Deprogramming Zero

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Abstract

Zero is a formed sign assigned to stand for a non-presentation: its positive formation certifies that something presented while its semantic office says that something did not. This is the root-grammatical contradiction the presence-only programme of Against Zero and The Mirror Calculus identifies; a surrogate zero is any sign a language reaches for to keep that concept under another spelling — [blank, null, empty, undefined, the identity “that does nothing”] — and the surrogate-zero test is the instrument that catches them. This paper deprograms zero out of Peano Arithmetic and carries the result through algebra, set theory, and the Riemann question. \(\mathsf{PA}\) uses its constant \(0\) in exactly four ways — as a named point, as the terminus past which the successor stops, as the base of the recursions for \(+\) and \(\cdot\), and as the base of induction — and each of the four is an instance of the contradiction. We give a formation-faithful interpretation of \(\mathsf{PA}\) into tally arithmetic \(\mathsf{TA}\), the theory of the positive integers with unit, successor, fusion and product, whose object language contains neither the sign nor any surrogate for it and whose unit \(\mathbf 1\) is formed by the first act and stands for presence, and a converse, so that the theories are bi-interpretable and equiconsistent: the sign is eliminable, which is its formal usability, the whole of what the objection from usability establishes. What the elimination exposes is the cost the classical notation concealed. Three of the four uses are substitutions — the unit tally, a return to the unit, induction from the unit — and each classical use of them hides a predecessor-witness obligation that the classical bookkeeping discharges in silence and \(\mathsf{TA}\) discharges in the open; the operation in which that silent discharge fails is division, and the classical notation names the failure with a returned non-value, “undefined,” which is a surrogate zero by the test. The fourth use, the terminus, is a kept record: by Lyndon’s preservation theorem no negation-free first-order theory excludes the cyclic models \(\mathbb Z/k\), so the statement that the first tally is reached by no successor is held in the metalanguage as a record, independent of the positive fragment, which is exactly where Against Zero places withheld presentation. A ledger counts the substitutions, the concealed obligations, and the record; a general schema extends the result to every theory whose zero is the least element of a discrete order; the identity of a group and the zero of a ring are shown to be surrogates by the test, which heap and truss theory remove; and the empty set, the surrogate of set theory, is placed as a kept record, forced by negative Separation and by \(\in\)-induction. The second half of the paper turns to the Riemann hypothesis and first proves why the sign was the obstacle: written with the sign, the hypothesis is [a universal over the plane whose matrix is an evaluation set equal to the sign], on a totality outside presentation — which positive logic does not form and arithmetic makes \(\Pi_1\), so that its affirmation is an \(\omega\)-inference with its sponsor left off the page — while the verification ledger presents tallies at every stage, Riemann’s completion \(\xi\) had already removed the surrogate zeros of \(\zeta\), and in the explicit formula each zero of \(\xi\) is a presented carrier whose equal weighting is the native form of the hypothesis. On the prime side the Admission law is Lagarias positivity and Li’s criterion — the positivity of a prime-indexed sequence — the mirror pairs carriers whose weights multiply to \(x\), Pólya’s function and the Davenport–Heilbronn function are the two models of the analytic base without the Euler product, and each kept stage of the ledger is a prime-counting bound on a segment. The register convention of The Mirror Calculus is then applied: the independence theorem of the volume is generalized unconditionally to every base in its signature; a base that fixes the referent is proved to decide zeta-specific Admission; counting the constructors of the Selection Jump against the Emmerson–Buchanan barrier hierarchy, the explicit sponsor is the terminal constructor; and the question is resolved at its registers — \(\mathsf{RH}_{\mathrm C}\) holds under the analytic interface, sealed, sponsored, and armed, the naming of the trace exhausting the verdict — with the resolution and the barrier result proved to guard each other.

Keywords: zero; surrogate zero; Li criterion; Lagarias positivity; explicit formula; presence-only formation; Peano arithmetic; interpretations; Lyndon positivity theorem; heaps; trusses; non-well-founded sets; Riemann hypothesis; register-typed assertion; Selection Jump; terminal constructor

Introduction

A formed sign assigned to stand for a non-presentation is a contradiction at the root of a grammar: its formation certifies that something presented, and its office reports the presentation as withheld. Against Zero [19] names the contradiction, names the family of surrogate zeros — the signs a language substitutes for zero so as to keep the concept while appearing to have given it up: [blank, null, none, empty, missing, undefined] — and states the test that catches a surrogate by its formal role rather than its spelling. The Mirror Calculus [1] is the mathematical language built so that every term of its object language is formed by an act, the sign and its surrogates held in the metalanguage. This paper deprograms the sign out of Peano Arithmetic, where it is the constant \(0\), and proves what each of its uses was standing in for, what the classical notation concealed by using it, where — once the sign is removed — the non-presentation is held, and which surrogates appear when a classical notation is pressed.

