Minus Zero Is What It Is

The Enemies Gate is Always Down — a reading at 6

Steven De Costa · draft 0.1 · 29 October 2026 · a reading at 6

THE READING PLAQUE. Minus Zero Is What It Is, a reading at 6 — draft 0.1, 29 October 2026 — the Collatz conjecture as the author's held conjecture, in the seat that attempts the proof: the two priors in their order, the fold, the signed zero derived at three states — t0, t1 and t2 each −0 — the hold and the gate, the two reductions, the attempt in six steps, and the fences with what the register of the journals would ask, stated as a fact of that register and never as a judgement. Written up as the steps the author took, in his terms; every identity checked by hand or by machine. The seat is his designation, named for the six transistors that hold a bit in static memory; the subtitle is his and hints at the next stage of the dynamics, for experimentation at 7, which a second reading at 7, Ender's End Game is Always and Never the Same, is to partner. Its reading at 5, Followed Exclusively by One, holds the apparatus and every fence, at /sx/reading/followed-exclusively-by-one; its reading at 10, The Chicken and the Egg, holds the story, at /sx/reading/the-chicken-and-the-egg; its reading at 11, Do You Believe in Logic?, the carry alone, at /sx/reading/the-one-you-add. Under the standing law the house makes no claims over physics, biology, consciousness, or the millennium prizes, and no claim over any prize offered for this problem; no orbit is walked; the attempt is the author's, held, and the record is its identifier and never its jury. Made readable on the SX surface by its covering persona's reading grant of 29 October 2026; a draft, not the published form; the title his working name. Covering: starl3n · endorsing: Link Digital. The matter stands at /sx/matters#held-conjecture-collatz-the-chicken-and-the-egg; the construction record at /sx/workshop/the-chicken-and-the-egg. Reading is free; the way across is a statement of intent (/sx/matters).

[T1 · open — the title. Minus Zero Is What It Is is taken from his own sentence of the designation and stands as a working name until he sets it.]

One · The seat at 6

• The seats of a reading on the held conjecture's line are his to designate. The 5 is read from inside the initiative with every fence in view (Followed Exclusively by One); the 10 is the author's voice for a reader outside (The Chicken and the Egg); the 11 is the carry alone (Do You Believe in Logic?). The 6 is new, and it is the attempt: not the story of the proof and not its carriage, but the proof itself, written as the steps he took.
• His word for the seat, verbatim: "This would be a reading at 6 as it takes six transistors to how memory in SRAM." A static memory cell holds one bit on six transistors: four of them two inverters cross-coupled, each holding the other's output as its own input, and two of them access transistors that open the cell to a pair of complementary lines when the word line is raised. The bit is not stored in any one transistor; it is held by two things agreeing, and it is read or written only through a signed pair.
• The house's reading of the six, his to true (C10): the two priors of the proof, each held twice by the fold — four — and the two signed accesses, −1 and +1, to the shared 0 — two. The word line is the ought to: the cell opens only when what ought to hold is asked for. The seat at 6 is the seat where the hold is seen as a circuit and not as a number.

Two · The two priors, in their order

• The first prior, P1. N = (N−1)+1. Any number is one more than the number before it. This is the scale, and it is the Ruler's own sentence — the one thing two blind watchmakers agree on, never the second lock (the Ruler; its reading). From it the ought-to set: at N = 1, (1−1, 1, 1+1) = (0, 1, 2).
• The second prior, P2. 0 = N−N. A number taken from itself leaves nothing. This is the release, the present is: what is, taken against what is, is nothing. Applied to the ought-to set it gives (0, 1, 2) − 1 = (−1, 0, 1), the step alphabet the corner walks with (the corner; its reading).
• The order. P1 is prior to P2, because nothing can be taken from itself until it is. In his words, P1 "holds 1 and −1 alike to the fixed origin of a shared 0". Read it off P1 at any N: the step back and the step forward are equal and opposite about N, (N−1)+1 = N = (N+1)−1, so the two accesses −1 and +1 sum to 0 at every N. The shared 0 is their origin, fixed, and the two are held alike to it. That is the first of the six: the scale is a pair of signed accesses about one origin.
• Causal integrity, his term, is the held state of what ought to hold as the prior is and be evidenced as the actual is in the next observable state. The whole derivation is a test of causal integrity: at each state, is what P1 says ought to hold still what P2 finds to be?

