# Null-Axis Probes and the Limits of Projection Reality Bit Engine 0.20.0 — 5 September 2026 Research direction: Anthony Michael Collado This release tests the proposal that pushing a state across a zero axis can reveal hidden coordinates and enable a complete higher-dimensional law. It gives exact conditions for successful recovery, complete ambiguity sets for a nonlinear probe model, and a written obstruction to closing a fluid law with any fixed finite list of energy derivatives. **It does not solve RH, P versus NP, or general Navier–Stokes regularity.** The results concern supplied mathematical models and synthetic observations. No physical grid, number of spatial dimensions, or new law of nature has been inferred. Observability is an established subject; see [Frazzoli's MIT lecture on reachability and observability](https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/43a02bc538e449c1e5eae487139b5030_MIT6_241JS11_lec20.pdf) and [the Cambridge discrete-time observability notes](https://www.statslab.cam.ac.uk/~rrw1/oc/L10.pdf). The proofs below specify what the engine implements, with no novelty claim. ## 1. Hidden information is a property of a map For a linear observation y=Cx, \[ Cx_1=Cx_2\iff x_1-x_2\in\ker C. \] The observation fiber is x0+ker(C). For C(x,y,z)=(x,y), the distinct states (1,2,3) and (1,2,-3) have the same observation (1,2). The coordinates did not disappear from the upper state; the observation discarded a distinction. This happens at nonzero observed coordinates too. Translating the origin can move an observed point to zero without changing what information is lost. Zero observation therefore does not establish geometric perfection. If a known vector u is added to both candidate states, \[ C(x_1+u)-C(x_2+u)=C(x_1-x_2). \] Thus a fixed linear sensor cannot reveal its kernel merely by applying the same known translation to each candidate. More generally, in a known linear system x[t+1]=Ax[t]+Bu[t], two candidates receiving the same input satisfy delta-x[t+1]=A delta-x[t]. The known additive input cancels. This statement does not cover nonlinear dynamics, state-dependent controls, or a control that changes A itself. At a geometric level, the boundary of a two-dimensional square is one-dimensional. A third coordinate is added by specifying an embedding or a larger state space. It is not implied by crossing that square's boundary. An embedded plane can divide three-dimensional space, but this fact does not identify the plane with the start of a new physical dimension. ## 2. When motion does expose the hidden state Consider permitted known maps A1,...,Aa on R^n and a fixed readout C. An experiment word w=(i1,...,ik) applies those maps in the listed order, giving \[ y_w=C A_{i_k}\cdots A_{i_1}x. \] Define the row space W generated by every row of every such observation, including the empty word C. Then the exact equivalence relation is \[ \boxed{x\sim x'\iff x-x'\in W^\perp.} \] Here W-perp is the annihilator under the ordinary coordinate dot product. If rank(W)=n, a basis of these observations determines the state. Otherwise the complete compatible states form an affine space with n-rank(W) hidden directions. These statements assume exact outputs and the declared maps. ### Finite computation of all-word observability Start with independent rows of C. For every retained row r and each allowed map Aj, consider r*Aj. Retain it only if it increases the span. Process each new row in the same way. Every retained row has an explicit experiment word: if r comes from word w, then r*Aj comes from first applying j and then w. This ordering matters for noncommuting maps. At most n independent rows can be retained. Once every retained row has been processed, the resulting span contains C and is invariant under right multiplication by every Aj. Induction on word length therefore puts every possible observation row in the span. Conversely, every retained row is an actual word observation, so no unobservable information has been invented. This proves that the construction gives W exactly. The algorithm processes at most m+a*n candidate rows, where m is the number of initial readout rows. It avoids enumerating all finite action words. This is an arithmetic-operation count; rational bit lengths and experimental costs still matter. The implemented budget is n<=16 and a<=8, not a claim that reality has that many dimensions or a fundamental ULP. The standalone checker reconstructs every witness row, checks linear independence, inclusion of C, closure under every Aj, and a full-rank basis of the annihilator. It imports no generator. Its accepted certificate covers all finite action words in that model, not all possible physical controls. The producer uses exact orthogonalization to choose rows; the checker uses exact row reduction to verify them. The annihilator is equivalently the largest subspace in ker(C) invariant under every permitted Aj. If a direction remains in that subspace, repeating these probes cannot reveal it. A different permitted map or sensor may change the