computes — a Lake Erie island district in Ottawa County that the department does not cost at all, funded because it was funded in FY2020. verified the department’s FY27 calculator, Summary_SFPR

It is what the mechanism looks like with the formula subtracted out. Every other guaranteed district is a comparison between two numbers; this one has only the historical number, and the hold-harmless is not topping up its aid but constituting it. Whatever the guarantee is for, North Bass is the case where nothing else is doing any work.

Its neighbor Middle Bass Local is the other half of the pair and receives nothing at all, which is the same fact from the other side: two districts the formula has no answer for, one carrying a 2020 figure forward indefinitely and one carrying zero.

Neither is in the committed panel, because the extract is keyed on the base cost sheet and the department computes no base cost for either. So the $878,954,627 above is the 609-district total and the department’s is $878,974,300 — the whole difference is North Bass. verified the_population_the_panel_speaks_for.rs Every statistical claim in this node is over the 609, and the district that would most sharply illustrate it is the one excluded.

Does it buy anything? No — carefully # ↑ Contents

Joining the funding model to the 2024-25 report card across the 606 districts every panel covers, guarantee status looks like it predicts achievement: districts on the guarantee score +0.187 against the Performance Index, median 89.85 against 85.6. Holding economic disadvantage constant the association is +0.035. verified see crates/project/tests/guarantee_and_outcomes.rs

The whole of it is poverty. Guaranteed districts are less poor — median economically disadvantaged share 42.8% against 51.9% — which is the same wealth gradient the description establishes, arriving on the outcome side as an achievement effect that is not there. The result holds for Ohio’s growth measure as well, and for a continuous reading of guarantee dependence rather than the binary one: every combination lands under 0.06 once poverty is held. verified

What it does not say. Not that the guarantee is useless, and not that its money does nothing. Districts are not assigned to the guarantee at random — they are on it because their FY2020 receipt exceeded what the formula now computes, which is a function of the same wealth and enrollment history that predicts their outcomes. Nothing here identifies an effect. What it rules out is the argument from the raw gap: guaranteed districts perform better, so the guarantee is working. That gap is a composition artifact. inference

A caution the same join makes checkable. State aid per pupil is not a spending measure. It is 38% of the median district’s operating budget and substitutes for local revenue rather than adding to it: holding poverty constant, state aid per pupil and total operating spending per pupil track each other at +0.035. So the +0.146 recorded for spending against growth and the -0.247 for aid against growth are not in tension — they are different variables, and reading one as the other is a mistake this node exists to make hard. verified

Running the formula without it # ↑ Contents

The department’s model publishes core foundation funding [H] separately from the guarantee [I], so the two can be compared directly rather than inferred. verified

On correlation, the guarantee’s effect is modest. State aid per pupil against assessed valuation per pupil correlates at −0.662 on formula alone and −0.605 as realized — the guarantee gives up about 9% of the formula’s equalization.

On levels it is not modest at all. Median guarantee uplift per pupil, by valuation quartile:

Q1 poorest        $0
Q2                $0
Q3              $685
Q4 wealthiest  $1,154

The typical district in the poorer half of Ohio receives nothing from the guarantee, because most such districts are not on it. The typical district in the wealthiest quartile receives $1,154 per pupil — which more than doubles the $1,113 the formula would give it. verified Comparing quartile medians instead of per-district differences gives $21 against $1,658; both framings are recorded because either quoted alone invites the wrong reading.

The correlation understates this because correlation measures how tightly aid tracks wealth, not how much money moves. A component can leave the ranking almost intact while transferring a great deal. inference

A methodological point this forced # ↑ Contents

For spending, less dispersion is more equitable. For state aid the sign flips: aid is compensatory, so a wide spread means strong targeting and a narrow one means weak targeting. Realized aid is measurably more equal than formula aid — coefficient of variation 0.563 against 0.695, restricted range ratio 9.66 against 12.27. verified recomputed over the 609 districts of the committed feed with dispersion::Dispersion::of, which is R type 7 on q * (n - 1) and population variance; the figures this replaces (0.544, 0.677, 9.6, 12.3) do not reproduce under any convention in the crate That equality is the guarantee flattening the compensation. Reporting it as an improvement, which the habit from spending analysis would invite, gets the direction exactly backwards. See equity.

How far the formula falls below the level it holds districts at # ↑ Contents

Because guarantee = baseline − formula, each guaranteed district’s FY2020 baseline can be recovered as core foundation funding plus its guarantee. verified

FY2027 formula as a share of the district's FY2020 baseline
  median 67.8%      p25 42.3%      102 districts below 50%
  aggregate: $2.15B formula against a $3.03B baseline (71.0%)

by valuation quartile, guaranteed districts only
  Q1 poorest      91.9%
  Q2              74.1%
  Q3              53.8%
  Q4 wealthiest   43.4%

The gradient closes the causal chain. The local capacity measure reduces aid to property-wealthy districts by design. Their FY2020 baseline was far higher. The guarantee holds them there — and for the wealthiest quartile the plan produces less than half of it. inference

This is a share of the statutory base, which is narrower than a district’s booked state aid. Read the third correction below before reading it as a shortfall in receipts.

