# Methodology This is written so a hostile reader can reconstruct every number on the site. Where something is uncertain it is marked as uncertain rather than smoothed over. --- ## 1. What is being computed Given a population figure for each of 36 units, a house size *H*, an allocation rule and a set of constraints, produce a whole number of seats for every unit summing **exactly** to *H*. The one deliberate exception is the Delimitation Commission method, which is applied literally and can miss *H* by a few seats (§3). Exactness is not decorative. Published projection tables usually do not sum to their own stated total: the PRS annexure this tool validates against totals 548 and 813 where it should total 543 and 815, because each state is rounded independently and a small-state exception is applied by hand to four units. That is a defensible way to present an illustration and an indefensible way to build a tool, because a reader cannot tell which seats are real and which are rounding. ## 2. The units 36 states and union territories, on the **present-day** administrative map. - Andhra Pradesh and Telangana are separate, splitting the undivided state's 42 seats 25 and 17. - Jammu & Kashmir and Ladakh are separate, splitting 6 seats 5 and 1. - Dadra & Nagar Haveli and Daman & Diu is **one unit holding two seats**, matching its legal status since 26 January 2020. That last decision is not cosmetic. Modelled as one unit with a one-seat minimum, it receives 1 seat at a house of 543 against the 2 it holds today, because its 586,956 people are about a quarter of a quota. Modelled as two units it would receive 2. We keep the legal status and disclose the consequence, rather than letting the data grant a rounding favour that the law does not. ## 3. The rules | Rule | Mechanism | Size bias | |---|---|---| | **Delimitation Commission method (1976)** | Union territories and states of 60 lakh or fewer keep their current seats. The rest share the remaining seats: one national quotient (their population ÷ those seats), and each state's population ÷ that quotient, rounded to the nearest seat. The quotient is not adjusted afterwards. | Neutral among the larger states. **Its total is not guaranteed to equal the house size**, and is shown as computed. The interface default. | | **Largest remainder, Hare quota** | Quota is total population over house size. Each unit takes the floor of its exact entitlement; leftover seats go to the largest fractional remainders. | Essentially neutral. Never violates quota. Can exhibit the Alabama paradox. | | **Huntington-Hill** | Every unit seeded with one seat, then each further seat to the highest population divided by √(n(n+1)). Used for the US House. | Mildly favours small units. | | **Sainte-Laguë (Webster)** | Divisors 1, 3, 5, 7; equivalently, one quotient for every unit, adjusted until the rounded seats add up exactly. | Least size-biased of the divisor methods. Reproduces the 1976 allocation, but is **not** the Delimitation Commission's procedure (see below). | | **D'Hondt (Jefferson)** | Divisors 1, 2, 3, 4. | Systematically favours large units. | | **Cube root of population** | Largest remainder applied to the cube root of population. | Compresses hard: a unit four times larger gets about 1.6 times the seats. | | **Base plus proportional** | A guaranteed base per unit, remainder distributed proportionally. The Cambridge Compromise shape used for the European Parliament. | Set by the base. A larger base protects small units at the direct expense of large ones. | | **Uniform scaling** | Seats in proportion to each state's *current* allocation. | **Not an apportionment rule.** Included because it is the shape of the uniform increase offered during the April 2026 debate, and because seeing that it moves nobody's share is more convincing than being told so. | ### The Delimitation Commission method, and the default India's allocation is not computed by any formula today: it is the allocation made on the 1971 Census and frozen in 1976. The last time a formula was applied, the Third Delimitation Commission (1972–76) worked as follows: 1. It set aside seats for the union territories and for states of six million people or fewer, which the proviso to Article 81 exempts from the population ratio. 2. It divided the remaining population by the remaining seats. 3. It rounded the result to a national quotient of 10,44,000 people per seat. 4. It divided each state's population by that quotient to find its seats. People per seat ended up ranging only from 10.31 lakh in Rajasthan to 10.67 lakh in Kerala. Sources: The India Forum, and the Centre for Policy Studies (August 2026). **Checked against what the Commission actually did.** Today's seats are that allocation, frozen, with later state splits dividing a parent's seats among its successors. Re-merging the successors gives the fifteen states above 60 lakh in 1971, which held 507 seats. Dividing their 1971 Census populations (Table A-02, recast to 2011 jurisdiction) by 10,44,000 and rounding to the nearest seat reproduces all fifteen exactly, for a total of 507. Rounding always up or always down does not, and neither does D'Hondt. The test suite asserts this (§16). **Why this is not Sainte-Laguë.