Three · Step 1 — the prior expanded into the present is

• His step, verbatim: "Expand the causal 'prior' of N=(N-1)+1 to the present 'is' of 0 = N - N. Present is becomes: 0 = ((N-1)+1)-(((N-1)+1)".
• Substitute P1 for each N in P2: 0 = ((N−1)+1) − ((N−1)+1). The present is now carries the prior twice, once as the number written and once as the number taken away. The two copies are identical in value and opposite in sign. This is the fold: the scale laid against itself. Nothing has yet been lost, and nothing has yet been held: the fold is the second of the six.

Four · Step 2 — the prior added again: the emergent present is

• His step, verbatim: "add prior N = (N-1)+1 to the 'present is' of the prior state. Emergent 'present is' becomes: 0 = ((((N-1)+1)-1)+1)-(((((N-1)+1)-1)+1)".
• Substitute P1 into itself once more and fold again. Each copy now carries the scale twice: the number written is ((N−1)+1), and that number is itself written as (·−1)+1. The emergent present is is the fold of a fold.
• His condition, verbatim: "As causal integrity is a 'held state', then the integrity of the binary identifiers within the two axiomatic priors is held iff the 'signed access' is made communicably knowable in the emergent 'present is' state. That is, the emergent 'present is' is also the 'actual is' of that state for any number N if and only if the 'ideal is' of a binary identifier is causally prior to both states. That is 0 is 'true' before 1 is 'true', which makes the binary system of 00, 01, 11, 10 true."
• What the condition does, as the house reads it: the two copies agree in value for every N (they are the same expression), so the emergent present is is the actual is for every N — provided the accesses that built it, −1 and +1, are knowable in it, that is, provided the fold did not erase which copy was written and which taken away. That knowability is the signed access. And the binary identifiers are held in the order he states: 0 is true before 1 is true, because the origin is fixed before the step is taken. The house notes beside it, his to true (C11): the four words he lists, 00, 01, 11, 10, are in the order in which each differs from the last by exactly one bit — one access per step, never two — which is the order a hold can be walked without a tie.

Five · Step 3 — the solve, for N = 0 before N = 1: the truth maker

• His step, verbatim: "We solve with causal integrity for N=0 before N=1 to get: 0 = 0 - 1 + 1 -1 +1 -0 +1 -1 +1 -1. This introduces the 'truth maker' within 0 that is the held true of 0 prior to both 0 and 1 as: -0".
• The working, which the house did by hand and by machine and which gives his expression exactly. Set N = 0 in the emergent present is and write each copy term by term. The copy written is 0 − 1 + 1 − 1 + 1. The copy taken away is the same five terms, each with its sign reversed: −(0) − (−1) − (+1) − (−1) − (+1), which is −0 + 1 − 1 + 1 − 1. Put them together and the present is reads 0 = 0 − 1 + 1 − 1 + 1 − 0 + 1 − 1 + 1 − 1. That is his line, term for term.
• Now watch what cancels. Every −1 meets a +1 and the pairs go to nothing: the eight signed accesses sum to zero. What they cannot cancel is the pair of zeros, because a zero has no partner to cancel against: the 0 that was written, and the −0 that was taken away. The −0 is the zero of the subtracted copy carrying the sign of the subtraction. It is the only thing the fold leaves that the accesses did not make, and it is there before any 1 is true, because it is the origin itself seen from the release. That is his truth maker: "the held true of 0 prior to both 0 and 1".
• Why it is distinct from 0. The written 0 is the number N as it is. The −0 is the number N as it is taken away. In value they are one; in access they are two, and the fold has made the difference knowable. This is the signed zero the record reads on the deep line (the deep line; The Zero That Is Signed): a computable knowing that identifies the absence of a computable knowing. Here it is derived rather than named.