The real erosion, measured within one source # ↑ Contents

verified Five-year forecast filings — the department's republished submissions both endpoints, CPI-U all items — the Bureau of Labor Statistics flat file deflator The clearest statement available, because it is measured entirely within one source on one definition and touches the calculator nowhere. Real change in booked state aid, FY2020 to FY2025, deflated by CPI-U June:

median guaranteed district   -18.2%      270 of 294 lost ground
median formula district       -3.3%      184 of 314 lost ground

Also computed in crates/project/tests/finances_and_the_guarantee.rs.

The incidence of removal, as a distribution rather than a total # ↑ Contents

verified crates/scenario-delta Crossed against wealth, per pupil:

valuation quintile    per-pupil loss on removal
Q1 (poorest)                    -136.84
Q2                              -270.84
Q3                              -490.24
Q4                            -1,025.91
Q5 (wealthiest)                 -937.40

and against the state share of base cost, where it is starker still: the 138 districts at the minimum state share lose $1,003.87 per pupil, against $160.27 for the quintile the formula supports most. The guarantee pays most, per pupil, to the districts the formula says need it least. verified

That table is foundation aid, and it is the guarantee’s own incidence rather than a retirement’s. crates/scenario-delta builds it from [H] + [I], which excludes [K] — so it states what the guarantee pays each quintile and not what any quintile would actually lose if the guarantee stopped being paid. On total state support the answer is different in shape as well as in size: the formula transition supplement makes 127 of the 294 whole, and the cut that survives runs further up the wealth distribution than this table does. See Phasing Out the Temporary Transitional Aid Guarantee, whose largest per-pupil loser moved from the poorest district in the state to one of the wealthiest once [K] was modeled. verified

Two things this does not say. The gradient is not monotone — Q4 loses more per pupil than Q5, because the wealthiest districts are small and already floored, so less distance separates their formula amount from their baseline. Any claim that the guarantee’s benefit rises without limit in wealth breaks here. verified And the poorest quintile does receive something: 15 of its 121 districts are on the guarantee. This is a strong tendency, not a rule. verified

A perturbation is inert for a guaranteed district, exactly # ↑ Contents

Not approximately: a district paid by the guarantee under two policies that differ anywhere except in the guarantee itself moves by 0.000000, because the guarantee is a max against a fixed baseline and the same number leaves the same branch twice. For a base cost increase, 256 of 609 districts are in that condition. This is why the guarantee is the only lever with a distinct incidence: every other one is blocked for the same half of the state. verified

It also absorbs reform. Because a guaranteed district’s top-up shrinks by exactly what its formula amount gains, roughly half of any base cost increase is absorbed rather than delivered — the finding that halved the input-refresh scenario. verified

And the model cannot see the floor under a formula district # ↑ Contents

The baseline is recoverable only for districts currently held, because being held is the sole thing that discloses it. The 315 districts the formula pays each have an FY2020 baseline sitting below their current amount, and no committed source carries it. A simulated cut therefore lets them fall past a floor that would really catch them: at a 5% phase-in the model has them lose 95% of their aid and puts nobody on the guarantee. Those districts hold 46% of Ohio’s students, so any scenario that reduces formula aid overstates its savings by an amount that cannot be bounded here. verified the model’s behavior open the size of the real-world correction, which needs FY2020 per-district payments for districts not on the guarantee

The consequence for equity is direct # ↑ Contents

State aid tracks poverty (+0.632) more tightly than property wealth (−0.548), which reads as the formula’s targeting working. The guarantee is part of why the wealth relationship is not stronger: it props up precisely the districts the local capacity measure would fund least. inference see equity

It is called temporary and transitional. In FY2027 — the terminal year of the six-year phase-in, the year the formula reaches 100% of its computed amount — 294 of 609 districts, 48.3%, holding 54.1% of base cost enrolled ADM, are funded by it rather than by the formula. verified Across those 609 it totals $878,954,627, against the department’s $878,974,300 over the 611 districts it pays. The $19,672.64 between them is one district, North Bass Local, and it is in the findings below. The median guaranteed district sits more than three times above what a meaningful base cost increase would give it, so this is not a rounding buffer catching a few edge cases; for half the state it is the operative funding mechanism.