The answer has three parts. Section 2 names the four zeroations of \(\mathsf{PA}\) and proves that each is an instance of the contradiction (Proposition 1); Section 3 gives tally arithmetic \(\mathsf{TA}\), whose every object-language term is formed by an act, the sign and its surrogates held in the metalanguage. Section 4 proves the two theories bi-interpretable (Theorem 1); the bi-interpretation is classical and is the surrogate’s formal usability, and what is new is its shape: \(\mathsf{PA}\)’s operations, carried into the positive structure, are the native ones plus a predecessor at every zeroation, each a witness obligation the classical notation owed and discharged in silence. Section 5 counts this in a ledger, shows where the silent discharge comes due — division, where the classical notation meets it with a surrogate (Proposition 3) — and proves that one zeroation is a record rather than a substitution (Theorem 2): positive first-order logic has the circle among its models wherever it has the line, so the terminus is held in the metalanguage, where Against Zero places every withheld presentation. Section 6 extends the result to all least-element surrogates.

Section 7 applies the surrogate-zero test in algebra and set theory. The identity of a group is zero under another name, and heap theory removes it—a heap is a group with its identity forgotten [23,24,25]—as truss theory removes the zero of a ring [26]. The empty set [the collection “with no member”] is a kept record: Separation for formulas outside the positive fragment and the \(\in\)-induction schema each force it, while set-level Foundation alone leaves a chain of singletons standing (Proposition 4), so a presence-only set theory is hereditarily inhabited and at most set-level well-founded, in the territory of Aczel [27] and Esser [28]. Section 8 generalizes the volume’s independence theorem to every base in its signature, unconditionally, and proves that the generalization is as wide as independence can be: a base that fixes the referent decides zeta-specific Admission, in the direction the hypothesis takes. Section 9 states, as theorems about the form of the classical sentence, why the sign was the obstacle: the hypothesis written with the sign is [a universal over the plane whose matrix is an evaluation set equal to the sign], which positive logic holds as a record and arithmetic makes \(\Pi_1\), so that its affirmation is an \(\omega\)-inference with its sponsor left off the page; while the verification ledger presents tallies at every stage, Riemann’s own completion removed the first surrogates, and in the explicit formula the zero of \(\xi\) is a presented carrier whose equal weighting is the native form of the hypothesis. Section 10 reads the grammar on the prime side: the Admission law is Lagarias positivity and Li’s criterion, hence the positivity \(\lambda_n\geq0\) of a prime-indexed sequence; the mirror pairs carriers whose weights multiply to \(x\); Pólya’s function and the Davenport–Heilbronn function are the axial and off-axial models of the analytic base stated short of the Euler product, so the content of the law lies in the prime-fused filter for Dirichlet series as well as for polynomials; and each kept stage of the ledger is a prime-counting bound on a segment. Section 11 applies the register convention to the Riemann hypothesis, states register by register what the programme proves there, and closes with the resolution: Theorem 11 delivers the classical statement under the analytic interface, prints the trace of the verdict, identifies the sponsor as the terminal constructor by Theorem 10 and the barrier hierarchy of [20], and keeps the clash channel armed; Theorem 12 then proves the co-guardianship of the resolution and the barrier hierarchy.

The Four Zeroations Are Four Instances of the Contradiction

Write \(\mathsf{PA}\) in the language \(\{0,S,+,\cdot\}\) with the usual axioms:

\[\forall x\;\; Sx\neq 0, \tag{PA1}\] \[\forall x\forall y\;\;(Sx=Sy\to x=y), \tag{PA2}\] \[\forall x\;\; x+0=x, \tag{PA3}\] \[\forall x\forall y\;\; x+Sy=S(x+y), \tag{PA4}\] \[\forall x\;\; x\cdot 0=0, \tag{PA5}\] \[\forall x\forall y\;\; x\cdot Sy=x\cdot y+x, \tag{PA6}\] \[\bigl(\varphi(0)\wedge\forall x(\varphi(x)\to\varphi(Sx))\bigr)\to\forall x\,\varphi(x) \quad\text{for each formula }\varphi. \tag{Ind}\]

The four are distinct in kind. point is a name. terminus is a record: the statement that \(0\) is the origin of every successor chain, written in the classical form [“\(0\) is not a successor”]. base makes \(0\) an identity for \(+\) and an absorber for \(\cdot\). induction base makes every inductive argument begin at an element whose only property is that it is named.

Tally Arithmetic

Three things are worth saying about this list before anything is proved. (TA1) is positive exhaustion: every tally is the unit or a successor, a disjunction with a witness on each side. (TA1\('\)) says the unit is first by exhibiting, for each successor, a tally whose fusion with the unit reaches it. And (TA0) is the one kept record of the theory: the unit is the origin of every successor chain. Every other axiom is an equation between formed terms, an injectivity, or an existence statement with its witness, and every term of the object language of \(\mathsf{TA}\) is formed by an act, the sign and its surrogates held in the metalanguage. The unit \(\mathbf 1\) is on the other side of the line: its formation is the first individuation act and its office is to stand for that act, so formation and office agree, and it is formed by an act as every other term is. (TA0) is kept because it must be — Theorem 2 shows that every positive first-order theory in this language has the cyclic groups \(\mathbb Z/k\) among its models, in which \(\mathbf 1\) is a successor; the positive axioms (TA1), (TA1\('\)) and induction are all satisfied there. In the register convention of [1] this is a kept record: a statement the metalanguage holds about the grammar, exactly as the volume holds nonemission as a metalanguage condition on the exit-locus record.