** Sainte-Laguë, largest remainder and Huntington-Hill *also* reproduce all fifteen, because no state's fraction fell near the rounding line, so the 1976 figures cannot distinguish them. The procedure itself differs from Sainte-Laguë in two ways that matter today: - **The quotient was fixed, not tuned.** Sainte-Laguë adjusts its divisor until the rounded seats add up exactly. The Commission applied one quotient once. Applied to today's data with the same set-aside, a fixed quotient with nearest rounding adds up at only 126 of the 358 house sizes from 543 to 900 on the 2011 Census (missing by at most 3 seats), and at 81 of 358 on the 2026 projection (at most 4). At a house of 815 on 2011 it produces 816. - **Not every unit is on the quotient.** Union territories and small states keep set-aside seats. At 815, Sainte-Laguë over every unit gives Delhi 11 seats and Jammu & Kashmir 8; the Commission's procedure keeps them at 7 and 5. **How the tool applies it.** Literally. Union territories and states at or below the threshold keep their current seats; the rest are rounded to the nearest seat against one quotient; and the total is shown as computed, with a notice when it misses the house size. The record does not say how the Commission would have settled a miss, and adjusting it silently would attribute a choice to the Commission that it never made. This is the one method exempt from the exact-total guarantee in §1. The other constraints do not apply to it, since it carries its own small-state rule. The defeated 2026 bills specified no allocation formula; the uniform increase reportedly offered during the debate was pro-rata scaling of current seats. Carnegie's May 2026 analysis uses Sainte-Laguë (Webster) explicitly. ### A change from the first implementation `baseProportional` originally found a divisor by bisection and then settled the leftover seats by cycling through units *largest first*. That settle-up was not neutral. Measured against an exact base-plus-proportional share at a house of 815 it gave Uttar Pradesh 127 seats against an ideal of 124.61, Bihar +1.12 and Maharashtra +1.05, while essentially every small unit undershot by about 0.8 of a seat — the opposite of what the method exists to do. It also produced 62 house-monotonicity violations between 500 and 900, none inherent to the method. It now gives every unit its base and distributes the remainder by largest remainder. That is exact by construction, needs no bisection, no guard loop and no silent failure path, and confines the size bias to the base itself where a reader can see it. Maximum deviation falls from 2.39 seats to 0.50, and no unit is off by a whole seat. ### Tie-breaking Every method is deterministic. Where two units have an equal claim the tie is broken **by unit code, ascending** — never by array order. This matters more than it looks: array order is the order of `units.json`, so an order-dependent tie-break would make the published allocation depend on how the data file happens to be sorted. Exact ties are rare with real population data but the property should not rest on that. ## 4. Constraints - **One seat per unit** always applies and cannot be switched off. - **No unit loses a seat**: floor at its current allocation. - **Small units do not lose seats**: the same, for units below a population threshold. The default of 60 lakh is the published assumption — the PRS footnote states the exception explicitly and applies it to Arunachal Pradesh, Goa, Manipur and Meghalaya. - **Maximum change per unit**: a cap of ±K seats against the current allocation. Unlike the others this is **two-sided**, so it needs ceilings as well as floors. Constraints are enforced by **locking**. Run the method; pin any unit that lands outside its band at exactly the bound it broke; re-run on what is left, for the seats that are left. The locked set only grows, so this terminates, and it works uniformly across every rule. ### Where locking is not enough, and what happens then With two-sided bounds the loop can finish with *every* unit pinned, and the sum of those pins need not equal the house size. This is not hypothetical: largest remainder at a house of 600 with a cap of 2 pinned 28 units at their ceiling and 8 at their floor, totalling 594. Returning that would break the guarantee in §1, so there is an explicit **repair pass**. The difference is settled one seat at a time, each going to — or being taken from — the unit currently furthest from the method's own *unconstrained* answer, with ties broken by code. That is the adjustment which deviates least from what the rule wanted, and it is fully deterministic. Global feasibility is checked before the loop runs, so the slack to do this always exists. When the cap makes the house size unreachable at all, the tool says so and offers the nearest feasible size rather than failing silently. A cap of ±5, for instance, cannot produce a house larger than **723** — well short of the 815 states-side figure in the 2026 bill. A phased transition and that expansion are not simultaneously satisfiable. ## 5. The Alabama paradox Largest remainder can take a seat away from a state when the house **grows**. This is a known defect of the method — see Balinski and Young, *Fair Representation* — and it happens on this data inside the politically relevant range. **It is not a defect of India's own method.