answer, but must be specified and justified. ### Executed examples | Model | Readout | Observable rank | Hidden dimension | |---|---|---:|---:| | Three coordinates, identity dynamics | First two coordinates | 2 | 1 | | Three coordinates, x-z quarter rotation allowed | First two coordinates | 3 | 0 | | Four coordinates, same rotation leaving the fourth untouched | First two coordinates | 3 | 1 | | Seven-coordinate shift chain | First coordinate | 7 | 0 | The seven-coordinate chain is only an example of the general n-dimensional theorem. A state initially supported on its seventh coordinate stays invisible for the first six readouts and then appears. A single zero snapshot could not have justified permanent invisibility. For a rotation, the signed hidden component can move into a visible one. That is a precise realization of the proposed directional probe. Using rows from several experiment words to reconstruct one initial state requires reproducible trials or a valid reset and measurement protocol. The code's synthetic exact inputs supply that condition; it is not inferred from data. ## 3. A two-sided nonlinear probe that recovers sign Let the sensor be the squared Euclidean distance h(x)=||x||^2. States x and -x have identical snapshots. With known probe displacement u, however, \[ y_+=\|x+u\|^2,\qquad y_-=\|x-u\|^2, \] and direct expansion gives \[ \boxed{y_+-y_-=4u\cdot x.} \] For u=delta*e_i with delta nonzero, \[ \boxed{x_i=\frac{y_+-y_-}{4\delta}.} \] Thus n independent coordinate probes recover all n signed coordinates in this model. The release recovers (3/5,-2/7,4/3) exactly with displacements of magnitude 1/8. The two readings in each pair must refer to the same initial x and the stated sensor law; their values cannot be borrowed from unrelated evolving states. If each squared-norm reading has absolute error at most eta, the coordinate error is bounded by eta/(2|delta|). This bound allows arbitrary correlation between the two errors. At eta=1/1000 and delta=1/8 it is exactly 1/250, and opposite extreme reading errors attain it. An ever-smaller push therefore does not guarantee ever-better resolution under fixed reading error. ### The complete fiber when the probes are insufficient The differences and sums contain distinct information: \[ d_i=(y_i^+-y_i^-)/4=u_i\cdot x, \] \[ s_i=(y_i^++y_i^-)/2-\|u_i\|^2=\|x\|^2. \] All s_i must agree on one s. Form U with rows u_i and let x0 be the minimum-norm solution of Ux=d, assuming these linear equations are consistent. Writing K=ker(U), the exact compatible set is \[ \boxed{\{x_0+v:v\in K,\ \|v\|^2=s-\|x_0\|^2\}.} \] **Proof.** Every linear solution is x0+v for v in K. Minimum norm means x0 is orthogonal to K, so ||x0+v||^2=||x0||^2+||v||^2. The sums impose exactly the displayed radius. Conversely these two identities reconstruct every original paired reading, so no additional information was discarded. Let q=dim(K) and rho^2=s-||x0||^2. The fiber is empty if rho^2<0, or if q=0 but rho^2 is nonzero. If rho^2=0, it is a unique point even if U has deficient rank. If rho^2>0 and q=1, there are two points; if q>=2, there is a sphere of compatible states. Inconsistent sums or linear contrasts are rejected before this representation is certified. The checker verifies x0, its orthogonality, the complete kernel basis, and the exact squared radius. In particular, it does not mistake two-sided data for unique reconstruction when only a single direction was probed. ## 4. Apply the projection criterion to fluid dynamics Earlier releases proved that an autonomous projected law exists only when the projected future is the same for every state in an observation fiber. The present construction shows exactly where a natural scalar description of fluid motion fails this criterion. Use the normalized energy on the periodic [0,2pi]^3 torus, \[ E(t)=\frac12\langle|u(t)|^2\rangle. \] For any finite collection of positive weights w_k and positive viscosity nu, the field \[ u(y,t)=\left(2\sum_k\sqrt{w_k}\,e^{-\nu k^2t}\sin(ky),\ 0,\ 0\right) \] is real, smooth, mean-zero, and divergence-free. Its advection term vanishes because it has no y-velocity and is independent of x. Each mode solves the heat equation, so the full unforced Navier–Stokes equation holds with constant pressure. It is a global smooth three-dimensional solution, even though the field only varies in one spatial coordinate. Orthogonality of distinct sine modes gives \[ E(t)=\sum_k w_k e^{-2\nu k^2t},\qquad E^{(j)}(0)=(-2\nu)^j\sum_k w_k k^{2j}. \] These are weighted moments of the squared wave numbers. They can hide different mode populations, rather than merely the orientation of a shape. ### An exact pair with the same first three energy coordinates At nu=1, take weights \[ w^A=(0,14,0,2),\qquad w^B=(7,0,9,0) \] at wave numbers k=(1,2,3,4). Then | Effect at time zero | Flow A | Flow B | |---|---:|---:| | E | 16 | 16 | | E' | -176 | -176 | | E'' | 2944 | 2944 | | E''' | -72704 | -52544 | The difference in the third derivative is -20160. These exact numbers are also independently reproduced by the engine's frozen full Fourier algebra, using an integer-amplitude realization. For each mode its four coefficients are (x-sine,x-cosine,z-sine,z-cosine), all depending only on y: | Flow | Wave number | Four coefficients | |---|---:|---| | A | 2 | (6,4,2,0) | | A | 4 | (2,2,0,0) | | B | 1 | (4,2,2,2) | | B | 3 | (6,0,0,0) | Their energy weight is one quarter of the sum of coefficient squares. The Fourier check verifies zero divergence, zero full nonlinear term, and the energy derivatives through order three using the product rule and successive Laplacians, rather than reusing the moment calculation. ### The obstruction holds for any fixed finite derivative count **Theorem.