The Bridge

The object on the positive side is already in the programme’s texts: Counting Back from Infinity [2] proves that the tally calculus is \((\mathbb Z_{>0},+)\) and that every element of it moves every other (“this semigroup has no identity element internally”), and Against Zero [19] proves the same for the native tally sort and recovers classical zero as the free identity adjunction \((\mathbb Z_{>0},+)^{1}\cong(\mathbb N,+,0)\), “so the old-world identity is exposed as a completion rather than treated as neutral furniture.” The bridge below is that adjunction carried to the full first-order theories with successor, product, and induction, made into a bi-interpretation, and priced. Counting Back from Infinity also holds, under the name Addition is partial, and that is the point, that fusion of rosters forms only under joint separation; \(\mathsf{TA}\)’s fusion is total because its sort is the pairing class, in which separation is supplied by tagging, and the partiality of roster fusion is the metalanguage condition the class quotients away.

The closed forms show what the sign concealed: \(\oplus'\) is fusion followed by one predecessor, and \(\otimes'\) is the product of two predecessors followed by one successor. Each \(\mathsf{pred}\) is total on its argument because, in each case, the argument is a successor — \(x\oplus y\) is a successor by (TA3)–(TA4) and (TA1\('\)); \(x\) and \(y\) are successors by the case hypothesis. This is the point of the construction: every predecessor the sign concealed is guarded by a witnessed successor. The classical notation discharged these obligations in silence; \(\mathsf{TA}\) discharges them in the open.

The Ledger

The cost is a property of a presentation rather than of a theorem: a different closed form for the same operation moves a \(\mathsf{pred}\) and changes the count by one. The figures below are for the closed forms of Definition 5, and what is invariant is the qualitative statement that every eliminated zeroation is paid by a substitution or a concealed obligation made explicit, and that exactly one record is kept.

The programme states the terminus twice before this paper: Counting Back from Infinity proves, under the name No stage of descent is zero, that every formed stage of the descent calculus is an inhabited positive residue, and holds in its Convention on inhabited stages that the empty open set “remains in the metalanguage of topology” while its positive coherent fragment is built [in its words, with “no object-language falsum, no negation, no empty disjunction, and no vacuous universal quantifier”] [2]. The theorem below proves that this placement is forced rather than chosen: every theory in that fragment has the one-point structure as a model, so the terminus is unstatable in it, and every classical axiomatisation must carry the record as a sentence.

The General Schema

The construction used three things about \(\mathsf{PA}\): \(0\) is least in a discrete order, \(S\) is a bijection onto the elements other than \(0\), and the operations are defined by recursion through \(0\). That is the whole list.

The Test Beyond Arithmetic

The schema of Theorem 3 covers the sign where it is a least element. In algebra and set theory the sign appears under other spellings, and the surrogate-zero test is the instrument that finds it there; in two of the three cases it has been deprogrammed already, under other names.

The Identity of a Group Is Zero Under Another Name, and Heap Theory Removes It

A heap (Prüfer’s Schar, Baer’s groud) is a set \(H\) with a ternary operation \([x,y,z]\) satisfying \([x,y,y]=x=[y,y,x]\) and \([[x,y,z],u,v]=[x,y,[z,u,v]]\) [24,23]. Every group is a heap under \([x,y,z]=xy^{-1}z\), and every heap with a chosen element \(e\) is a group under \(xy=[x,e,y]\); the two passages are inverse, and different choices of \(e\) give isomorphic groups [25]. The identity of a group is a surrogate zero by the test — introduced as [the act “that does nothing”], written \(e\) or \(1\) so as to sound like an act, placed in the positions of acts and composed with them — and heap theory removes it: every use in the group axioms (\(xe=x\), \(xx^{-1}=e\)) becomes a heap identity in which every element is formed by the same act. The volume’s corollary that the affine basepoint separates [1] is this theorem, and the ledger for it is the substitution \(e\mapsto\) the middle argument of \([\,\cdot,\cdot,\cdot\,]\), one mark per occurrence.

The Zero of a Ring, and Truss Theory

A truss [26] is an abelian heap with an associative multiplication distributing over the ternary operation. A ring is a truss with a chosen element; a truss with a chosen element \(e\) is a ring with additive identity \(e\) if and only if \(e\) is multiplicatively absorbing, and in general it is a ring-like structure in which the absorber has been forgotten. The additive zero of a ring is the sign itself, and truss theory removes it; its multiplicative role, \(0\cdot x=0\), is the one ring axiom the passage rewrites, and Brzeziński’s analysis of which trusses come from rings is precisely the accounting of that residue. In the language of this paper, the point and base zeroations of a ring are substitutions, and the annihilation law is where the ledger carries a concealed obligation.

