If cash and price are both rounded, test both ranges.
Displayed cash of $200.00 and price of $22.22 can be consistent with nine exact shares under our chosen rounding rules. Allowing both positive inputs to vary inside their half-up cent ranges gives a share interval strictly between 39999/4445 and 40001/4443. Both outer endpoints are excluded. This is an existence test for assumed display rules, not proof of an actual SCHD transaction.
The single-rounded-price worksheet treats cash as exactly $200. Here the cash display also discards digits. The candidate quantity remains exact; fees, tax adjustments, currency conversion and separate share rounding are excluded. A real account comparison needs those fields and their documented meanings.
Start with two input intervals
With a chosen $0.01 increment and positive half-up ties, cash displaying $200.00 can lie in [199.995, 200.005). Price displaying $22.22 can lie in [22.215, 22.225). Each lower endpoint is included; each upper is excluded because it rounds to the next cent. These are independently allowed input ranges in this fixture, not measured uncertainty about a real fill.
For Q = C/P, the smallest possible ratio is approached by lowering cash toward 199.995 and raising price toward 22.225. Its lower limit is 199.995/22.225 = 39999/4445, but that price endpoint is unavailable. The upper limit is 200.005/22.215 = 40001/4443, but that cash endpoint is unavailable. Consequently neither extreme share ratio is attained.
Show a pair of values that actually works
For nine shares, choose exact cash C = 200 and exact price P = 200/9. Their quotient is exactly 9. Cash rounds to $200.00, and price rounds to $22.22. This pair is a witness: it demonstrates compatibility without pretending it is the customer's hidden execution record.
The independent feasibility check works backward from any positive candidate q. A permitted price must be in the intersection of [22.215, 22.225) and [199.995/q, 200.005/q). If the intersection has positive width, choose its midpoint P and calculate C = qP. Then round both constructed values and verify both displayed inputs. An empty intersection supplies no witness and the candidate fails this model.
Why the exact-cash test can give a different answer
When cash is fixed at exactly $200, the previous worksheet excludes candidate 8000/889 shares: it would require price $22.225, which rounds to $22.23. With cash allowed to vary within the new interval, that same candidate is compatible. A slightly smaller cash amount and a price just below $22.225 can satisfy both displays.
This is a change in assumptions, not a correction to the earlier arithmetic. The exact-cash interval is (8000/889, 40000/4443]; the two-rounded-field interval is (39999/4445, 40001/4443). The new lower limit is smaller by 1/4445 share and the new upper limit is larger by 1/4443 share. The CSV preserves fractions so those small differences are not lost in displayed decimals.
Reproduce the boundary and witness matrix
| Case | Candidate shares | Compatible? |
|---|---|---|
| nine | 9 | True |
| lower_extreme | 39999/4445 | False |
| upper_extreme | 40001/4443 | False |
| old_lower | 8000/889 | True |
| old_upper | 40000/4443 | True |
| below | 8.99 | False |
| above | 9.01 | False |
| zero_display_zero_qty | 0 | True |
| zero_display_positive_qty | .0001 | True |
| positive_cash_zero_qty | 0 | False |
| missing_qty | Unknown | unknown_candidate |
Download displayed inputs and exact candidates · Download bounds, compatibility and witness values
The new lower and upper extreme ratios both fail. Nine shares, and the two earlier exact-cash boundary quantities, pass. Candidates 8.99 and 9.01 remain outside. Each successful positive-quantity case saves an exact witness price and cash value. Decimal half-up rounding of those witnesses independently reproduces the requested displays.
A compatible result means at least one pair works, not that every pair in the input rectangle works. Nor does it assign a probability to any pair. The fixture contains no distribution of possible fills, and it does not rank candidate values by likelihood.
A displayed zero is different from known exact zero
For nonnegative cash displayed as $0.00, the chosen half-up interval is [0, 0.005). At a price displayed as $22.22, zero shares are compatible, but a small positive candidate of 0.0001 share is also compatible: choosing price $22.22 gives cash $0.002222, which displays as $0.00. This is different from the earlier fixture with cash known to be exactly zero.
Do not turn a missing field into that zero display. Missing cash or price leaves the test unknown; missing quantity leaves compatibility unknown even if bounds can be calculated. Negative cash or shares and a nonpositive price are rejected. A price such as $22.222 is invalid under the declared cent grid; the validator does not silently round an input a second time.
What evidence is still needed?
The Schwab One Account Agreement distinguishes statement precision and settlement rounding. It does not establish our chosen half-up cent-display rules for both fields. Obtain the actual precision, tie convention, cash basis and share field before applying an interval test to an account. If the quantity is also rounded, this exact-candidate method needs another interval rather than a guessed exact quantity.
A public DRIP pricing discussion asks how to tie reinvestment prices to transaction records. It motivates the traceability problem; none of its account figures enters this worksheet. This page cannot authenticate or challenge a particular broker execution.
Evidence and scope
Sources read October 28, 2026. The displays, candidate quantities and rounding convention are explicit hypothetical inputs. Exact-ratio bounds are crosschecked against independently constructed feasible pairs and Decimal display rounding. The new contribution is the two-variable feasibility and zero-display matrix, relative to the freshly read fixed three-site sample and our earlier exact-cash test. It is not a global originality or investment-performance claim.