** Under the Delimitation Commission's procedure a larger house only lowers the quotient, so no state's rounded figure can fall, and the divisor methods cannot produce it either. The finding matters for projections and analysis built on largest remainder, not for a Delimitation Commission working the way the last one did. Across house sizes 543 to 900 on the 2011 series there are **32** instances. Uttarakhand falls from 5 to 4 at a house of 548, and 548 is itself the total of the published unchanged-strength column. Odisha falls from 21 to 20 at 595. **Why you should believe it.** Finding the paradox with the same code that implements the rule is circular: a bug in the implementation, or a floating-point comparison of nearly equal remainders, could manufacture one. The test suite therefore re-derives it from scratch in **exact integer arithmetic**. For unit *i* the entitlement is *wᵢH/W*; the floor is BigInt integer division and the remainder ordering is the ordering of *(wᵢH) mod W*, with no division and no floating point anywhere. The one-seat locking is reimplemented too, so the comparison is like-for-like. The two agree seat for seat at every house size from 400 to 1100. Two qualifications that belong with the claim: - The count is **32, not 33**. An earlier figure of 33 was computed on the superseded Telangana population; Haryana 12→11 at 560 was in that list and is not in this one. Nothing in the interface hardcodes an instance — they are recomputed from whichever series is loaded. - **Cube root and base-plus-proportional exhibit it too**, because both distribute by largest remainder. The honest framing is "the largest-remainder family", not "largest remainder alone". Huntington-Hill, Sainte-Laguë and D'Hondt produce no instances at all. ## 6. Quota A unit's **quota** is the pair of whole numbers bracketing its exact proportional entitlement. Largest remainder never leaves that bracket; divisor methods can. Cube root and base-plus-proportional leave it almost everywhere, which is expected, since neither is trying to be proportional to population. The interface reports violations, and distinguishes a unit that is outside quota because of the *rule* from one that is outside because a *constraint* pinned it. The engine reports both and does not guess which is which. ## 7. Population series **1971–2011** come from Census of India Table A-02, *Decadal variation in population 1901–2011*, in which — by the Registrar General's own statement — "the population figures for all the previous censuses in this table are adjusted to the 2011 jurisdiction". That is why Jharkhand, Chhattisgarh and Uttarakhand have figures back to 1971 without any hand-built district concordance: the boundary reconciliation has already been done officially. The two splits A-02 does not make are made here at district level, which is the only defensible method: - **Telangana** = Census district codes 532–541 of undivided Andhra Pradesh - **Ladakh** = codes 003 (Leh) and 004 (Kargil) of undivided Jammu & Kashmir **2016–2036** come from Table 21 of the *Report of the Technical Group on Population Projections* (National Commission on Population, MoHFW, July 2020), printed pages 259–260. Figures there are printed in thousands, so every projected population is a multiple of 1000 and carries up to ±500. ### Years that are refused 1901–1961 are excluded and the extractor fails its own invariants if they are restored. Goa, Daman and Diu were Portuguese territory until 1961 and sit outside the Indian totals before then; Arunachal Pradesh and Puducherry are not separately enumerated in several columns; and the Andhra Pradesh district rows do not sum to the state row before 1941, so the Telangana split is not derivable. Excluding a year is better than fudging it. ### Cells that are empty Assam was not enumerated in 1981. Jammu & Kashmir and Ladakh were not enumerated in 1991. Those cells are `null`. A-02 does print official interpolations and they are preserved in `scripts/raw/census_series_a2.json`, but they are not populations anyone counted, so the tool will not allocate on them. An allocation run on 1981 or 1991 is missing a unit, the interface says so prominently, and the shares are of a house that excludes it. Sikkim acceded to India in 1975 and is outside the published 1971 national total. ## 8. The Telangana figure Two figures circulate for Telangana's 2011 population