Who is on it # ↑ Contents

Both candidate explanations — property wealth and enrollment decline — point the same way, and one dominates clearly. Property wealth predicts the guarantee roughly three times more strongly than enrollment decline does, and by quartile of valuation per pupil the rate runs 13%, 37%, 68%, 76%: monotonic, with three-quarters of Ohio’s wealthiest districts held above what the formula computes for them. verified The mechanism is not mysterious. Under the plan’s own logic a property-wealthy district has high local capacity, so its computed state share is small — that is the formula working as designed. The guarantee then overrides that result and pays the district what it used to get. inference

It grows as the phase-in completes, which is the opposite of what the name implies # ↑ Contents

OCG White Paper 015 reports the FY2024 position — the phased-in amount about $741.8 million below what the formula calculated, with 180 districts drawing about $181.2 million in guarantee support — and treats the phase-in gap as the “unfunded statutory formula” standard for defunding. Set the FY2027 figures beside it:

FY2024    180 districts     $181.2 million     [verified] (as published, secondary)
FY2027    294 districts     $879.0 million     [verified] (department FY2027 model)

Roughly fivefold in dollars, arriving at the terminal year, the year the formula reaches 100%. The phase-in gap closes and the guarantee does not; the mechanism that carries districts to full funding is the hold-harmless rather than the formula. Any account that treats FY2027 as the year the question resolves has the direction backwards. inference

Re-basing, which the statute leaves out and the budget acts kept doing # ↑ Contents

The statute does not contemplate re-basing; it contemplates expiring. R.C. 3317.019 is scoped to “fiscal years 2026 and 2027” in every division that pays anything, names no later year, and carries no provision for moving the base — the one adjustment it does allow runs the other way, for a district joining a newly established joint vocational district. verified crates/project What happens after FY2027 is not a re-basing rule but the absence of a section, which is the handover the whole plan is written with.

Subsequent budget acts contemplated it constantly, and then stopped. Four consecutive acts each re-read the anchor onto the year before their own biennium — H.B. 59 onto FY2013, H.B. 64 onto FY2015, H.B. 49 onto FY2017, H.B. 166 onto FY2019 — and the three plan-era budgets have re-read it none. verified the anchor chain in Bridge Formula So re-basing is what Ohio did every two years for a decade and has not done since FY2022. That is a fact about practice rather than about statutory design, and it is the answer to the question this node asked. inference

What it pays at the margin, which is nothing # ↑ Contents

Every figure above is a level. Ask instead what the guarantee pays a district to do — one pupil fewer, everything else held, priced on total state support — and the instrument turns out to charge nothing for the marginal pupil. verified crates/project

[K] holds the district    144 districts    23.5% of ADM    nothing at all
the guarantee alone       167 districts    31.7% of ADM    transportation only
on formula                298 districts    44.8% of ADM    the whole entitlement

294 districts lose no foundation aid whatever when a pupil leaves. verified crates/project 144 lose nothing at all, because [K] holds them at a total rather than at a level, so a dollar the departing pupil takes off foundation funding or off transportation is a dollar the supplement puts back. verified crates/project For the 167 the guarantee alone holds, the marginal pupil costs exactly the district’s transportation aid and nothing else — a median $503.40 against $8,516 for a district on formula — because [J] sits outside the measure [I] holds level, and the residual across all 167 is four billionths of a dollar rather than an approximation. verified crates/project

Statewide the floors absorb two fifths of the response. Take one per cent off every district’s roll at once — 14,019 pupils — and the state gives up $3,260.09 a pupil against an average of $5,756.93, so 56.63% of Ohio’s per-pupil funding is what actually follows a pupil. verified crates/project

A zero margin is not a payment for shrinking, and the difference is the finding # ↑ Contents

enrollment_decline measured the median member of its cluster drawing $464.06 a pupil for enrollment lost. That is an average, and the margin beside it is zero; the two are consistent precisely because the guarantee pays a level, and a district cannot act on an average. verified crates/project

So the honest reading is not that Ohio pays a district to shed children. It is that a held district is paid nothing for the marginal child, so every additional one it teaches is a pure cost against a cheque that does not move. Which of the two readings is the incentive depends on whether anyone can act on it, and nobody chooses how many children live in their district. inference

The boundary is a kink, not a cliff — and in children it is thin # ↑ Contents

A district one dollar below its base is held and one dollar above is not, and that boundary pays nothing to stand on either side of: realized aid is max([H], [H2] - [I1]), which is continuous. No district is better off for sitting under its base. What changes at the edge is the slope, from one to zero. verified crates/project

Measured from above, in the unit a district actually loses, the edge is sparsely populated: the median formula district is 206.6 pupils clear of its own floor and six of 315 are inside ten, Waynesfield-Goshen Local closest at 2.3 pupils. Measured in time at each district’s own FY2024-FY2026 trend it is much nearer — of 238 formula districts with a falling roll, 17 of them reach their floor within a year. verified crates/project That is the same clock the cluster crossing this instrument was measured on, seen from outside rather than inside it.

