Quantum, Quant, Quark :
Quantum, Can’t, Crack
on rebuilding the tables of one through eleven, the unit as a choice, & a small dream about confinement
there is a sequence i kept returning to.
11, 25, 39, 52, 66, 80, 94, 107, 121 …
it looks like the table of eleven and it isn’t. the table of eleven is 11, 22, 33, 44. this one drifts. it walks past its own multiples and then, every so often, lands back on one exactly. 11 is 11×1. 66 is 11×6. 121 is 11×11. 176 is 11×16. the landings are themselves a table .. 1, 6, 11, 16 .. a table of five sitting inside a table of eleven like a smaller animal inside a larger one.
and the residues, taken mod 11, do this:
**0, 3, 6, 8, 0, 3, 6, 8, 0, 3, 6, 8 …**
forever. no drift, no decay, no eventual scramble. four states, closed.
this is my blog & not a paper, so i’m allowed to follow it where it goes rather than where it can be defended. what follows is one third arithmetic i checked / one third structure i built on purpose / one third a dream about particle physics that i know is a dream. i’ll mark which is which as i go. worth keeping the habit even off the record.
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## i. the closure condition
start with the only thing that is really happening here.
take a modulus m. take a finite list of steps s1, s2, … sk. build a sequence by starting somewhere and adding the steps in cyclic order, forever.
then the residues mod m are purely periodic, with period dividing k, **if and only if**
> Σ si ≡ 0 (mod m)
that is the whole theorem. it is not deep. page one of any modular arithmetic course. but it is the exact reason the sequence above behaves, and stating it plainly turns a curiosity into a construction … because once you know the condition you can *build to it* instead of hunting for it.
for eleven the steps are 14, 14, 13, 14. sum: 55. and 55 = 5 × 11 ≡ 0. that’s it. that’s why it closes.
reparametrise. write each step as si = m + di. the d values are the drift .. how far each step overshoots the modulus. then the residues are just the running partial sums of the drifts, and the closure condition becomes the cleanest possible statement:
> **Σ di ≡ 0 (mod m)**
for eleven: d = (3, 3, 2, 3), summing to 11. residues 0, 3, 6, 8, then +3 = 11 ≡ 0. closed.
so the sequence *looks* like the table of eleven, every step roughly eleven, but the residues walk a fixed closed orbit determined entirely by how you partition eleven.
**the drift pattern is the object. the sequence is its shadow.**
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## ii. reigniting one through eleven
now do it for every modulus from one to eleven. take the drifts to be a partition of m into four parts, as equal as the number allows. steps become m + di. the sequence looks like that number’s table, walking with a limp.
reading as: **m / drift / steps / sequence / residue cycle**
1 / (1) / 2 / 1, 3, 5, 7, 9, 11 … / 0
2 / (1,1) / 3, 3 / 2, 5, 8, 11, 14, 17 … / 0, 1
3 / (1,1,1) / 4, 4, 4 / 3, 7, 11, 15, 19, 23 … / 0, 1, 2
4 / (1,1,1,1) / 5, 5, 5, 5 / 4, 9, 14, 19, 24, 29 … / 0, 1, 2, 3
5 / (2,1,1,1) / 7, 6, 6, 6 / 5, 12, 18, 24, 30, 37 … / 0, 2, 3, 4
6 / (2,2,1,1) / 8, 8, 7, 7 / 6, 14, 22, 29, 36, 44 … / 0, 2, 4, 5
7 / (2,2,2,1) / 9, 9, 9, 8 / 7, 16, 25, 34, 42, 51 … / 0, 2, 4, 6
8 / (2,2,2,2) / 10, 10, 10, 10 / 8, 18, 28, 38, 48, 58 … / 0, 2, 4, 6
9 / (3,2,2,2) / 12, 11, 11, 11 / 9, 21, 32, 43, 54, 66 … / 0, 3, 5, 7
10 / (3,3,2,2) / 13, 13, 12, 12 / 10, 23, 36, 48, 60, 73 … / **0, 3, 6, 8**
11 / (3,3,2,3) / 14, 14, 13, 14 / 11, 25, 39, 52, 66, 80 … / **0, 3, 6, 8**
look at the last two rows.