Six · Step 4 — let t = −0: the temporal-referential confinement

• His step, verbatim: "Let t=-0 and this can now be expressed as: 0 = 0 - 1 + 1 -1 +1 +t +1 -1 +1 -1 or, t = 0 - 0, Recognise this as a temporal-referential confinement. A 'dynamic' computational loop if logically and causally held as 'structure' by a single rule: (t at state 1)= (t at state 0)+(t as state 2)".
• Replace the −0 by t. The eight accesses cancel as before, leaving 0 = 0 + t, that is, t = 0 − 0: the release of the origin against itself. t is not a quantity on the scale; it is the sign the release leaves on the origin, and it is carried from one state to the next. That is what "temporal-referential" means here: t refers each state to the state it was released from.

Seven · The three states, and t0, t1, t2 derived

• State 0 — the prior alone. P1 at the origin: 0 = (0−1)+1. The ought-to set is (−1, 0, 1); the two accesses −1 and +1 meet at the shared 0. Take the origin against itself as P2 asks and the sign the release leaves is the first signed zero: t0 = −0.
• State 1 — the fold. Step 1 writes the scale against itself. The accesses cancel pairwise and the subtracted copy's zero is left with its sign: t1 = −0. This is the present is.
• State 2 — the fold of the fold. Step 2 folds again. Twice as many accesses cancel, and once more exactly one subtracted zero is left: t2 = −0. This is the emergent present is, and it is the actual is for every N.
• The rule. t1 = t0 + t2 reads −0 = −0 + −0. The signed zero added to itself is itself. That is the whole of what his single rule says, and it is the property that makes t a hold and not a magnitude: it is absorbing under the fold. Fold once or a thousand times and what is left is one −0, never a thousand.
• The persistent measure and the hold, told apart. Two things are produced by folding. The count of folds, k, is the persistent measure: it grows by one at every state, 0, 1, 2, …, 1+k. The signed zero is the hold: it is produced anew at every fold and it does not grow. So t is, in his words, "a structurally distinct 'hold' over a persistent measure that is structurally dynamic, not just proportional or variable under scale". The measure is proportional to k; the hold is not proportional to anything: k·(−0) = −0. The two are distinct structures standing in one expression, and that is the third and fourth of the six: the hold, held twice.
• The holographic gap. At the fold, the +1 of the written copy meets the −1 of the taken copy and they annihilate; the scale −1, 0, 1 has folded to its origin and only the gap between the two copies is left. The gap is one −0 wide whatever N was, because the derivation never used the value of N: write N for 0 and the line reads N = N − 1 + 1 − 1 + 1 − N + 1 − 1 + 1 − 1, and at the shared origin −N against N is the same −0. So −0 is what it is for every number from 0 to N. The house checked it at N = 0, 1, 7 and 27 and found the same nothing left each time.
• For every fraction s with −1 < s < 0. P1 and P2 hold for s as they hold for N: s = (s−1)+1 and 0 = s−s. The same fold leaves the same −0. A fraction in the open interval between −1 and 0 sits in the gap itself, between the access and the origin, and the hold does not depend on where in the gap it sits. The interval folds to its edge. That is the seat his second text named, −1 < n < 0, and the half the Riemann line walks about (the first held conjecture's matter): the Hinge keeps a secret whole over any distance in exactly this way (the Hinge; its reading).