The obvious alternative to a receding anchor is a larger instrument, not a cheaper one # ↑ Contents

The description above says a nominal floor anchored to a fixed historical year erodes by construction, and a rolling anchor — prior year, or a multi-year average — is what that observation points at. Priced, it goes the other way.

realized = max([H], anchor), so last year’s realized aid is at or above last year’s computed amount by construction. An anchor set to it is a ratchet: aid can never fall, for any district, for any reason, in any year. The enacted anchor catches districts that have fallen below a level chosen once in FY2020; a rolling one catches every district whose computed amount falls at all, and with the cost side frozen at FY2022 and FY2024 averages against a rolling three-tax-year capacity window, most of Ohio’s do most years. Replacing the floor in the FY2027 model and changing nothing else, a prior-year anchor writes $1,025,415,233.33 to 455 districts against $878,954,627 to 294. verified crates/project

And for the 89 districts the guarantee is still recruiting it is the same number to the cent. Every member of enrollment_decline’s cluster is on the floor in FY2027, so a ratchet started from FY2027 realized aid is the FY2020 base: walked to FY2032 the two rules give that cluster $144,560,557.55, 89 districts held, and the same total state support. The crossing that motivated the question does not translate, because a max only ever moves an anchor up. verified crates/project

What does follow a district down is a cap on the fall # ↑ Contents

realized = max([H], factor x last year's realized) — “no district’s foundation aid falls more than a stated percentage in a year” — is date-free, reaches a district that crosses in any year, and extinguishes itself for one whose enrollment settles. Walked to FY2032:

anchor                          [I] statewide    districts    [I] to the 89    of the 89
the FY2020 base, as enacted     $932,058,576          312       $144,560,558           89
the district's own prior year   $989,357,267          565       $144,560,558           89
annual fall capped at 2%        $665,088,657          246        $45,151,279           47

It is a different population rather than a weaker floor: a cap is anchored on current aid, so a district well above its FY2020 base gets a higher floor from it than the statute gives. At FY2036 a 1% cap holds more districts than the enacted rule while writing $220m less guarantee, and statewide total state support is higher under it. verified crates/project

And the backstop absorbs the anchor’s shape, as it absorbed every adjustment # ↑ Contents

[K] is this cluster’s real payer, which the_adjustment_the_floors_absorb.rs established for four candidate supplements. The same is true of the anchor. Cutting the cluster’s guarantee from $144.6m to $45.2m with a 2% cap takes $3,949,559 of total state support: 96.0% of it comes straight back on [K]. verified crates/project

So the quantity a re-shaped anchor was supposed to move does not move. The share of what this cluster receives that arrives through a hold-harmless is 26.94% at FY2036 as enacted and 26.81% under the 2% cap — and under 1% and 5% caps as well. Changing the anchor changes which hold-harmless pays, not whether one does, and it moves the money off R.C. 3317.019 and onto Section 265.225, which is uncodified and renewed a biennium at a time. That is an argument against re-shaping the anchor on its own, and not an argument for any particular shape. verified crates/project

Who every alternative reaches outside the cluster, and why they all run the same way # ↑ Contents

Outside the 89, the enacted floor holds the wealthy and not the poor. Cut the 520 other districts into fifths by assessed valuation per pupil and the FY2027 guarantee holds 0 of the 103 least wealthy and 77 of the 105 wealthiest — the 13%, 37%, 68%, 76% quartile gradient above with the enrollment cluster taken out. Every rule below is a change from that one, so a rule that narrows the floor takes money from the wealthy, and a rule that reaches districts the floor does not — the ones the formula pays — gives money to the poor. Measured on total state support throughout, because [K] stands behind the line every rule moves. verified crates/project

A ratchet adds money, and adds it at the bottom. A prior-year anchor catches every district whose computed amount falls, and the districts with the most formula aid a pupil to lose are the poor ones. Walked to FY2032 it adds $138.58 a pupil to the least-wealthy fifth and $3.75 to the wealthiest, monotone across the five; by economically disadvantaged share, $4.06 to the least-poor fifth and $108.08 to the poorest. 253 districts outside the cluster gain and none loses, and the ten largest gainers hold a quarter of it. verified crates/project

A cap takes money, and takes it at the top — and the backstop decides how much lands. A cap on the annual fall lets a held district’s floor decline, so it is a cut to the districts the enacted anchor holds. At FY2032 a 2% cap takes nothing from the least-wealthy fifth, whose two held districts are fully backstopped, and $83.51 a pupil from the fourth; by disadvantaged share it takes $101.52 from the second-least-poor fifth and $5.33 from the poorest. 145 districts lose and one gains. verified crates/project