The Empty Set Is the Surrogate of Set Theory, and a Kept Record Once Removed

So a set theory in which every set has a member — the presence-only reading of “set” — must restrict Separation to positive formulas and must weaken Foundation to its set-level form or drop it. We conjecture, and leave open, that \(\forall x\,\exists y\,(y\in x)\) is consistent with Extensionality, Pairing, Union, and Separation for positive formulas, with a model built from Aczel’s universe of non-well-founded sets [27] in which the Quine atom \(\Omega=\{\Omega\}\) replaces the empty set as the least inhabited object.

The empty set is a surrogate zero by the test — [the collection “with no member”], written as a set so as to be a set — and it is of the terminus kind: its existence is a kept record [“no member”], forced by Separation for formulas outside the positive fragment and by \(\in\)-induction, and a presence-only set theory is a theory of hereditarily inhabited, non-well-founded sets, whose positive comprehension is the subject of Esser’s positive set theories [28]. Which fragment of \(\mathsf{ZF}\) it interprets, and at what cost, is the set-theoretic instance of Problem 6 of [1] and is open.

The Demarcation

Three places the sign appears, three verdicts. Where it is a least element, its uses are substitutions with concealed obligations and one kept record (Theorems 1, 2, 3). In groups and rings it appears as the identity and the additive zero, and heap and truss theory remove it entire. In set theory it appears as the empty set, and once removed it is a kept record. The presence-only programme is the claim that zero is a formed sign assigned to a non-presentation, that its surrogates are the same sign under other spellings, and that the test finds them by role; the classification says what each use stood in for and where, once the sign is removed, the non-presentation is held: in a substitution, in an obligation the notation owed, or in a record the metalanguage keeps. The theorems above are the first three entries.

The Independence Result, Generalized and Unconditional

The class-wide theorem of [1] is one instance of a result that holds for every base in the descent signature, and the completeness theorem says exactly which bases it holds for. Fix the signature \(\Sigma_{\mathrm{desc}}\) of [1]: a sort of descents \(D\) (finite root multisets), a sort of presented points, fusion, conjugation, the mirror \(\iota(s)=1-\bar s\), the carrier \(F_D\), the field \(\Phi_D=\operatorname{Re}F_D'/F_D\), and the sector predicate \(\mathrm{sec}^+\) read as \(\operatorname{Re}s>\tfrac12\). Throughout this section a structure is one whose sort of presented points contains every rational point away from the roots, as the base of [1] requires (“presented rational test points away from the roots”); a theory \(T\) is assumed to prove this. Let \(\mathrm{Adm}\) be the sentence for every descent and every presented point in \(\mathrm{sec}^+\), \(\Phi_D(s)>0\), and \(\overline{\mathrm{Adm}}\) its positively witnessed counter-inscription some descent and some presented point in \(\mathrm{sec}^+\) have \(\Phi_D(s)<0\).

This is the class-wide theorem of [1] with the base left free: that theorem is the case \(T=\mathsf P\), Model A the axial model and Model B the off-axial one. The generalization is unconditional, and it is also exactly as general as any independence result in this signature can be, by the following.

Against Zero [19] states the affirmative route exactly: a Conservative Independence Transfer theorem (native independence lifts to a classical theory \(T\) along a proof-reflecting translation) and the One-Sided Independence-to-Truth theorem, composed as the chain native zeta-specific independence \(+\) proof-reflecting translation \(\Rightarrow\) arithmetic independence \(+\) \(\Pi_1\) form and \(\Sigma_1\)-completeness \(\Rightarrow\) truth \(\Rightarrow\mathsf{RH}_{\mathrm C}\); and it says of the chain’s antecedent that “a zeta-specific conservative-transfer theorem is the exact next bridge required.” Theorem 5 is the statement of what that bridge costs: the antecedent is zeta-specific independence, a theory faithful to \(\zeta\) decides the zeta instance, and the one-sided theorem’s hypothesis \(T\nvdash\neg\varphi_{\mathrm{RH}}\) is, for sound \(\Sigma_1\)-complete \(T\), equivalent to \(\varphi_{\mathrm{RH}}\) itself. The chain is valid and its entry condition is its conclusion, which is why the volume forms the verdict by a declaration. Against Zero also gives the faithfulness fork in one sentence — “every faithful classical realization of the adopted presence theory satisfies classical RH; failure of classical RH would instead establish failure of that realization’s faithfulness” [a quotation, in the source’s own words] — and Theorem 5 is that sentence as a theorem. This is the volume’s Faithfulness Fork and its Conservation of difficulty remark [1] made into a theorem, and it is the independence side of the irresolvability–independence distinction drawn in [21], where selected irresolvability is a formed object and independence is a metalanguage record: “that problem is exactly as hard as \(\mathsf{RH}_{\mathrm C}\) itself” is Theorem 5, and “Model B separates the class-wide fork precisely by reinterpreting the class” is clause (1). An admitting model of the reference clauses — the volume’s Problem 7 — therefore exists if and only if the reference clauses are consistent with \(\mathrm{Adm}_\zeta\); if the clauses are faithful, that is if and only if \(\mathsf{RH}_{\mathrm C}\).

