# Missing mathematical arguments: reductions and obstructions **Reality Bit Engine 0.13.0 — 4 September 2026** **Research framework:** Anthony Michael Collado **Status:** Written mathematical arguments with exact computational support. No novelty, formal-kernel verification, or Millennium solution is claimed. This pass addresses the mathematical gaps identified in earlier releases. It supplies a reduction removing the two-prime restriction from a bounded Lagarias verification, an obstruction to a proposed all-integer lower-bound strategy, a sharp information bound on a particular SAT summarization strategy, and a broader obstruction to universally decreasing quadratic fluid energies. | Target | Argument obtained | What remains missing | |---|---|---| | RH | Monotonicity and canonical/record reductions cover all integers up to a specified cutoff; the run certifies through \(10^{24}\) | An argument covering the remaining infinite record sequence | | P versus NP | Exact summaries preserving every continuation need enough distinguishable states; a particular fixed-capacity route fails | A general algorithm with a polynomial worst-case bound, or a lower bound covering every permitted algorithm | | Navier–Stokes | Every positive-order homogeneous Sobolev quadratic energy fails universal monotonicity; a general amplitude argument explains why | A continuation-strength estimate for all data, or an admissible breakdown example | The RH reduction is already present in the literature. In particular, [MacArevey's February 2026 preprint](https://arxiv.org/html/2602.15905v2) proves monotonicity of the normalized Lagarias expression and reduction to superabundant numbers. We give a discrete proof suitable for this engine, rather than treating its rediscovery as novelty. No priority or verification-record claim is made for the finite bound below. ## 1. A discrete monotonicity proof Write \[ G(h)=h+e^h\log h,\qquad B_n=\frac{G(H_n)}n,\qquad R_n=\frac{\sigma(n)}n. \] The Lagarias inequality is \(R_n\le B_n\). Its all-integer version is equivalent to RH by [Lagarias's theorem](https://arxiv.org/html/math/0008177v2). ### Theorem 1 \(B_{n+1}>B_n\) for every integer \(n\ge1\). **Proof for \(n\ge64\).** For \(h\ge1\), \[ G'(h)=1+e^h\left(\log h+\frac1h\right), \qquad G''(h)=e^h\left(\log h+\frac2h-\frac1{h^2}\right)>0. \] The standard integral comparisons for the decreasing function \(1/x\) give \[ \log(n+1)\le H_n\le1+\log n. \tag{1} \] Set \(U(x)=1+\log x\). We first show \(x>2U(x)^2\) for \(x\ge64\). An exact logarithm bound gives \(\log2<7/10\), so \(U(64)<26/5\) and \[ 64-2U(64)^2>64-2(26/5)^2>0. \] Moreover \(x-4U(x)\) is positive at 64 and increasing thereafter, because its derivative is \(1-4/x>0\). It follows that \[ \frac{d}{dx}\bigl(x-2U(x)^2\bigr)=1-\frac{4U(x)}x>0, \] proving the desired bound for all such \(x\). For \(h=H_n\), (1) now gives \(n>2h^2\). Since \(\log hh. \] Also (1) gives \(e^h\ge n+1\). Therefore \[ \begin{aligned} nG'(h)-(n+1)G(h) &=n-(n+1)h+e^h\left(\frac nh-\log h\right)\\ &>n>0. \end{aligned} \tag{2} \] Convexity of \(G\) and \(H_{n+1}=H_n+1/(n+1)\) imply \[ n\bigl(G(H_{n+1})-G(H_n)\bigr) \ge\frac n{n+1}G'(H_n)>G(H_n). \] Rearranging proves \(B_{n+1}>B_n\). **Finite base cases.** The lab calculates rigorous interval bounds for \(G(H_n)\), \(1\le n\le64\), and checks \[ n\,\underline{G(H_{n+1})}-(n+1)\,\overline{G(H_n)}>0 \] for all 63 adjacent pairs. Every lower bound is strictly positive. The same integer-interval arithmetic certifies \(\log2<7/10\). These finite certificates complete the proof, subject to the documented interval implementation and its execution. \(\square\) ## 2. From arbitrary integers to canonical prime profiles Call an integer canonical here if \[ n=2^{a_1}3^{a_2}5^{a_3}\cdots p_k^{a_k}, \qquad a_1\ge a_2\ge\cdots\ge a_k\ge1, \] where the primes are consecutive; include 1 as the empty profile. ### Theorem 2 For every integer \(n\ge1\), there is a canonical integer \(m\le n\) with \(R_m\ge R_n\). **Proof.