ten & eleven produce the *same residue cycle*, 0, 3, 6, 8, from different drifts, different steps, different sequences. the orbit i started from is not a property of eleven. it is a property of the shape (3, 3, 2, …) applied to any modulus large enough to hold it. eleven closes it with a final 3 / ten closes it with a final 2. same orbit, two different exits.
that is worth sitting with. i had assumed the pattern belonged to eleven. it belongs to the partition. eleven was the room it happened to be standing in.
also note seven & eight, both giving 0, 2, 4, 6. eight gets there with perfectly uniform drift (2,2,2,2), an arithmetic progression that closes exactly. seven gets there limping (2,2,2,1). different machines, indistinguishable from outside if you only ever look at the residues.
**many machines / one output.** hold that. it comes back twice. once as beauty & once as a knife.
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## iii. changing the identity of one
> oṃ pūrṇamadaḥ pūrṇamidaṃ pūrṇāt pūrṇamudacyate |
> pūrṇasya pūrṇamādāya pūrṇamevāvaśiṣyate ||
> that is whole. this is whole. from the whole, the whole arises.
> take the whole from the whole … the whole alone remains.
> ( īśopaniṣad, śānti-pāṭha )
i have heard this recited my whole life as a statement about the infinite & only recently as a statement about *units*. because read arithmetically it is a claim that the one is not diminished by subtraction. which is exactly false for the integers and exactly true for the thing i want to build.
what if a number is not its value. what if a number is *the list of everything it leaves behind*.
take n. compute its residue against every modulus from one to eleven. that vector
> r(n) = ( n mod 1, n mod 2, n mod 3, … , n mod 11 )
is a complete alternative name for n. by the chinese remainder theorem, keep only the pairwise-coprime moduli, say 8, 9, 5, 7, 11, and the map is a bijection onto a range of size 8 × 9 × 5 × 7 × 11 = 27,720. every integer below 27,720 has exactly one such name & the name determines the number.
and inside this system, **one is no longer one.**
> r(1) = ( 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 )
one has become a vector. it is not the unit any more in the sense of being the thing you count with. it is one address among many, distinguished only by being the address where almost every coordinate happens to read 1. the multiplicative identity survives, coordinatewise multiplication still fixes it, but its *primacy* is gone. it is no longer the seed from which the others are built by repetition. it is a peer.
that is the reidentification. the unit stops being the generator & becomes a member. take the whole from the whole & the whole remains … but it is no longer standing where it was.
now collapse harder. throw away the residue values. keep only whether each residue is zero. one bit per modulus:
> b(n) = ( [1|n], [2|n], [3|n], … , [11|n] )
reading as **n / word**, moduli 1 through 11 left to right:
1 / 1 0 0 0 0 0 0 0 0 0 0
2 / 1 1 0 0 0 0 0 0 0 0 0
3 / 1 0 1 0 0 0 0 0 0 0 0
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4 / 1 1 0 1 0 0 0 0 0 0 0
5 / 1 0 0 0 1 0 0 0 0 0 0
6 / 1 1 1 0 0 1 0 0 0 0 0
7 / 1 0 0 0 0 0 1 0 0 0 0
8 / 1 1 0 1 0 0 0 1 0 0 0
9 / 1 0 1 0 0 0 0 0 1 0 0
10 / 1 1 0 0 1 0 0 0 0 1 0
11 / 1 0 0 0 0 0 0 0 0 0 1
12 / 1 1 1 1 0 1 0 0 0 0 0
13 / 1 0 0 0 0 0 0 0 0 0 0
there is the parallel 0/1 system. every integer becomes a binary word. addition is gone, the encoding is not additive, but divisibility becomes *visible*, sitting on the surface as a bit.