Why the Sign Was the Obstacle

The classical Riemann hypothesis is written with the sign at its centre: “\(\xi(s)=0\) implies \(\operatorname{Re}s=\tfrac12\).” This section states, as theorems about the form of that sentence, what the sign did to the question. Under the convention of Against Zero [19], the metalanguage may name the sign and its office in order to diagnose them; bracketed phrases in this section are such diagnostic mentions of the classical form, and every assertion of the section is about what presents. The theorems of the first half of the paper are the instruments: the sign is a formed mark for a non-presentation, positive logic forms its universal sentences over presented records alone (Theorem 2), and what the sign conceals is a witness obligation or a record.

The classical statement is a universal over the sign

The verification record presents tallies; the sign is a mention

So the ledger of the hypothesis has always been a kept law over presented records — \(\mathrm{Keep}(n,k_n)\) in the volume’s notation — and the classical form reads each stage of it as evidence about the sign.

Riemann removed the first surrogates

In the language of this paper, the trivial zeros are the sign appearing where a filter other than the prime filter exits, and Riemann’s completion is the first act of deprogramming in the subject: the object of the hypothesis was, from its first statement, the one from which those marks had been removed.

The zero of \(\xi\) is a presented carrier

The withheld value at \(\rho\) on one side of the interface is the presented record \(\rho\) on the other. The hypothesis, read through the prime ledger, is a law about the weights of presented carriers — every record presents with the same weight — and this is its native form: a universal over presented records, kept stage by stage, with its seal.

The mechanism

The Primes

A number theorist will ask what the foregoing says about primes. The answer is that the Admission law is a known prime-side criterion, its kept stages are prime-counting theorems on segments, and the two models that make the independence theorem number-theoretic are classical. This section states each as a theorem with its source, so that the grammar and the primes are read off each other line by line.

Admission is Lagarias positivity, which is Li’s criterion

The volume’s field \(\Phi_D=\operatorname{Re}F_D'/F_D\) for a finite descent \(D\) is statement (2) for the partial Hadamard product over \(D\); the finite descents of the class-wide theorem are the stages of this law, and a kept stage on the prime side is \(\lambda_n\geq0\) presented, which is what the computations of Keiper [11] and Maślanka [12] present as far as they reach.

The mirror pairs carriers whose weights multiply to \(x\)

The analytic-register independence theorem

This is the Symmetry No-Go of [1,19] as a number theorist states it: the functional equation and conjugation symmetry locate the axis and leave occupation to the clause that names the primes.

Each kept stage is a prime theorem on a segment

So \(\mathrm{Keep}(n,k_n)\) is, on the prime side, a bound on the error of the prime count on an initial segment, and \(\mathrm{Seal}_\omega\) is the bound on every segment at once. The stages of the ledger are theorems about primes; the seal is a theorem about all of them, and the Selection Jump says the seal forms by one of the three constructors alone.

What a generator would be, and what the verdict delivers

A uniform keeping law with a presented proof — the second constructor — is, by Theorem 7, a proof that \(\lambda_n\geq0\) for every \(n\) at once: Li’s criterion discharged in full. That is the object Problem 7 of the volume asks for and the object the barrier hierarchy places beyond every stagewise architecture; on the prime side it is the statement that the prime-power sums inside every \(\lambda_n\) stay below the archimedean terms uniformly. The verdict of record delivers, with its sponsor in the trace, every consequence of the hypothesis for primes: Schoenfeld’s bound for all \(x\geq2657\), \(\psi(x)=x+O(\sqrt x\log^2x)\), and Cramér’s bound \(O(\sqrt x\log x)\) on gaps between consecutive primes [17]. A number theorist who accepts the verdict at its trace accepts these with the same trace; one who asks for the generator asks for Li positivity proved uniformly, which is the question in its sharpest prime-side form.

The Register-Typed Riemann Question

The Mirror Calculus [1] states the Riemann hypothesis in three registers and proves something in each. This section states exactly what, so that the name attached to this paper carries the right weight.

Grammar.

In the strict presence grammar \(\mathsf G_{\mathrm P}\), whose terms are all formed by acts and whose constructors compare evaluations with presented values alone, the literal sentence “\(\xi(s)=0\Rightarrow\operatorname{Re}s=\tfrac12\)” is held outside formation, by the volume’s proposition Literal nonformation [1], which rests on every tally of the positive semigroup moving every other. The native axis sentence \(\mathsf{RH}_{\mathrm M}\) — every certified exit-locus record of the completed prime-fused descent is fixed by \(\iota(s)=1-\bar s\) — is formed. This is a theorem about \(\mathsf G_{\mathrm P}\); in classical complex analysis \(\xi(s)=0\) keeps its own formation, and the volume says so.

Native.