** Let \(S_a(r)=1+r+\cdots+r^a\). The divisor-sum identity is \[ R_n=\prod_{p^a\parallel n}S_a(1/p). \] If a prime \(q\) occurs while a smaller prime \(p\) is absent, replacing \(q^a\) by \(p^a\) decreases the integer and increases \(R_n\), since \(S_a(r)\) is increasing for \(r>0\). If smaller and larger primes \(p0\). To see the latter fact, write \[ \frac{S_b(r)}{S_a(r)}=1+\sum_{j=a+1}^b\frac{r^j}{S_a(r)}. \] Each summand has positive derivative: \[ \frac{d}{dr}\frac{r^j}{S_a(r)} =\frac{r^{j-1}}{S_a(r)^2}\sum_{\ell=0}^a(j-\ell)r^\ell>0. \] Continue any available prime-gap replacement or exponent exchange. Each step strictly decreases a positive integer, so the process terminates. At termination no gap or increasing exponent pair remains, and the integer is canonical. Its divisor ratio has never decreased. \(\square\) ### Corollary 2.1: record reduction A superabundant number is an integer whose divisor ratio is strictly larger than every earlier one. If any Lagarias counterexamples exist, the smallest must be superabundant. **Proof.** If a smallest counterexample \(n\) were not superabundant, some \(mB_n>B_m, \] making \(m\) an earlier counterexample, a contradiction. \(\square\) ### Corollary 2.2: finite coverage Fix \(N\). Enumerate every canonical integer at most \(N\), and scan them in order to retain strict record values of \(R_n\). If the Lagarias inequality holds at every retained record, it holds at every integer from 1 through \(N\). **Proof.** For any \(n\le N\), Theorem 2 supplies a canonical \(m\le n\) with \(R_m\ge R_n\). The finite record scan supplies a retained \(r\le m\) with \(R_r\ge R_m\). Then \[ R_n\le R_m\le R_r\le B_r\le B_n. \] The last inequality is Theorem 1. \(\square\) ### Completeness of the enumeration The recursion starts from 1. At each node it appends the next consecutive prime with every positive exponent no larger than the previous exponent, stopping when the product exceeds \(N\). The first exponent is bounded by \(\lfloor\log_2N\rfloor\), computed from integer bit length. Every canonical prime profile has exactly one such path, and every generated path is canonical. A prime list extending through the first primorial larger than \(N\) suffices, because even exponent one at each of those primes would exceed \(N\). Divisor sums use exact geometric sums. Record comparisons use integer cross multiplication; no floating logarithms or approximate ratios decide candidate generation or record status. ## 3. The bounded run and its arithmetic With \(N=10^{24}\), the lab generated **172,513 canonical candidates** and retained **148 record cases**. Every record case was certified with exact integer interval bounds, with equality at 1 and strict positive lower margins otherwise. Under the written reduction proofs, this establishes the Lagarias inequality for **every integer \(1\le n\le10^{24}\)**, rather than only the individually listed records. The last retained record is \[ 985496152350226952635200. \] The final gap from that record to \(10^{24}\) is covered by the same record-dominance and monotonicity argument. This result is a finite computer-assisted verification, not RH, and is not claimed as a new verification record. ### Harmonic numbers without summing \(10^{24}\) terms Let \(M=4096\). The program computes \(H_M\) directly with outward rounding. For \(n>M\), decreasing-integrand comparisons give \[ H_M+\log\frac{n+1}{M+1} \le H_n\le H_M+\log\frac nM. \tag{3} \] Indeed, compare each \(1/j\), for \(M+1\le j\le n\), with its integrals over \([j,j+1]\) and \([j-1,j]\). Summation gives (3). For a positive rational logarithm argument, the program writes \(x=2^k r\) with \(1\le r<2\) using integer comparisons, then encloses \(\log x=k\log2+\log r\). This range reduction avoids a slowly converging logarithm expansion at a huge argument. The underlying interval code uses a 192-bit tick scale and 256-term exponential/logarithm bounds, with outward rounding and documented tails. It evaluates the Lagarias right-hand side monotonically over the harmonic bracket. For an independent check of the reduction implementation, an ordinary full divisor sieve through 100,000 found exactly the same 24 record cases as the canonical recursion. Separate tests compared all canonical candidates through 10,000 against independently factored integers and compared the integral harmonic brackets with exact rational sums through 512. Completeness beyond those small comparison domains rests on the written enumeration proof and correct execution