and notice 13. its word is identical to 1's word. thirteen is prime & larger than eleven, so it leaves a remainder against everything in the window. inside this eleven-modulus system, **13 & 1 are the same object.** the encoding cannot see the difference. neither can it see 17, 19, 23 … every prime past the window collapses onto the same word as unity.
this matrix of divisibility bits is not new. it is the Redheffer matrix wearing different clothes. the object whose determinant equals the Mertens function, whose behaviour ties straight to the Riemann hypothesis. i did not go looking for that. i went looking for a way to change the identity of one & arrived at a structure number theory has been staring at for a century. which is roughly what always happens. no one invents anything. you walk far enough in a direction and find footprints.
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## iv. the ladder \ one to nine, then eleven, then back down through eight
here is the architecture. this part i built deliberately rather than found.
the circuit runs **1 → 9, then 11, then back from 8.** not arbitrary. each leg does a different job & the reason is base ten.
**first leg, one through nine.** these are the digits. every symbol the base has. building the unit against each of them in turn is section ii, the eleven tables, the drift partitions. nine tables / nine closed orbits / the unit expressed nine ways.
**then nine & eleven together, because they are the mirror pair.** this is the piece i think is genuinely lovely & it is old arithmetic. in base ten:
10 ≡ **+1** (mod 9) … so every power of ten is ≡ 1, and n mod 9 is just the **digit sum**. casting out nines. the digital root.
10 ≡ **−1** (mod 11) … so powers of ten alternate +1, −1, and n mod 11 is the **alternating digit sum**.
nine & eleven are the two neighbours of the base, one below and one above, and they are the only two moduli that see the whole number’s digit structure at once. one sees it plainly / one sees it in alternation. a number & its reflection.
> 9 \ 10 / 11
**ten itself is blind.** n mod 10 is the last digit & nothing else. the base cannot see itself. it is the one modulus in the neighbourhood that tells you almost nothing. which is why the ladder steps over it. you go 9, then 11, and you skip the thing you are standing on.
**then the fall back through eight.** 1000 = 8 × 125, so every power of ten from the fourth onward vanishes mod 8, and n mod 8 is exactly the **last three digits**. eight is where decimal hands you back to binary. 8 = 2³, and from eight you fall to four, to two, to one. the powers of two. the 0/1 system from section iii, arrived at from above instead of built from below.
so the circuit closes:
> **9** gives the digit sum \ **11** gives the alternating digit sum \ **8** gives the binary tail \ and 8 → 4 → 2 → 1 returns you to the unit, changed.
and the three are pairwise coprime, so CRT applies clean:
> 8 × 9 × 11 = **792**
three residues .. one digit sum, one alternating sum, one three-digit tail, all computable by hand faster than you could do the division .. pin down any integer modulo 792 uniquely. that’s the whole loop in one number.
my earlier work kept running aground @ mod 8 specifically. there’s an impossibility result i proved there that i spent weeks resenting. it reads differently now. mod 8 isn’t a wall in the ladder. it’s the rung where the ladder turns around.
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## v. quantum to quark
now the dream. flagged as a dream.
**a quantum is the smallest amount of a thing you can have.** the word means *how much*. it is fundamentally a measure. Planck’s discovery was that energy comes in countable lumps & the lump is a quantity you can, in principle, isolate and count.