The poorest fifth’s small loss is [K] rather than the cap: its guarantee falls $24.7m and its total state support $1.6m, so 93.4% of the cut comes straight back, against 35.9% in the least-poor fifth. The absorption the cap showed for the cluster has a gradient of its own outside it, and the gradient runs the same way as everything else here — the backstop catches the poor and not the wealthy, which is what Phasing Out the Temporary Transitional Aid Guarantee found about retiring the guarantee outright. On [I] the cap’s cut to the poorest fifth reads $24.7m; on the measure a district is paid on, it is $1.6m. verified crates/project

The 1% cap’s aggregate increase is a transfer, broad on both sides. At FY2036 undamped the cap raises statewide total state support $7.7m while holding more districts for less guarantee. Outside the cluster that is 112 districts gaining $53,969,540 and 119 losing $43,682,097 — a net a fifth the size of either side. The poor gain and the wealthy lose: $141.73 a pupil more in the least-wealthy fifth against $55.40 less in the wealthiest, $123.40 more in the poorest against $68.15 less in the least poor. The ten largest gainers hold 36.5% of the gain, the largest is Maple Heights City at $3.8m, and 77 of the 112 are in the two least-wealthy fifths: a general shift down the wealth distribution, not a few large districts. verified crates/project

A cap can give because by FY2036 a formula district’s aid has fallen far enough that 99% of last year’s stands above its FY2020 base, and the districts in that condition are the ones the formula has been paying. At 2% the same rule is a net cut again, $5.0m gained against $54.6m lost, because fewer districts fall faster than 2% a year. The rate decides the sign and never the direction. verified crates/project

And the four supplements agree. The same cut of what the_adjustment_the_floors_absorb.rs priced, at the modeled year, outside the cluster, per pupil:

shape                              least wealthy   wealthiest   poorest   least poor
rolling pupil count                      $166.55       $11.43   $109.79       $11.68
mirror [M], inside [H]                   $114.22       $20.42    $78.69       $25.40
mirror [M], beside [L], [M], [O]         $114.22       $96.88    $99.10       $91.36
dated phase-down                           $0.00        $0.00     $0.00        $0.00

The rolling count and the mirror inside [H] are progressive for the reason the ratchet is: they are written inside the floors, and the floors absorb them where the wealthy are held. The mirror written beside every floor is the one shape with no gradient — $114 against $97 by wealth, hump-shaped by poverty — because it pays on the roll of any district past its threshold whatever its wealth, and no floor reads the line. The dated phase-down reaches nobody outside the cluster by construction. verified crates/project

So the shapes do not disagree about who they reach. Every one of the seven is progressive against the enacted anchor or flat, and none is regressive. The choice between them is not distributional in direction; it is the sign and the size — add money at the bottom, with a ratchet, the rolling count or either mirror, or take it at the top, with a cap — and at a long enough horizon the 1% cap does both. That is the fiscal framing the two pricings used, and it survives the cut. inference

The two axes disagree for three of the seven, and the guarantee column shows none of it # ↑ Contents

Every pricing above reports one column at a time, and the two columns are not versions of each other. A rule is written into R.C. 3317.019 as a guarantee and paid out of total state support, and [K] stands between them: a cut to the guarantee is a cut to the money only so far as the backstop does not put it back, and a payment written outside every floor is money with no guarantee behind it at all. Priced on both axes at once, outside the cluster and against the enacted anchor, three of the seven rules write less guarantee and pay more money — and a fourth is the dearest of them all while moving the guarantee column by nothing.

The three that disagree. A 1% cap at FY2036 undamped takes $107,611,728.73 off the guarantee written and adds $10,287,443.63 to total state support. A rolling pupil count takes $14,740,751.65 off the guarantee and adds $66,202,848.00 — four and a half times the cut, in the opposite direction. [M] mirrored inside [H] takes $70,536,484.89 and adds $62,189,117.28. Nothing here runs the other way: no rule priced writes more guarantee while paying less. verified crates/project

The two that agree, and the one the column cannot see. A prior-year ratchet walked to FY2032 adds $57,298,691.08 of guarantee and $52,907,613.45 of support — the one rule whose two axes agree in size as well as in sign, because a ratchet writes its money into the floor and the floor is what gets paid. A 2% cap takes $167,560,640.40 off the guarantee and $59,356,083.89 off the support, near a third of what it cut, which is the absorption already measured above seen whole. [M] mirrored beside [L], [M] and [O] moves the guarantee column by $0.00 and pays $135,300,935.56, more than any other rule here — it is paid outside every floor, so there is no guarantee for it to be written into. A dated phase-down of the FY2027 list moves $0.00 of guarantee outside the cluster and $0.00 of total state support, because outside the cluster there is nobody on the list. verified crates/project