of its implementation. These arguments have not been checked by a formal proof kernel. The v0.12 portable verifier retains its original scope; its previous acceptance does not certify this new, broader computation. ## 4. A proposed all-integer extension is false The earlier two-prime proof used \(H_n>\log n+1/2\). A natural extension tries a fixed constant \(c\) and the lower expression \[ L_c(n)=\log n+c+n e^c\log(\log n+c), \] on sufficiently large integers where the logarithms are defined. For a suitable harmonic lower bound, this is below the true Lagarias right-hand side. Proving \(\sigma(n)\le L_c(n)\) everywhere would then suffice. ### Theorem 3: fixed constants below \(\gamma\) cannot close this route For every fixed \(c<\gamma\), the inequality \(\sigma(n)\le L_c(n)\) fails at infinitely many integers. **Proof using a known theorem.** Gronwall's maximal-order theorem, as stated in [Lagarias's paper, Theorem 2.2](https://arxiv.org/html/math/0008177v2), gives \[ \limsup_{n\to\infty}\frac{\sigma(n)}{n\log\log n}=e^\gamma. \] Directly from the definition, \[ \lim_{n\to\infty}\frac{L_c(n)}{n\log\log n}=e^cL_c(n)\) infinitely often. \(\square\) The exact arithmetic also records a concrete failure at \(c=1/2,n=12\): the upper bound on \(L_{1/2}(12)\) is below \(\sigma(12)=28\), while the actual Lagarias inequality at 12 is satisfied. This is a counterexample to the proposed sufficient bound, not to RH. Choosing a more accurate fixed constant below \(\gamma\) does not remove the infinite obstruction. The coefficient and error terms must be handled with a genuinely sharper argument. **Remaining RH gap:** bound the record values for the entire infinite sequence, with an error estimate strong enough to compare them with \(B_n\). Monotonicity of \(B_n\) does not determine whether a future jump in \(R_n\) could cross it. ## 5. The SAT information bottleneck One possible engine strategy is to summarize an earlier partial assignment and then use that summary to answer every possible continuation. Its capacity can be bounded exactly. ### Theorem 4: equality requires \(2^n\) distinguishable summaries Let \(x,y\in\{0,1\}^n\), and suppose a deterministic summary \(\alpha(x)\in S\) and evaluator \(g\) satisfy \[ g(\alpha(x),y)=\mathbf1_{x=y} \quad\text{for every }x,y. \] Then \(\alpha\) is injective and \(|S|\ge2^n\). Any fixed-width binary summary needs at least \(n\) bits. **Proof.** If distinct \(x,x'\) share a summary, choose \(y=x\). The same evaluator input would have to return 1 for \(x\) and 0 for \(x'\), a contradiction. Thus every distinct \(x\) needs a distinct summary. \(\square\) This equality relation has a CNF expression with only \(2n\) clauses: \[ \bigwedge_{i=1}^n (\neg x_i\lor y_i)\land(x_i\lor\neg y_i). \] The lower bound therefore already occurs for a simple constraint family. It does **not** prove an exponential-time SAT lower bound: the summary needs \(n\) bits, equality is easy to compute, and the displayed formula is easy to satisfy. The theorem concerns one exact one-way summarization architecture, not every possible algorithm or symbolic representation. For \(d\) coordinates with \(b\) bits each, the necessary capacity condition is \(db\ge n\). Relabeling a fixed number of states as dimensional cycles cannot increase that capacity unless additional winding or precision bits are stored. The lab preserves explicit collision witnesses for summaries with \(n-1\) bits at sizes 1 through 12; the proof covers arbitrary summary maps and all \(n\). **Remaining P-versus-NP gap:** show that a particular algorithm handles all allowed inputs in polynomial time, or prove a lower bound against every algorithm in the relevant model. A limitation of this summarization strategy is not such a lower bound. ## 6. A broader Navier–Stokes obstruction Work with smooth, mean-zero, divergence-free fields on the three-torus of side \(2\pi\), without forcing, with viscosity \(\nu>0\). Use normalized spatial means. ### Theorem 5: a necessary condition for a universal quadratic Lyapunov functional Let \(Q\) be a differentiable homogeneous quadratic functional. At a smooth initial field \(v\), write its Navier–Stokes derivative under amplitude scaling as \[ \frac{d}{dt}Q(Av)\bigg|_{t=0}=A^3 C_Q(v)-\nu A^2 D_Q(v). \tag{4} \] If this derivative is nonpositive for every smooth initial field and every real amplitude, then \(C_Q(v)=0\) for every such \(v\). **Proof.