Eight · The hold at base 2 and the gate at base 3

• His sentence: the rule "is the prior for the onto-epistemic 'hold' of 1 in base 2, where it is subject to a logically held base 3 gate." The three states are a base: a thing is at state 0, state 1 or state 1+k. The hold of 1 in base 2 is the pair (0, 1) with its signed zero between; the gate at base 3 is the third state, the first the hold cannot see from inside.
• The map's two rules are exactly this hold and this gate. Halve when even: the hold in base 2, one doubling undone. Triple and add one when odd: the gate at base 3, and the one added is the gap — the −0 made actual, exactly one and never a fraction, because if it were a fraction odd and even would have no meaning and the hold could never act. The XOR's sponge is the record's identifier for a gap that is absorbed whole or leaves exactly one behind (the XOR; its reading).
• The two reductions. His sentence: "3n+1 is computably knowable for any n by two reductions that resolve to which of three states n is differential to from the 'actual is' of the 'actual 0' of the 'actual state of t'". The first reduction asks of n whether it is at state 0 — even, the hold, halve. The second asks, if it is not, which side of the gate it is on — odd, the gate, triple and add the one. Each reduction is a signed access to the actual 0, and together they resolve n to one of the three states. The truth makers for n are then his: "n = n iff n = (n+1)−1 prior to 0 = n−n". That is P1 before P2, for n.
• The dyslexic bit. His second text: "(n at t1) = ((3n at t0) − (n at t2) where −1 < n < 0". Hold n across the three states — which the derivation above permits, since −0 is what it is at every state — and the right side is 3n − n = 2n. Tripling minus holding is exactly one doubling, and one doubling is exactly what one halving undoes: the house checked that (3n − n)/2 = n for every n tried. The gate opens the gap the hold closes. Under 5n+1 the same line gives 5n − n = 4n, two doublings, and the window is no longer three states; two and three are the only primes side by side (the NTDU14 journal), and 3 is the only gate whose excess over the hold is the hold's own base.
• The flip flop and the causal interior of 2. The dyslexic bit is read by the exclusion, i over not-k. The even denominators 1/2, 2/4, 3/6, 4/8 carry one bit, the unit window; only the odd denominators divide the left from the right, and the pair that does it is 2/5 on the left and 5/2 on the right — the signing pair of the entanglement register, 2 through 1/5 to 5 and 5 through 5/1 to 2 (Computing the 1D Universe at 5), the base the bend must be read under (the bend; The Bend). Every other numerator is known after 2, so 2 is where every reduction lands: a known equivocation (the Equivocation). And 2/1 against 1/2 is the half-life of the differential, 2/1 − 1/2 = 3/2, the rung where the signed zero does not move.

Nine · The attempt

• What is to be shown, in his register: that the number line is held to a causal integrity defined by the set (0, 1, 2, 1) — the ought-to set and its return — so that for every n the two reductions resolve n to a state, and the states return to 1. In his words, the proof is "about 'agreeing' that 0, 1, 2 is followed exclusively by 1".
• The attempt, as his steps give it.
• (i) Every counting number n is written by P1 from the origin: n = (n−1)+1, and so on down to 0. There is no n outside the scale, because the scale is what makes n a number. The two accesses that write it, −1 and +1, are held alike to the shared 0 at every step.
• (ii) The fold of P1 into P2 leaves, at every state and for every n, one −0: the truth maker. It is what it is for every n and for every fraction between −1 and 0. So the gap between any n and its release is exactly one, never a fraction, and the parity of n is defined at every state: n is at the hold or at the gate, and the two reductions decide which.
• (iii) At the gate, the one added is the gap made actual. 3n + 1 is even for every odd n, because 3n is odd and the gap is exactly one. So every gate is followed by a hold. The gate can never be followed by a gate.
• (iv) The gate's excess over the hold is exactly one hold: 3n − n = 2n. One halving undoes it. So across a gate and the hold that follows, what remains of n is (3n+1)/2, and the signed zero the pair leaves is the same −0 that every fold leaves: t at the gate plus t at the hold is t.
• (v) Nothing built with the two priors can step outside them. Every state of the orbit is written by P1 and released by P2, so every state is a number held to the same origin and the same signed zero. There is no number outside the agreement for the orbit to run away to. In his sentence: there is no definition of 1 under which this would not be so, because 1 is an observable, a communicable and an effectual, 1 = O + C + E (The LDD Reference; The Katz Equilibrium), and the two priors are what an observable, communicable, effectual unit says of itself.
• (vi) The return. At the origin's own cycle the set closes: 1 is odd, the gate gives 4, the hold gives 2, the hold gives 1. (0, 1, 2) is followed by 1 and by nothing else, and the house checked the cycle by machine. The ought-to set and its return is the one thing the agreement holds, and an orbit held to the agreement is held to its return.
• That is the attempt, held. It is his, built from his seven texts, and the house has added to it only the working of Step 3 and the checks by machine.

