A portal · four layers, one predicament

The Answer Filed Before the Question

Sometimes the deadline is shorter than the computation. Then nothing computes the answer. Something was filed in advance, before the question existed, and whatever accuracy is missing has to be bought back in correction rounds. Four of this site's layers do exactly that, and the exchange rate between the two is not a matter of taste: for the numerical pair it can be measured, and it comes out at one extra round square-roots the table.

This page teaches none of these four subjects. Each has its own full layer and every station below links straight there. What this page adds is the quantity none of them carries: what you file in advance sets the rate at which correction rounds pay off, and the four are four different answers to the question of what can be filed.

The bench below is the argument. Choose how many correct bits you need and how many correction rounds you are willing to pay, and it computes what has to be filed. Every figure is recomputed in your browser from the closed forms printed in the check panel, and again offline by the verifier.

The exchange bench

What you file

a table of answers 0x5f3759df an arctan ladder

A table of 2^k precomputed values of 1/√x, the best one that size can be.

Table size: 2⁶ entries

Each doubling of the table buys exactly one more correct bit in the seed.

Correction rounds paid: 2

One round of Newton for a table or a constant; one rung of the ladder for CORDIC.

Correct bits (vertical) against correction rounds paid (horizontal). Recomputed live. The flat top is the ceiling of the arithmetic, not of the method.

The frontier, in one table

Fix the accuracy you need and ask what it costs. Every row below is a way to reach the same answer. Nothing here is typed in: each row is solved live from the closed forms.

Correct bits required: 24

24 bits is a float32 significand. 53 is a float64 one.

Ways to reach the required accuracy. The plain rows are tables of every size; the two coloured rows are the real systems, and neither of them is a plain table.
Rounds paid	Seed must supply	Entries filed	Ratio to the row above

Each extra round takes the square root of the table, near enough. The closed form predicts a shortfall of 0.21 in the exponent, and the fact that a table size must be a whole power of two rounds it away.

Four things you can file

The bench measures two of these four. The other two are reported in their own units, and this page makes no attempt to convert milliseconds or receptor event-keys into bits. What the four share is not an axis. It is a predicament.

The Reflex Too Fast to Think

files a response

The purest case, and the one that explains why anything is ever filed. The doctor taps the tendon and the leg kicks in about 18 ms. Any route through the cortex costs at least 49 ms by the same conduction arithmetic, and a fast voluntary response to a go signal, measured in biceps brachii, is about 90 ms. So the deadline is shorter than the consultation, and the number of correction rounds available is zero at any price. What is filed is a single response with no index at all: the arc has no question to look up, because the same stretch always produces the same contraction.

filed one arc
rounds available 0
deadline 18.3 ms
cheapest consultation 48.9 ms

The Sine That Never Multiplied

files an alphabet

A chip with no multiplier cannot refine an estimate, because refining is multiplying. So CORDIC files something that is not an answer: a ladder of angles, arctan 1, arctan ½, arctan ¼, and grinds the answer out one signed digit at a time. Measured over rungs 8 to 40, the rate is 1.0001 bits per rung, flat, and no rung ever buys two. The tell that this is an alphabet and not a table: two unrelated angles read the identical constants in the identical order. Only the signs differ. The question lives in the sign word the machine writes at run time, never in the table.

filed 1 constant per bit
rate 1.0001 bits/rung
to 24 bits 24 filed, 24 rounds
gain K = 0.60725

The Number That Undid the Root

files a rule

Quake III filed one number, 0x5f3759df, and got an approximation to 1/√x good to 3.4376%, which is 4.862 correct bits, for a table with no entries in it. It can do that because a float's exponent field already is a logarithm, so the constant does not store answers, it stores the rule that rebuilds one. Measured against the best possible table of the same accuracy: it seeds better than 16 entries and worse than 32, so it is worth a table of between 16 and 32 entries and files 32 bits where that table would file 1,024. Because a multiplier is available here, each Newton round doubles the correct bits, and three rounds finish a float32.