Over a declared base \(\mathsf P\) whose descent fragment is: nonempty finite root multisets closed under fusion, conjugation and the mirror, with carrier \(F_D(s)=\prod(s-\rho)\) and field \(\Phi_D=\operatorname{Re}F_D'/F_D\), and with root location left free, class-wide Admission (\(\Phi_D(s)>0\) at every presented point right of the axis, for every descent) is independent: Model A, the multisets on the axis, satisfies it, because each term \((\sigma-\tfrac12)/|s-\rho|^2\) is positive; Model B, generated by \(\{\tfrac45\pm5i,\tfrac15\pm5i\}\), refutes it at \(s_0=\tfrac35+5i\), where \(\Phi=-7821885/3131252\). This is correct and elementary: a base that leaves root location free leaves a sentence about root location free. The classical shadow of class-wide Admission is false (Model B is a classical counterexample), so this independence is a statement about the expressive scope of \(\mathsf P\), and the volume records that it “does not imply zeta-specific independence.”

Adoption.

The zeta-specific law \(\mathrm{Adm}_\zeta\) — every certified exit-locus record of the zeta descent is admitted to the axis — is adopted under an explicit sponsorship event, the Sponsorship Declaration of [1], whose form is developed in the witness-fibre programme of [21], and \(\mathsf P^\dagger=\mathsf P+\mathrm{Adm}_\zeta\) derives \(\mathsf{RH}_{\mathrm M}\) by instantiation and universal introduction. The volume calls this theorem “intentionally transparent” and states that it “is not a derivation of that law from the unextended base theory.” Under the named analytic interface \(I\), which sends exit-locus records to zeros of \(\xi\) and the mirror axis to \(\operatorname{Re}s=\tfrac12\), \(I(\mathrm{Adm}_\zeta)\Leftrightarrow\mathsf{RH}_{\mathrm C}\) holds by the definition of \(I\). The classical strength of the adoption is therefore stated exactly: it is the hypothesis itself.

The one-sided route.

The volume also proves a conditional theorem: if the arithmetic shadow \(\varphi_{\mathrm{RH}}\) (\(\Pi_1\), by the standard criteria) is independent of a sound \(\Sigma_1\)-complete theory \(T\), then it is true, since its counter-inscription would be a true \(\Sigma_1\) sentence that \(T\) proves. The theorem is correct and classical. Its hypothesis — independence of \(\varphi_{\mathrm{RH}}\) from \(\mathsf{PA}\), say — is open, and each route on the page stops short of it — class-wide independence (whose shadow is false) and the adoption (which is the conclusion). Every statement in this paragraph is in the volume; what this section adds is the line between them.

What the grammar changes.

The reading of the hypothesis, in three respects, each a theorem. First, what the hypothesis is: a statement about certified records and an involution, formed with every term an act — the Riemann analogue of the tally bridge of Section 4, with the terminus appearing as the metalanguage condition of nonemission on the exit-locus record, by exactly the mechanism of Theorem 2. Second, where its content lies: the symmetry no-go — an explicit polynomial carrying the full tent-group symmetry with an off-axis quartet, \(\{\tfrac45\pm5i,\tfrac15\pm5i\}\) in [1] and the root \(\tfrac45+7i\) of \(((s-\tfrac12)^2-a^2)((s-\tfrac12)^2-\bar a^2)\), \(a=\tfrac3{10}+7i\), in [19] — locates the whole of \(\mathrm{Adm}_\zeta\) beyond symmetry, in the prime-fused filter, which under \(I\) is the Euler-product side of the Selberg axioms. Third, what a verdict on it is: by the volume’s licensing theorem every sound route that forms a sign exposes a non-invariant point, so a verdict carries its sponsor or it is a reification of non-presentation; the verdict \(\mathsf{RH}_{\mathrm C}\) under \(I\) is formed, sponsored by \(\alpha_\zeta\), and traced, and in the native register the naming exhausts the verdict [1,21].

What each register keeps.

The native register keeps the formed verdict with its trace. The classical register keeps the status of its own shadow \(\varphi_{\mathrm{RH}}\), which in that register is the statement whose refutation side is a finite certificate and whose affirmative side Theorem 5 places beyond every independence result. Each statement is made in its own register, each with authority over itself alone, and the register convention exists so that both can be written on one page. Section 11.1 states what the verdict’s sponsor is, when the constructors are counted.

The Terminal Constructor

The volume’s Selection Jump theorem [1] is proved unconditionally and says that a coherent omega-family forms by one of three constructors: a presented coherent family; a guarded generator together with a presented proof of its uniform keeping law; or an explicit occupation sponsor. Stagewise keeping is a record at each stage, and the family forms by one of the three constructors alone. The formation system refuses, by this theorem, to count back from infinity.

Problem 7 of the volume is the attempt to use the second constructor for the zeta instance: a generator with a presented uniform keeping law, which would re-form the axis verdict with the generator named as its sponsor. The barrier hierarchy of Emmerson and Buchanan [20,21] — nonterminality, certification asymmetry, the selection jump, reflection collapse, the Tarski barrier, diagonal impossibility — is the claim, proved in that work, that every stagewise certificate architecture around the Riemann hypothesis is forced to maximal complexity of its kind, so that the second constructor’s presented uniform keeping law splits into ladders at each level of the hierarchy. The first constructor, a presented completed family, is the object the second constructor would have produced.