**a quark is the smallest thing you can never have alone.**
that’s colour confinement & it’s the strangest fact in the standard model. no experiment has ever produced a free quark. try to pull two apart and the field between them doesn’t weaken with distance the way gravity or electromagnetism do. it forms a flux tube whose energy *grows* as you separate them. keep pulling & eventually the tube holds more energy than it costs to make a new quark-antiquark pair, so the vacuum simply manufactures one. the tube snaps. you now have two bound pairs. you have never, @ any point, held a single quark.
and the inverse holds too. **asymptotic freedom.** the closer two quarks are, the *weaker* the force between them. nearly free when nearly touching / bound hardest when far apart. exactly upside down from everything else.
so, the wordplay, & the actual thought under it:
**a quantum can be isolated & counted. a quark cannot be isolated @ all. what would it mean to make a number behave like a quark?**
it would mean a unit that has no meaning outside its relations. r(1) is not a value, it is a position held simultaneously against 8, 9 & 11. pull it out of the system to examine it on its own and there is nothing there … the way 13 & 1 collapse to the same word the moment your window is too small. the information doesn’t live in the object. it lives in the tension between the moduli. cut the tube & you don’t get the unit, you get two new relations where there was one.
that is the confinement dream. not “encrypt the number” but *make the number un-isolatable in principle*.
and here is the honest part. one paragraph, then i stop.
it doesn’t work. Shor’s algorithm doesn’t care what notation you factor in. factoring is a property of the integer, not of how you wrote it down, & a change of representation is a change of clothes. worse: if the metaphor did hold it would hold backwards. asymptotic freedom says the closer you look the freer the thing gets, and looking closely is precisely what a quantum computer is for. the picture argues against itself if you push it.
but as a statement of *what you would want* .. not a stronger lock but a secret with no isolable interior .. i think confinement is the right shape of wish. real cryptography does something adjacent in its way. in lattice problems the hardness isn’t in any coordinate, it’s in the relation between all of them @ once, & no single coordinate leaks anything. that isn’t confinement. but it rhymes.
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## vi. what it actually gave
not a lock. i should say that plainly since i started out wanting one.
a small theorem about closure, which generalised to every modulus & turned out not to belong to eleven @ all. a ladder from one to nine to eleven & back through eight that closes on 792 and is, as far as i can tell, a pleasant way of noticing what base ten already looks like from the side. a reidentification of one that dissolved the unit into a vector & then collapsed every prime past the window onto that same vector. and a bit-matrix that walked me into the Redheffer matrix without asking.
**and the knife.** back in section ii, seven & eight produce the same residue orbit by different means. i read that as richness. an attacker reads it as: the output space is smaller than the machine space, so guessing the output is cheaper than guessing the machine, & for most attacks the output is all you need. same fact, twice. which one it is depends entirely on which side of it you’re standing.
the error i keep making & keep finding instructive is mistaking **unfamiliarity for hardness.** a system nobody has looked @ isn’t strong. it’s unexamined. those feel identical from the inside & are opposite from the outside, and the only instrument that separates them is other people attacking your work in public. eight years of that is what the post-quantum standards went through. several candidates were broken during the competition, by rivals, in the open. SIKE survived years of scrutiny & then fell in an afternoon to Castryck and Decru, on a laptop, after everyone had already believed in it.
so the closing note, & the reason i keep returning to one particular verse:
> caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām |
> ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ ||
> add four to a hundred, multiply by eight, then add sixty-two thousand:
> this is the *approach* to the circumference of a circle of diameter twenty thousand.
> ( āryabhaṭa, gaṇitapāda 10 )
62832 / 20000 = 3.1416. and the word he chooses is **āsanna** … *near, approached, come close to*. not “this is π.” near to π.
the honesty is inside the notation. not a caveat appended @ the end, not a disclaimer in smaller type. the epistemic status is a grammatical feature of the claim itself. fifteen hundred years ago somebody built doubt into the sentence structure & we have been steadily removing it since.
my own architecture proposal has a line i’m fonder of than anything else in it: *every number in this document that is not cited is a target.*
same instinct. fifteen centuries apart. i’ll take that as the lineage.
the residues close @ 0, 3, 6, 8. that part is true. i checked it, & then i checked what it was true *of*, and it was true of less than i wanted and rather more than i expected.
that’s usually the trade.
Always a humble navigator in the vast ocean of mathematics : Narayan pandit as said.
Suggest read: “Ganit Kaumaudi”
/rrr
