(a) compute the maximum permissible assessments for the years nineteen hundred eighty-one through nineteen hundred eighty-five pursuant to the limitations contained hereinabove as if the provisions of this subdivision were in effect and applicable for assessment rolls prepared in the year nineteen hundred eighty-one;

(b) subtract the maximum permissible assessment for the year nineteen hundred eighty-one as computed in paragraph (a) hereof from the actual assessment entered on the assessment roll completed and applicable for the year nineteen hundred eighty-one and divide such result by four;

(c) the assessment for the year nineteen hundred eighty-two shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (b) hereinabove;

(d) the assessment for the year nineteen hundred eighty-three shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (b) hereinabove;

(e) the assessment for the year nineteen hundred eighty-four shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (b) hereinabove;

(f) the assessment for the year nineteen hundred eighty-five shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (b) hereinabove;

(g) commencing and applicable in the years nineteen hundred eighty-six through nineteen hundred ninety, the limitations contained hereinabove shall be computed as measured against the maximum permissible assessments for the years nineteen hundred eighty-one through nineteen hundred eighty-five as computed in paragraph (a) hereinabove and as measured against the actual assessments for the years occurring after nineteen hundred eighty-five. 2. The assessment roll of a special assessing unit wholly contained within a city shall identify those parcels classified in class two which have fewer than eleven residential units. The assessor of any such special assessing unit shall not increase the assessment of any parcel so identified in any one year, as measured from the actual assessment on the previous year's assessment roll, by more than eight percent and shall not increase such assessment by more than thirty percent in any five-year period. The first such five-year period shall be measured from the individual assessment appearing on the assessment roll completed in nineteen hundred eighty-one provided that, if such parcel would not have been subject to the provisions of this subdivision in nineteen hundred eighty-one had this subdivision then been in effect, the first such five-year period shall be measured from the first year after nineteen hundred eighty-one in which this subdivision applied to such parcel or would have applied to such parcel had this subdivision been in effect in such year. If, in respect to any parcel so identified on the assessment roll completed and applicable for the year nineteen hundred eighty-four, the actual assessment for the year nineteen hundred eighty-three exceeds by more than thirty percent the assessment for the year nineteen hundred eighty-one, such assessor shall compute the actual assessments to be entered on assessment rolls applicable to the years nineteen hundred eighty-four through nineteen hundred ninety-two as follows:

(a) compute the maximum permissible assessments for the years nineteen hundred eighty-two through nineteen hundred eighty-seven pursuant to the limitations contained hereinabove as if the provisions of this subdivision were in effect and applicable for assessment rolls prepared in the year nineteen hundred eighty-two;

(b) subtract the maximum permissible assessment for the year nineteen hundred eighty-two as computed in paragraph (a) hereof from the lesser of the actual or transition assessment entered on the assessment roll completed and applicable for the year nineteen hundred eighty-two;

(c) subtract the maximum permissible assessment for the year nineteen hundred eighty-three as computed in such paragraph (a) from the lesser of the actual or transition assessment entered on the assessment roll completed and applicable for the year nineteen hundred eighty-three;

(d) add such results of the computations prescribed by paragraphs (b) and (c) hereof as are greater than zero and divide such sum by four;

(e) the assessment for the year nineteen hundred eighty-four shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (d) hereinabove;

(f) the assessment for the year nineteen hundred eighty-five shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (d) hereinabove;

(g) the assessment for the year nineteen hundred eighty-six shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (d) hereinabove;

(h) the assessment for the year nineteen hundred eighty-seven shall be the maximum permissible assessment for such year as computed pursuant to such paragraph (a) minus the quotient determined pursuant to paragraph (d) hereinabove;