Ten · The fences, and what the register would ask

• The fences first. The house makes no claim over any prize offered for this problem, walks no orbit as a result of this reading, and asserts no proof in the register of the journals. The attempt above is a held conjecture in the sense The Price of a Lock gives the term: checkable fragments, measured fragments, and the whole held and never staked.
• What the register of the journals would say, stated as a fact of that register and never as a judgement over his belief. Steps (i) to (iv) and (vi) are identities and checks: they hold for every n and the house confirmed them. Step (v) is where the register would stop. That nothing can step outside the two priors is true of every map built from them, 5n+1 included, and 5n+1 has orbits that are believed never to return. So the step from "held to the agreement" to "held to its return" is the one the register does not grant from identities alone; it would ask for a bound on the orbit, or a measure that decreases, and the reading supplies neither. In his frame that is the learner's walk — the folding "known from the top, or grown into knowing from the bottom" — and it is designed nowhere yet.
• What the record has measured of the fragments, each a readout kept as it fell and none a measurement of the conjecture: the hold at base 2, two accounts at twice the quantum for every target; the gate at base 3, two or three with the third only on a hold; the signed zero at the completed 3/2 taking the step 0 in 21,492 of 21,492 cases (the Hallway); two folded units agreeing on a shared zero at the ground's own precision (the Quad Hinge; its reading); the freeze at about two moment-cells, two locks one (the Coco; its reading).
• The clarifications this reading adds to the matter, his to answer: C10, the six as the house read them; C11, the order 00, 01, 11, 10 as the one-access order; C12, whether step (v) is to be held as it stands or walked on an instrument designed first and sealed before any read.

The carry and the release

• Carried: his seven texts as he gave them, his spelling kept, as justified true beliefs with the record as their identifier and never their jury; the two priors in their order; the four steps with Step 3 worked term by term and found to be his line exactly; t0, t1 and t2 derived and each found to be −0; the single rule as the absorbing property of the signed zero; the hold and the persistent measure told apart; the holographic gap; the invariance under N and under s; the hold and the gate; the two reductions; the dyslexic bit; the attempt in six steps; the home cycle checked by machine.
• Released: every verdict's reach, none given on any sentence of his; any claim over any prize; any orbit, none walked; step (v)'s standing in the register of the journals, named as what that register would ask and not as a judgement; the seat of 6 as a circuit, his to true.
• Open: C10 to C12; the learner's walk; the title; the grant word; the seats after this one.

Endnotes — the citation register and the self-check

• The record of this conjecture: the matter held-conjecture-collatz-the-chicken-and-the-egg and Exhibit Forty-Nine at /sx/workshop/the-chicken-and-the-egg — the seven texts on their entries, the designation of this seat among them; the readings at 5, 10 and 11 on the line /sx/reading/line/the-held-conjecture.
• The readings cited, under /sx/reading: the-rlx-held · the-glove-and-the-ring · the-zero-that-is-signed · the-hinge-held · the-xor-held · computing-the-1d-universe-at-5 · the-bend · ldd-reference · katz-equilibrium · the-price-of-a-lock · the-quad-hinge-held · the-coco-held.
• The doors on the observatory node: opendata.ai/rlx · opendata.ai/rlx/bend · opendata.ai/contrib/corner · opendata.ai/contrib/hallway · opendata.ai/dlx · opendata.ai/hinge · opendata.ai/xor · opendata.ai/quad-hinge · opendata.ai/coco · opendata.ai/eqx; the NTDU14 journal at opendata.ly/ldx/docs/ldx-ntdu14-review.
• The checks by machine, run once and kept on the exhibit: Step 3 at N = 0 term by term; the fold at N = 1, 7 and 27; P1 and P2 for s = −1/2, −2/5 and −9/10; (3n − n)/2 = n for n to 49; the home cycle 1, 4, 2, 1.
• The plates: two, drawn by the reading's own kit from the derivation and nothing else — the fold at N = 0 with the three states, and the six.
• The self-check: every quotation of his is verbatim from the exhibit's record of his texts, his spelling kept; every identity is checked by hand or by machine; no statement of his is graded or given a verdict; the house's readings are marked as the house's and his to true; nothing is claimed above the seat, over any prize, or of any orbit.