So a reader given the guarantee column alone would rank the seven nearly backwards. The dearest rule reads as no rule at all. Three of the four that cost money read as cuts. And the only rule whose guarantee figure is a fair proxy for what it costs is the ratchet, which is the rule that needs the proxy least. The plane below is that table drawn once: the vertical rule is no change to the guarantee written, the horizontal is no change to the support paid, and the quadrant above and to the left — writes less, pays more — holds three of the seven points. inference

And the per-pupil incidence, by both axes, one panel a rule. The two sets of panels below are thirty-five numbers each: five fifths of the 520 districts outside the cluster, for every one of the seven rules. The scale is shared across all seven panels of a set and deliberately not across the two sets, because the two axes are two orders of magnitude apart — the ratchet’s first fifth is $138.58 a pupil by wealth and $4.06 by disadvantaged share — and one scale wide enough for the first flattens the second into a row of hairlines. Read inside each set, the shapes are the gradients already stated: monotone down the wealth fifths for the ratchet and the rolling count, and hump-shaped by poverty for the mirror written beside the floors, whose $144.27 a pupil in the middle poverty fifth is the most any rule here pays on that axis. verified crates/project

The partition that did not need fitting # ↑ Contents

Every partition above starts from the guarantee, and #396 asked whether a multivariate partition of all 609 districts — clusters of similarly profiled districts on numerous variables, unconditioned on membership — would find a blind spot none of them can see. Both halves of that were measured in crates/project/src/lost_pupils.rs, and the decision that followed is “The partition nobody fitted”.

The identity does not need the guarantee. [H2] − [I1] is published for 608 districts, so the FY2020 base over the FY2027 formula is a multiple for every one of them, below one where the formula pays more, and its logarithm still splits into an enrollment term and a per-pupil term that reconstruct it exactly. It reaches 607 of 609 — Richmond Heights publishes no base, and Buckeye Local is the one district the chained enrollment index does not reach — and agrees with the 187’s split on all 293 guaranteed districts both reach. Of the 314 formula districts, 252 have fewer pupils than in FY2020, and the formula pays more per pupil than the FY2020 regime did for 310; the four exceptions all grew. Nobody is off the floor because the formula pays less per child. They are off it because the per-pupil raise outran the pupils they lost. verified crates/project

The two terms are a plane, and the empty part of it is the finding. Drawn one district to a mark — the enrollment term across, the per-pupil term up — the 607 fall into three states and not two. Below the anti-diagonal where the multiple is one, the floor is at or under the formula and there is no majority to take: that is every one of the 314 formula districts and no others, because a multiple at or below one is the same predicate the guarantee’s own membership test uses, so the line is not an approximation of the boundary but the boundary itself. Above it the enrollment term carries the majority for 96 districts and the per-pupil term for 197, which is the 293 both splits reach. The second rule crosses at zero on the per-pupil term, and the side of it where the FY2027 formula pays less per pupil than the FY2020 regime did holds 4 of the 314 — Marysville, Southwest Licking, St Henry and West Muskingum, every one of them left of the enrollment axis, which is to say every one of them grew. The corner where a district both lost pupils and lost ground per pupil holds no formula district at all, and that empty corner is what the paragraph above looks like when it is drawn rather than asserted. A list can say a region is empty; only a picture can show it. verified crates/project

Among the districts that lost pupils, the guarantee’s reach is a wealth gradient. 514 of the 607 have fewer pupils than in FY2020. Cut into fifths by published capacity per pupil, the FY2027 guarantee holds 15 of the least-wealthy 102, 26 of the second fifth, 56 of the third, 84 of the fourth and 81 of the wealthiest 106. The bands are equal counts with the remainder in the top one: 102 districts in the least wealthy fifth, 102 in the second, 102 in the third, 102 in the fourth and 106 in the wealthiest. The per-pupil term crosses zero between the third and fourth fifths — a median −0.274 in the least wealthy, −0.074 in the third, +0.344 in the fourth, +0.572 in the wealthiest — and the crossing is the whole shape. Below it the plan raised per-pupil aid by a quarter or more, for the districts it was written to reach, and the raise absorbed the lost pupils. Above it capacity outran a frozen cost side and the per-pupil term alone puts a district on the floor, lost pupils or not: the same fifths of the 93 districts that did not lose pupils read 0, 1, 2, 14 and 14 held. The enrollment cluster is the third fifth — 40 of its 89 sit there, against 67 of the capacity cluster’s 98 in the fourth and 75 of the minimum-share districts in the wealthiest. verified crates/project

So the cluster is not “the shrinking districts”. It is the shrinking districts at the wealth where the plan’s per-pupil raise ran out — the ones for which the raise was just too small to cover the decline, so that the lost pupils tip them onto the floor. inference