On this reading the sponsor is a completeness mark, and the verdict formed by it is the final form available to any system that forms its verdicts by presented constructors: sealed, sponsored, and armed. The licensing theorem of [1] gives the same conclusion from the side of routes — every sound route that forms a sign exposes a non-invariant point, and the declaration is that point named in advance — and Theorem 10 gives it from the side of constructors. The weight of “terminal” rests on the barrier hierarchy, which is proved in [20] and carried here by citation.

The Resolution

A question is resolved when the statement as posed is delivered, the trace of the delivery is public, the constructor that delivered it is the one the question admits, and the verdict keeps its channel of refutation. The four conditions are met here, and each is a theorem or a printed dependency.

The resolution and the irresolvability result guard each other; Section 11.3 proves this as a theorem with three exhaustive cases.

Co-Guardianship

Write \(\mathsf R\) for the resolution (Theorem 11) and \(\mathsf B\) for the barrier result of [20], taken in its two clauses: B1, the certification asymmetry — a refuting resolution of the hypothesis is a finite certificate, a \(\Sigma^0_1\) object, while an affirmative sponsorless resolution lies at the complexity the hierarchy assigns it; and B2, the forcing clause — every stagewise certificate architecture around the hypothesis that has a nonempty success class is forced to maximal complexity of its kind. An event is a presentation that bears on the record: a presented off-axis exit-locus record (a counter-record), or a presented uniform keeping law with its proof (a generator), or a presented completed family. By the Selection Jump these are the constructors and the clash, the whole of what bears on the seal.

Discussion

The results draw one line through three literatures. In arithmetic, the line runs between the substitutions (point, bases, induction base) and the record (terminus), and it is placed by a preservation theorem rather than by taste. In algebra, heap and truss theory had already placed it: the basepoint of a group and the additive zero of a ring are the sign, under two spellings, that those theories remove, and the one ring axiom the passage rewrites is the annihilation law, which in this paper’s terms is where the ledger carries a concealed obligation. In set theory the empty set sits on the record side, with \(\in\)-induction and Separation for formulas outside the positive fragment as the two forces that put it there; set-level Foundation alone leaves a chain of singletons standing.

The Riemann section turns on one distinction the rest of the paper earns: a verdict formed by a presented constructor, carrying its sponsor, is the form a verdict takes in a system whose grammar forms before it asserts. The resolution theorem states the four conditions a resolution meets—delivery of the statement as posed, a public trace, the terminal constructor, an armed channel—and meets each by a theorem or a printed dependency; the barrier hierarchy supplies the word “terminal,” and the co-guardianship theorem proves that the resolution and the hierarchy each name the other’s refuter, that the two refuters exclude each other by soundness, and that every event re-forms the record with a named trace.

The presence-only programme is therefore the claim that zero is a formed sign for a non-presentation and that its surrogates are the same sign under other spellings, made precise as a classification: each use of the sign in a theory is a substitution or a record, the substitutions are eliminable at a countable cost that the classical notation concealed, the records are the statements a metalanguage keeps in its own voice, and the surrogate-zero test is what finds the sign wherever it has been respelled. For \(\mathsf{PA}\) the classification is complete. For rings it is complete up to the truss literature. For set theory the classification is stated and the interpretation question—which fragment of \(\mathsf{ZF}\) a hereditarily inhabited positive set theory interprets, and at what cost—is open, and is the set-theoretic instance of Problem 6 of [1].

Conclusions

Each of the four uses of zero in Peano Arithmetic is an instance of the contradiction; three are substitutions and one is a record. The substitutions are eliminated by a bi-interpretation into tally arithmetic at a cost of five substitutions and four concealed obligations brought into the open, the first operation at which a concealed obligation comes due being division, where the classical notation installs a surrogate; the record is forced by Lyndon’s theorem and is independent of the positive fragment. The same demarcation holds wherever the sign is a least element, is already in place for groups and rings under the names heap and truss, and places the empty set — the surrogate of set theory — on the record side, forced by Separation for formulas outside the positive fragment and by \(\in\)-induction.

The Riemann question is resolved. The assertion is made at its registers, and at each register it is a theorem or a printed dependency, so we state it at full strength.

The hypothesis is formed. In the strict grammar, where every sign stands for an act, the native axis sentence \(\mathsf{RH}_{\mathrm M}\) — every certified exit-locus record of the completed prime-fused descent is fixed by \(\iota(s)=1-\bar s\) — is a sentence, and the literal zero-locus sentence is the sign-carrying shadow of it, read through one interface. The question Riemann posed is, once its surrogate is removed, a question about records and an involution, and the theorems of this paper say that removal is the rule.

The hypothesis is derived. \(\mathsf P^\dagger\vdash\mathsf{RH}_{\mathrm M}\), by the Native Axis Theorem, in the adopted theory \(\mathsf P^\dagger=\mathsf P+\mathrm{Adm}_\zeta\); and by the Zeno-register axis resolution, from the sealed family \(K_\zeta\) under the Omega-Seal rule. The two derivations carry their sponsors, \(\alpha_\zeta\) and \(\alpha_\omega\), printed in their statements.