and **both reconcile** to the undivided Andhra Pradesh total of 84,580,777, which is why no arithmetic check can choose between them: ``` TS 35,193,978 + AP 49,386,799 = 84,580,777 TS 35,003,674 + AP 49,577,103 = 84,580,777 ``` They are not rival estimates of one quantity. They measure two different territories. - **35,193,978** is the sum of the ten districts *as constituted at the 2011 Census*. Census Table A-02 for Andhra Pradesh, district codes 532–541, gives exactly this. - **35,003,674** is **Telangana as actually constituted**. The Andhra Pradesh Reorganisation Act 2014, as amended by Act 19 of 2014, transferred seven mandals of the Bhadrachalam division of Khammam district to Andhra Pradesh for the Polavaram project. The difference is exactly **190,304**, and residual Andhra Pradesh moves the same amount the other way — which is precisely why both pairings reconcile. **This site uses 35,003,674**, because the unit holding 17 Lok Sabha seats is present-day Telangana, not the 2011 districts. The Registrar General's own Technical Group report independently prints Telangana at 35,004 thousand and Andhra Pradesh at 49,577 thousand. The arithmetic is closed to the person. Act 19 of 2014 transfers seven whole mandals *less thirteen named revenue villages*. From the Census 2011 village directory for Khammam: ``` seven mandals, 338 revenue villages 225,233 less the eleven retained rural villages 34,929 net transferred 190,304 ``` Sarapaka and Bhadrachalam are also retained but are enumerated as towns, so they fall outside the rural village directory on both sides of the subtraction. Run `python scripts/verify_bhadrachalam.py` to reproduce it. Note that the *unamended* Act named a different, shorter list: Seetharama Nagar is not retained under the amendment, and including it would break the reconciliation by exactly its own population of 1,332. ### The residual caveat Table A-02 is on 2011 **district** boundaries. For 34 of 36 units that is today's map. For Andhra Pradesh and Telangana it is not: the seven Khammam mandals sit inside the historical Telangana figures. The effect is 190,304 people in 2011 and a smaller, unquantified amount in earlier years. It is left visible rather than patched, because patching earlier years would require ratio-splitting a district, which the concordance file rules out as indefensible. ## 9. Map depiction The map shows India's external boundaries in accordance with Survey of India standards: the whole of Jammu & Kashmir as claimed including Pakistan- administered territory and Gilgit-Baltistan, Aksai Chin as part of Ladakh, and the whole of Arunachal Pradesh. Most open GeoJSON of India in circulation derives from Natural Earth, GADM or similar international sources and draws the Line of Control and Line of Actual Control as international boundaries. That depiction is not acceptable. The widely used DataMeet/GADM file fails this test decisively: its northern extent stops at 35.5°N and Gilgit-Baltistan, Muzaffarabad, Mirpur, Aksai Chin and Shaksgam all fall outside its India polygon. Compliance here is established by **testing the geometry**, not by trusting a provenance claim. Fifteen landmarks are checked by point-in-polygon on the final simplified file — twelve that must be inside and three controls that must be outside — and the test also asserts that Aksai Chin, Shaksgam and Gilgit-Baltistan are owned by *Ladakh specifically* and PoK by *Jammu & Kashmir*. This runs as part of `node test/run.js`, so a regression fails the build. **Honest limitation.** The upstream repository states that its data is curated from publicly available sources, declares no licence, and makes no Survey of India claim. Our confidence rests on our own geometry test rather than on an upstream assertion. A citable government-hosted equivalent would be better and has not been found. ## 10. Known limits - The 2016–2036 series are **projections**, made in 2019 and published in 2020. They predate both the 2027 Census and COVID-19 mortality. - Projected figures carry up to ±500 from thousands-rounding. - Regional blocs (South, West, Punjab & J&K, Hindi-belt, East, North-East) drive the share panel and colour the chamber, and are **not official**. They follow language and region rather than party, since the delimitation argument divides that way and party alignments change between elections. The official grouping carried in `units.json` is zonal council membership, which cuts differently. Membership of the analytical groups is listed in the interface. - Vote weight is people per MP against the national average. The smallest units always look extreme because a one-seat minimum guarantees it: Lakshadweep's 64,473 people elect one MP under every rule. They are excluded from the headline comparison for that reason, and the exclusion is stated on the page. - Seat counts are the total for each unit. This tool says nothing about constituency boundaries *within* a state, which is the other half of what a Delimitation Commission does.