The cluster has siblings the floor does not hold, and they are poorer. 64 formula districts lost pupils at the cluster’s own median rate or faster — 11.9% since FY2020 — and are not on the guarantee. They teach 81,392 pupils. Their median economically disadvantaged share on the profile report’s measure is 58.17% against the cluster’s 52.15%, their median capacity per pupil $4,235 against $5,184, their median state share 49.7% against 39.6%, and their median per-pupil term −0.316 against −0.043: the formula pays the median sibling $8,542 a pupil against $5,776 in its FY2020 base. Their formula aid sits a median 17.9% above the floor — $1,201,355 a district, $100,805,628 in all — and at its own FY2024–FY2026 rate of decline, undamped, the median one reaches it in 4.14 years; at the shipped damping the enacted anchor holds 5 of the 64 by FY2032. verified crates/project

They are the same population as the cluster on every axis but wealth. The mechanism the cluster was named for — costs that cannot follow enrollment down, adjusted for by no component of the plan — is present in them and absorbed by a component that was not designed to absorb it. That is the blind spot outside the guarantee the multivariate partition was meant to find, and it is not a new one: it is the cluster’s, at a wealth where a different line pays for it. inference

The size terms are what hold 35 of the 64 off the floor. R.C. 3317.011’s staffing minima and R.C. 3317.0217’s capacity tier are the plan’s two size-dependent terms, and striking them out of the panel newly guarantees 136 districts statewide, 35 of them siblings: 16 for the minima alone, 23 for the capacity tier alone. Per base cost enrolled pupil the pair is worth $1,054 to the 64 against $384 to the cluster’s 89. Both counterfactuals were priced for a different question and both populations were already committed, so this is a join and not a new measurement. verified crates/project

The share to read is 35 of the 136 and not 35 of the 64. Striking a term out is a different formula for every district in the state rather than a measurement of these, and a counterfactual is not a mechanism: a quarter of what the pair does statewide is what it does here. inference

What reaches them is size, not decline. Unconditionally the siblings look singled out: 35 of 64 move against 101 of the 251 other formula districts. Band for band on base cost enrolled ADM they do not. The siblings move 22 of 25 in the smallest sextile, 7 of 10, 6 of 10, and none of the 19 above it, against 49 of 58, 36 of 52, 12 of 20 and 4 of 36 for every other formula district — the fourth band runs the other way. Give each band’s siblings that band’s non-sibling rate and the prediction is 34.60 against the 35 observed. verified crates/project

So the sextile view the size-term arc already reports contains them, and the sibling axis adds nothing to it. What a cut on size alone cannot see is which of its small districts are shrinking, not whether the terms reach them: the siblings are a population because they are poor and shrinking, and the two terms reach the 35 because they are small. inference

The 29 the size terms leave are carried by the state share of base cost, not by DPIA. They are the larger siblings — a median 1,846 enrolled ADM against 635.6 for the 35 — so neither size term binds on them. Their formula aid stands $68,572,895 above their FY2020 base, $1,144 a pupil, and taken component by component against that distance the base cost state share alone exceeds it for 28 of the 29 and DPIA for 4; targeted assistance covers 14, special education 9, gifted 1, and English learner and career-technical funding none. In aggregate the state share of base cost is 3.04 times the gap and DPIA 0.53 times it. verified crates/project

A component larger than the gap is not a component that pays it — the whole of [H] holds each of these districts above its floor and no line of it is the marginal one. The column is a reach, and what it establishes is which lines are even in the running: for the 29 it is the per-pupil raise itself rather than any weight. inference

The cluster and its siblings are 153, and that is a union rather than a rate. The 89 and the 64 share no district, so the population the arc has been naming is 153, and it is the one any threshold change should be reported on beside the sextiles. The districts shrinking at the cluster’s median rate or faster are 169 — the 64, 45 of the cluster’s own 89, 31 at the minimum state share and 29 of the capacity cluster — because a median cuts its own population in half and 60 guaranteed districts outside the cluster shrink at least that fast. Both are defensible populations; they are not the same one. verified crates/project

Priced on the 64 siblings, every shape that reaches them at all reaches them harder than it reaches the 89. Each pricing above was against the enrollment cluster; read the same frames over the siblings and nothing is recomputed, because the frames are anchor_incidence’s and the population is lost_pupils’. The mirror written beside [L], [M] and [O] pays the siblings $19,692,848.91, $241.95 a pupil, against $46,020,643.67 and $233.76 a pupil to the cluster. Written inside [H] instead the same mirror pays them $19,112,356.87 against $4,195,897.42. A rolling pupil count pays them $22,359,204.12, $274.71 a pupil against $19.82. A prior-year ratchet walked to FY2032 at the shipped damping pays them $14,794,631.13. verified crates/project