filed 32 bits
seeds 4.862 bits
worth a 16 to 32 entry table
to 24 bits 3 rounds

The Receptors We Meet Twice

files a generator

An immune system cannot iterate towards a receptor for a pathogen it has not met, so it files a machine that manufactures the whole space in advance. In the toy model that layer declares, the filed inventory is 48 × 2 × 13 = 1,248 segment combinations carrying 9.048 bits of choice, and the assembled event keys carry 43.712 bits. The difference, 34.66 bits, is manufactured at assembly time by junctional trimming and insertion and was never filed at all. On first contact the rounds paid are zero: the receptor that meets the pathogen was built before the pathogen arrived.

filed 1,248 combinations
filed entropy 9.048 bits
assembled entropy 43.712 bits
rounds on first contact 0

What the bench actually shows

The interesting thing is not that precomputing helps. Everyone knows precomputing helps. The interesting thing is that the kind of object you filed decides the exchange rate you are then allowed to trade at, and the two rates are not close together.

File an answer, and a correction round has an estimate to work on. One Newton round squares the relative error, so correct bits double, and the ledger reads bits(r+1) = 2 · bits(r) − log₂(3/2). The shortfall of 0.585 of a bit is exact, and it is the 1.5 in the Quake line y*(1.5 − 0.5*x*y*y) showing up as a tax. Run that backwards and every extra round you are willing to pay takes the square root of the table you have to store: 16.7 million entries, then 4,096, then 64, then 8.

File an alphabet, and a round has nothing to square, because there is no estimate yet, only a partly written word. Each rung contributes one digit and therefore one bit, forever, and no larger table improves the rate. That is not a defect. CORDIC is built for a machine with no multiplier, and Newton needs one, so the frontier the table and the constant sit on is simply not available to it. Comparing them on storage alone would be the mistake this page is trying not to make.

And file a generator large enough and you can skip the rounds altogether, which is what both biological stations do, from opposite ends. The reflex files one response because there is no time to look anything up. The repertoire files a machine that makes 10¹³ answers because there is no way to know the question. Same predicament, opposite extremes of how much gets filed.

Not the same page as Only by Running

Only by Running asks whether a shortcut exists at all: whether a system's future can be known faster than by simulating it. This page assumes the shortcut exists and asks what it costs, which is a different question with a different answer shape. There, the finding is a wall. Here, the finding is a price list.

One relation on this page is already published and is not claimed as new: CORDIC and the fast inverse square root already point at each other, as kin in spirit and opposite in method. What neither of them carries, and what this page contributes, is the measured rate that makes the opposition quantitative.

Show the check

Every figure on this page comes from one of three places, and they are not equally strong. Derived means a closed form proved here and recomputed in your browser. Measured means a number obtained by running code over a dense sweep. Carried means a figure taken from the member layer's own verifier, cited rather than re-derived. Nothing on this page is asserted.

The closed forms this page computes from

The seed. For the best possible table of 2^k constants covering x in [1,4), the worst relative error is exactly (√b − 1)/(√b + 1) with b = 1 + 3·2^−k. This is exact, not fitted, and the reason is that bins are uniform in x while 1/√x is steepest at the left, so the leftmost bin is always the worst one and the best constant in a bin is the harmonic mean of the two endpoint values. As k grows the bit count tends to k + log₂(4/3), that is k + 0.4150, from above. The verifier checks the closed form against a brute-force sweep over every bin for k = 0 to 20 and against a 300,000 point dense sample, and confirms by exhaustion that bin 0 carries the maximum.

The round. Writing y = (1+e)/√x, one Newton round takes e to exactly −1.5e² − 0.5e³. So the correct bits go to 2·bits − log₂(3/2), and the measured shortfall over 29 grid transitions is 0.5935 bits against a predicted 0.5850, with the residue coming from the small cubic term. The verifier reports both.

The ladder. CORDIC's rate is measured, not derived: 1.0001 bits per rung, mean over rungs 8 to 40, with a maximum single-rung gain of 1.022 bits.