(i) commencing and applicable in the years nineteen hundred eighty-eight through nineteen hundred ninety-two, the limitations contained hereinabove shall be computed as measured against the maximum permissible assessments for the years nineteen hundred eighty-three through nineteen hundred eighty-seven as computed in paragraph (a) hereinabove and as measured against the actual assessments for the years occurring after nineteen hundred eighty-seven. 3. If the assessment appearing on an assessment roll completed on or after January first, nineteen hundred eighty-two for any parcel not subject to the provisions of subdivision one or two of this section, other than a parcel classified in class three, is greater than the assessment appearing on the previous year's assessment roll the assessor shall determine a transition assessment for such parcel for the first assessment roll on which such greater assessment appears and for each of the succeeding four assessment rolls by computing the difference between such greater assessment and the assessment appearing on such previous year's assessment roll and adding the following percentages of such difference to the assessment appearing on such previous year's assessment roll: in the first year, twenty percent; in the second year, forty percent; in the third year, sixty percent; in the fourth year, eighty percent; and in the fifth year, one hundred percent. If the assessment of a parcel is increased during a period for which transition assessments have been established because of any prior assessment increases, such new increase shall be phased-in over a five-year period as set forth in this subdivision, and such phased-in increases shall be added to the transitional assessments previously established for the prior increase; provided, however, that if in any year any such transition assessment exceeds the actual assessment for such year, taxes imposed on such parcel for such year shall be based on such lesser actual assessment. Notwithstanding the foregoing, during the period of any such transition, the assessment roll shall contain an entry of the full amount of such greater assessment which shall be used by the commissioner in its determination of class ratios pursuant to paragraph (b) of subdivision one of section twelve hundred two of this chapter. In establishing state equalization rates, class equalization rates, special state equalization rates and special state equalization ratios under article twelve, article twelve-A and article twelve-B of this chapter, the commissioner shall use the transition assessments as provided for in this subdivision in its determinations, or where the actual assessment is the lesser, such actual assessment shall be so used. 4. Notwithstanding the provisions of subdivision three of this section no transition assessment shall be computed for any individual parcel for a year in which subdivision one or two of this section were applicable to such parcel, nor for any real property owned by the state and which is subject to taxation pursuant to title two of article five of this chapter, nor for any real property for which a statute provides that a state equalization rate or special equalization rate shall be employed to determine the assessed value or the taxable assessed value. 5. Nothing in this section shall prevent placing on the assessment roll new property, additions to or improvements of existing property or formerly exempt property or the full or partial removal from the roll of property by reason of fire, demolition, destruction or new exemption and such increase or decrease in value shall not be included in the computation of the limitations prescribed by this section. Any parcel which would be assessed at a greater amount but for the provisions of subdivision one or two of this section shall be excluded from any survey or computation made by any body or officer for the purpose of determining a level of assessment to be used in the administrative or judicial review of assessments including, but not limited to, class ratios computed pursuant to paragraph (b) of subdivision one of section twelve hundred two of this chapter, ratios computed by selection of parcels or from actual sales of real property under the provisions of subdivision three of section seven hundred twenty of this chapter, and residential assessment ratios computed pursuant to section seven hundred thirty-eight of this chapter, but not including state equalization rates or class equalization rates. In the event that a parcel appearing on the assessment roll completed in nineteen hundred eighty or any subsequent roll is subdivided or consolidated with another, such assessor shall make an appropriate allocation of the assessed value of such parcel in the determination of the limitations prescribed in this section. 6. Notwithstanding any provision of law to the contrary, in any special assessing unit wholly contained within a city, beginning with the assessment roll completed in two thousand five and for each subsequent assessment roll, the assessor shall not increase the assessment of any existing property classified in class two that has fewer than eleven residential units, with respect to any increase in value attributable to additions to or improvements of such property that were not reflected on the assessment roll for a previous year, by more than one-third of the amount that such assessment would increase, in the absence of this subdivision, with respect to any increase in value attributable to additions to or improvements of such property that were not reflected on the assessment roll for a previous year. Notwithstanding the provisions of subdivision five of this section, the remainder of the increase in value attributable to such additions or improvements that were not reflected on the assessment roll for a previous year shall be subject to the limitations on increases provided in subdivision two of this section. This subdivision shall not apply to the construction of a new building or structure. This subdivision shall not apply where, as a result of such additions to or improvements of the existing property, there are more than ten residential units in such property.