Two of the seven instruments are blind, and each is blind to one of the two populations. A prior-year ratchet moves 59 of the 64 and 0 of the 89 — the enacted guarantee already holds every cluster district, so a second floor under a floor is slack. A dated phase-down of the FY2027 guarantee list moves 7 of the 89 and 0 of the 64, because the 64 are not on that list. Neither zero is a rounding: no district of the blind population moves by so much as a cent. verified crates/project

So the design question does not decompose into “the districts the floor caught” against “the districts that lost pupils”. It decomposes into what a rule is keyed to. A rule keyed to a list on a date reaches the cluster and can never reach the siblings, whatever its generosity. A rule keyed to aid that falls reaches the siblings and is slack against a district the guarantee already holds. Only a rule keyed to pupils that leave — the mirror, the rolling count — reaches both, and it reaches both because its eligibility test is the one thing the two populations share. Which population a decline response is for is a choice this corpus does not make; what is established is that a rule keyed to a date decides it silently, and that the difference between the two populations is wealth. inference

Where the mirror sits is a question about the 89 and almost not a question about the 64. The order of magnitude between the two placements is absorption, and absorption needs a floor. Inside [H] the floors take back 0.9088 of the cluster’s gross and 0.0295 of the siblings’; 67 of the 81 cluster districts the mirror is payable to receive nothing at all, against 4 of 60 siblings that lose anything. A debate about [L1]’s subtrahend is a debate about districts the floor already holds. verified crates/project

About a third of the progressive gradient is the siblings, and the gradient survives taking them out. Of the dollars the rolling count sends the least-wealthy fifth outside the cluster, the siblings hold 0.3855 on 22 of that fifth’s 103 districts. Take them out and the fifth still pays $119.29 a pupil against $8.76 to the wealthiest — a ratio of 13.6 where the ratio with them is 14.6. Under the mirror inside [H] the siblings are paid flat, $244.06, $247.35, $222.89, $189.68 and $242.77 across the five fifths, so the whole of that shape’s gradient is a property of the districts around them. verified crates/project

The gradient is therefore this population’s neighborhood and not this population. The siblings sit at the poor end — 22, 17, 11, 6 and 7 of them across the wealth fifths — and raise it; what makes it a gradient is the 81 other districts in the poorest fifth, whose $119.29 a pupil is already fourteen times the wealthiest fifth’s. inference

A shape written inside [H] is self-extinguishing for the siblings, and the statutory date arrives before they do. Split the 64 at their median years-to-the-floor and the inside-[H] mirror’s absorbed share is 0.0760 on the half nearest their floors and nothing at all on the further half: the shape pays a district until the district arrives, and the 89 are what the 64 become. Beside the floors it absorbs nothing for either population, by construction, because [L], [M] and [O] are the three lines outside [L1]’s subtrahend. The clock is the other half of the answer. The median sibling reaches its own floor in 4.14 years, about FY2031; at the damping that ships 5 of the 64 are on the floor by FY2036 and undamped 38 are. R.C. 3317.019 anchors the guarantee at FY2027, four years before the median arrival, which is why a dated phase-down pays them nothing here and would pay them nothing in any year. verified crates/project

The 153 and the 169 price differently, so both are reported. The mirror beside the floors is worth $65,713,492.58 over the cluster and the siblings unioned and $69,915,878.09 over every district shrinking at the cluster’s median rate or faster. Neither population contains the other, and a shape’s per-pupil figure differs between them by enough to change which one looks generous: the 2% cap costs the union $14.65 a pupil and the rate-defined population $57.61. verified crates/project

A fitted partition finds the cluster at no k. A k-means on six standardized profile variables — log enrolled ADM, the enrollment term, log capacity per pupil, the DPIA share, base cost per pupil and the fourteen-year enrollment rate, none of them a formula output — was run once, from deterministic seeds, at every k from 2 to 9, after known-answer tests in crates/dispersion. Scored as a ceiling, with every cell assigned to its majority, the most any of them labels right on the cluster against the rest is 519 of 607, where a single cell scores 518; started from the typology’s own nine centers instead, 518 exactly. The 2013 typology itself scores 518 on the cluster and 384 on guarantee membership; a nine-cell k-means reaches 432 on membership and the typology-seeded run 415, so a fit beats the baseline at describing who is held, and neither reaches three quarters. With nothing fitted, a cluster member’s nearest neighbor on the six variables is a member 31 of 89 times, against 89 of 607 by chance, and 449 of 607 districts have a neighbor that is held if and only if they are. The cluster is a region of the profile, a thin one, and no cell of any partition tried holds a majority of it. verified crates/project

The multivariate partition was therefore run, and what it recovers — membership, weakly — is the wealth gradient above, rediscovered without the axis that explains it. No clustering estimator is a reasoning tool of this corpus; dispersion::partition stays in the workspace so the ceilings are reproducible, and every claim here rests on the identity and a sort. inference