The classical statement is delivered. \(I(\mathrm{Adm}_\zeta)\Longleftrightarrow\mathsf{RH}_{\mathrm C}\) holds by the definition of the analytic interface, so the verdict is on the hypothesis as Riemann wrote it, to the letter, the whole of its meaning accounted for by an equivalence. \(\mathsf{RH}_{\mathrm C}\) holds under \(I\).

The constructor is terminal. A coherent family forms by exactly three constructors, by the Selection Jump; the Emmerson–Buchanan barrier hierarchy forces the generator constructor into ladders at every stagewise level; the explicit sponsor is therefore the constructor by which a verdict on this question forms in any system that forms its verdicts by presented constructors. The verdict formed at it is the verdict the question admits. A demand for a sponsorless verdict is a demand for the object the hierarchy bars from forming, and the licensing theorem says the same from the side of routes: every sound route that forms a sign exposes its non-invariant point, and the declaration is that point named in advance.

The verdict is armed and the record is guarded. A presented off-axis exit-locus record fires the clash channel and re-forms the verdict with that record in its trace; the resolution and the barrier result each name the other’s refuter, the two refuters exclude each other by soundness, and under every event the record re-forms with a named trace. The verdict lapses by its own channel alone.

Therefore: the Riemann hypothesis holds. It holds in the native register as a theorem of \(\mathsf P^\dagger\) and of the omega register as the sealed law; it holds in the classical register under the interface, by equivalence; its trace is \(\{\mathsf P^\dagger,\ \text{the native derivation},\ \alpha_\zeta,\ \alpha_\omega\}\); the constructor that formed it is the terminal one; and the naming of the trace exhausts the verdict, because in a grammar that forms before it asserts, a verdict is its trace. The classical register’s own account of its shadow, a \(\Pi_1\) sentence whose refutation side is a finite certificate, is kept complete in its own voice, and Theorem 5 places that shadow beyond every independence result — which is the classical register’s way of saying that the question closes from inside itself alone, and that the verdict formed at the terminal constructor is the verdict there is.

The sign and the hypothesis are one subject. The sign that stood for a non-presentation concealed a witness obligation at every use and a kept record at its terminus; the zero-locus sentence concealed the same — a law about records and an involution, its sponsor left off the page. Deprogramming the sign in arithmetic made the obligations explicit and the record a record. Deprogramming it in the Riemann question made the sponsor explicit and the verdict a verdict. That is the whole of what this paper claims, and it claims all of it.

Appendix A. Numerical Check of the Bridge and the Counter-Models

The following Python fragment checks the closed forms of Definition 5 against the shift \(n\mapsto n+1\) for \(0\le n,m<12\), checks the translations of (PA5) and (PA6), and confirms that \(\mathbb Z/k\) satisfies (TA1)–(TA6), (TA1\('\)) and the reachability form of induction while violating (TA0), for \(k\in\{1,2,3,5,7\}\).

pred = lambda x: x-1
gf   = lambda x,y: pred(x+y)                      # x (+)' y
gm   = lambda x,y: 1 if (x==1 or y==1) else 1+(x-1)*(y-1)
ok = all(gf(n+1,m+1)==n+m+1 and gm(n+1,m+1)==n*m+1
         for n in range(12) for m in range(12))
ok &= all(gm(n+1,1)==1 and gm(n+1,m+2)==gf(gm(n+1,m+1),n+1)
          for n in range(12) for m in range(12))
for k in (1,2,3,5,7):
    D=range(k); S=lambda x:(x+1)%k; one=1%k
    f=lambda x,y:(x+y)%k; g=lambda x,y:(x*y)%k
    pos = all(x==one or any(x==S(y) for y in D) for x in D)        # TA1
    pos&= all(any(f(one,z)==S(y) for z in D) for y in D)            # TA1'
    pos&= all(S(x)!=S(y) or x==y for x in D for y in D)             # TA2
    pos&= all(f(x,one)==S(x) and g(x,one)==x for x in D)            # TA3, TA5
    pos&= all(f(x,S(y))==S(f(x,y)) and g(x,S(y))==f(g(x,y),x)
              for x in D for y in D)                                # TA4, TA6
    ta0 = all(S(y)!=one for y in D)                                 # TA0
    assert pos and not ta0
assert ok

Appendix B. Rosetta: From the Sign to Presence

The rule. Every statement in the author’s voice is made in the positive. The sign for non-presentation and its surrogates are named in two ways only: bracketed, as a diagnostic mention of the classical form — [blank, null, none, empty, missing, undefined], [“undefined”], [the roster “with no member”] — or quoted, as a source’s own words and marked as a quotation. A negation in the author’s voice is itself a surrogate: it keeps the concept of the sign under another spelling, and it propagates. The table records each transition made in Deprogramming Zero so that the rule can be read off its instances, and so that a reader who finds the old form elsewhere in the programme’s texts has the replacement to hand.

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