** The inviscid evolution is quadratic in velocity and the derivative of a quadratic functional is linear in its velocity argument, giving the cubic term. The viscous evolution is linear, giving the quadratic term. If \(C_Q(v)\ne0\), choose the sign of \(A\) to make \(A^3 C_Q(v)>0\) and then increase \(|A|\). The cubic term eventually dominates the finite quadratic term, contradicting universal nonincrease. \(\square\) This is a necessary condition for this particular global-monotonicity strategy. It is not a classification of all invariant functionals or a statement that every useful estimate must be monotone. ### Theorem 6: every positive-order homogeneous Sobolev energy fails that strategy Let \[ Q_s(u)=\frac12\langle u\cdot(-\Delta)^s u\rangle, \qquad s>0. \] For every real \(s>0\) and every \(\nu>0\), some smooth divergence-free initial field has \(Q_s'(0)>0\). **Proof.** Use \[ u_0=A(\sin y,0,\cos x-\cos(x+y)). \] More generally, let a self-adjoint isotropic Fourier multiplier assign weights \(m_1\) and \(m_2\) to the squared frequencies 1 and 2. These are the only squared frequencies present in \(u_0\). Orthogonality gives \[ Q(u_0)=A^2\left(\frac{m_1}{2}+\frac{m_2}{4}\right). \] The pressure contribution vanishes because the multiplier preserves incompressibility. Writing \(w=\cos x-\cos(x+y)\), the only nonzero advective component is \[ A^2\sin y\,w_x =\frac{A^2}{2}\bigl(\cos x+\cos(x+y)-\cos(x-y)-\cos(x+2y)\bigr). \] Only its first two modes pair with the initial field. The resulting initial derivative is \[ \boxed{Q'(0)=\frac{A^3}{4}(m_2-m_1)-\nu A^2(m_1+m_2).} \tag{5} \] For \(Q_s\), take \(m_1=1\), \(m_2=2^s\). Since \(2^s>1\), choosing \[ A>4\nu\frac{1+2^s}{2^s-1} \] makes (5) positive. This proves the statement for every positive real \(s\). \(\square\) The exact Fourier implementation checks (5) at integer orders 1 through 8, including sign reversal of the amplitude. At order zero it recovers decreasing ordinary energy as a control. These finite checks support the implementation; the algebra above supplies the all-real-order argument. For \(m_1=1+\lambda,m_2=1+2\lambda\), (5) also reproduces the earlier \(E+\lambda Z\) counterexample. The new argument therefore explains a broader class of failures instead of testing one weight after another. The local smooth solution framework for these initial derivatives is part of the standard theory described in the [official Navier–Stokes problem statement](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf). Initial growth of \(Q_s\) is not a finite-time singularity. **Remaining fluid gap:** obtain an estimate strong enough for continuation for arbitrary permitted data, potentially using nonlinear functionals, time-dependent estimates, or other structure. Universal monotonicity of these fixed quadratic energies cannot supply that missing argument. ## 7. Reproduction and proof boundary From the project root: ```sh python3 tools/proof_gap_lab.py --limit 1000000000000000000000000 --output results/my-proof-gaps.json python3 -m unittest discover -s tests -p 'test_proof_gaps.py' -v ``` On Windows, use `py -3` in place of `python3`. The default cutoff is already \(10^{24}\). The implementation caps the permitted cutoff there to bound this prototype's resource use; raising that cap would require a fresh coverage and precision assessment. All nine new test methods passed. The frozen result `results/proof-gaps-v013.json` stores exact record data, interval margins, canonical-list identity, finite monotonicity checks, lower-bound failure, separator witnesses, fluid probes, and source/document digests. Arbitrarily large integer values are serialized as strings. The mathematics in this supplement is not asserted to be new to the field. The finite coverage claim relies on its written reduction proofs, exact interval bounds, and correct code execution. The new lab shares the previously audited interval and Fourier routines; the v0.12 standalone verifier has not been extended to formally certify these new universal arguments. No dimensional or theological premise substitutes for an unproved step.