Every figure on this page, and where it came from

Figure	Value	How obtained
Seed accuracy of a 2^k table	k + 0.4150 bits	Derived, closed form above, checked against a sweep
Newton shortfall per round	log₂(3/2) = 0.5850	Derived from the error recursion, measured at 0.5935
0x5f3759df seed error	3.4376%	Measured in float32 over 400,000 points, agrees with the member layer's verifier
0x5f3759df seed accuracy	4.862 bits	Measured, the same run
Its table equivalent	between 16 and 32	Measured: better than 4.542 bits, worse than 5.481
CORDIC rate	1.0001 bits/rung	Measured over rungs 8 to 40
CORDIC gain K	0.6072529350088813	Carried from the CORDIC layer's verifier, recomputed here
Reflex loop time	18.33 ms	Recomputed here from t = L/CV + one synapse, at 1.70 m and 85 m/s
Cortical detour floor	48.9 ms	Recomputed here, cord conduction both ways plus 20 ms of cortex
Voluntary response to a go signal	about 90 ms	Carried from the reflex layer, at the size that layer's own correction of 2026-08-01 left it: a fast voluntary response in one muscle, biceps brachii, not a record of any kind
V(D)J filed inventory	48 × 2 × 13 = 1,248	Carried from the V(D)J layer's declared toy model
V(D)J entropies	9.048 and 43.712 bits	Carried from the same verifier; the 34.66 bit gap is computed here

Every free choice, named

The reduction range is x in [1,4). That is one full period of 1/√x in binary, since 1/√(4x) = ½ · 1/√x exactly, so a seed over [1,4) is a seed for every positive float and the exponent adjustment is exact. A different range would move every seed figure.
Table entries are uniform in x, which is what a hardware seed table indexed by leading mantissa bits actually does. A table with bins uniform in log x would be slightly better and would remove the log₂(4/3) offset. This page does not claim its tables are optimal over all possible binnings, only over all possible values for a given binning.
Each bin holds the constant that minimises the maximum relative error, the harmonic mean of the endpoint values. That makes these figures an upper bound on what any constant table of that size can do, so the comparison with 0x5f3759df is generous to the table, which is the direction that makes the finding safe.
Accuracy is counted as −log₂ of the worst relative error over the range. Average error would flatter every method and is not reported.
The Newton curves are computed from the exact error recursion, so they keep climbing past the point where an actual float32 or float64 pipeline would stop. The bench draws the float32 and float64 ceilings for exactly this reason, and the verifier measures the real float32 pipeline hitting 23.03 bits.
CORDIC's 1.0001 bits per rung is measured over the angle range |θ| ≤ 1, not over its full convergence range of 1.7433 radians. Near the edge of the range the first rungs behave differently.
The frontier table rounds table sizes up to a whole power of two. That is why the measured halving of the exponent sometimes reads 0.00 and sometimes 0.50 rather than the 0.21 the closed form predicts.
The reflex figures use the member layer's model: a round-trip nerve path of 0.8817 times height, one synapse at 0.7 ms, and a cortical floor of 20 ms of processing. Those are the layer's declared constants and this page recomputes rather than re-derives them.
The V(D)J numbers are from a declared toy model, not a measured human repertoire. The layer is explicit about this and so is this page. The 1,248 is a segment count in that model, and real segment counts vary with the counting convention.
The linked V(D)J layer carries no affinity maturation step, checked on disk by the verifier. Somatic hypermutation is real immunology and is exactly the correction-round leg this page would want, but it is not on that layer, so this page places the repertoire at the zero-rounds end and says so rather than borrowing biology the layer does not carry.
Milliseconds, rungs, bits and receptor event keys are never combined. There is no four-way axis on this page and no number claims to span the seam.

Run it yourself. Zero dependencies, no network: node research/filed-before-the-question/verify.mjs. It recomputes the closed forms, the sweeps, the frontier ladder, the reflex arithmetic, and it checks that the numbers printed on this page match the ones it just computed.

What this page does not establish

It does not establish that precomputation is the right choice anywhere. It measures what a choice costs, given that the deadline forced it. A system with time to spare should compute the answer.

It does not establish a common currency across the four stations. Two of them are measured in bits and rounds on a shared bench; the other two are reported in their own units and placed beside them because they face the same predicament, not because they sit on the same axis. Anyone who wants a four-way number will not find one here, and that is deliberate.

It does not establish that the seed tables here are the best possible tables. They are the best possible values for a uniform-in-x binning, which is what hardware does. A better binning exists and would shift the offset by a fraction of a bit without touching the exchange rate, which is the finding.

It does not establish anything about biological optimality. That a reflex has no time for a correction round is arithmetic. That this is the best design an organism could have is not a claim this page makes.