** For every integer m>=0 there are two such global smooth shear solutions with equal E^(j)(0) for 0<=j<=m and unequal E^(m+1)(0). **Proof.** Set N=m+2 and choose distinct positive squared wave numbers lambda_i=i^2, for i=1,...,N. Define \[ v_i=\frac1{\prod_{\ell\ne i}(\lambda_i-\lambda_\ell)}. \] Lagrange interpolation of z^j at these N nodes gives \[ z^j=\sum_i\lambda_i^j \frac{\prod_{\ell\ne i}(z-\lambda_\ell)} {\prod_{\ell\ne i}(\lambda_i-\lambda_\ell)} \quad(0\le j\le N-1). \] Compare coefficients of z^(N-1). It follows that \[ \sum_i v_i\lambda_i^j=0\quad(0\le j\le m),\qquad \sum_i v_i\lambda_i^{m+1}=1. \] Split v into its positive and negative parts. Both parts are nonzero, because their difference sums to zero and v is nonzero. Use these positive parts as the two energy weight sets. The corresponding energy derivatives agree through m, while their next difference is (-2*nu)^(m+1), which is nonzero. The exact shear construction realizes both sets as smooth flows. Consequently no single-valued autonomous differential law \[ z_m'=F(z_m),\qquad z_m=(E,E',\ldots,E^{(m)}), \] can describe all these shears: the same z_m at time zero would require two different last derivative components. This is a precise failure of law descent through that projection. It is not a claim that every finite set of observables or every possible intervention fails, and it is not a fluid singularity example. Both constructed solutions remain globally smooth. The written interpolation proof covers every finite m. The executable checks orders 0 through 12 within its declared resource budget. It does not infer the theorem from that finite sample. This is a finite-moment ambiguity argument, without a claim of mathematical novelty. ### What a justified finite grid would buy If exactly N distinct wave numbers are known and no others are possible, the first N energy derivatives determine the weights through an invertible Vandermonde matrix. The four-mode example is recovered by the observability engine with C=(1,1,1,1) and A=diag(-2,-8,-18,-32). Here A represents differentiation of the energy components, not a physical control that someone applies to the fluid. The recovered weights are exactly (0,14,0,2) and (7,0,9,0). This recovers the energy evolution in that shear model; it does not recover every velocity phase or solve a general interacting flow. Assuming an exact finite cutoff without proving its adequacy changes the all-data continuum problem. The full problem's hypotheses remain those in [Fefferman's official statement](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf). ## 5. Test the corresponding shortcuts on RH and P versus NP ### RH: paired zeros need not sit on the symmetry line One proposed lift would pair hidden coordinates across a central axis and then identify the pair with the axis itself. A symmetric set need not consist of fixed points of its symmetry. Consider w=s-1/2 and the entire polynomial \[ F(s)=((w-1/4)^2+1)((w+1/4)^2+1) =w^4+\frac{15}{8}w^2+\frac{289}{256}. \] It obeys F(s)=F(1-s) and F(conjugate(s))=conjugate(F(s)), but has zeros s=1/2 +/- 1/4 +/- i, all off the central line. The engine verifies all four roots by exact Gaussian-rational arithmetic and the tests use the factored form independently of the expanded polynomial. This polynomial is not the Riemann xi function and is not a counterexample to RH. It refutes only the inference from those two symmetries alone to critical-line zeros. A valid RH proof must use enough of the actual zeta function's structure to exclude such off-line possibilities. See [Bombieri's statement](https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf) for the target; adding a coordinate does not establish its required property. ### SAT: a positive local error need not have an improving nearby move For n>=3 Boolean variables, consider \[ \left(\bigvee_{i=1}^n x_i\right) \wedge\bigwedge_{i\ne j}(\neg x_i\vee x_j). \] At an assignment with k true variables, the number of violated clauses is \[ \boxed{H(k)=\mathbf1_{k=0}+k(n-k).} \] The all-zero assignment violates one clause. Every one-bit move produces n-1 violations, strictly more than one, while the all-one assignment satisfies the whole formula. The engine freezes this strict local minimum and a satisfying witness. Tests enumerate every assignment for n=3,...,8. Therefore a rule that stops when no one-bit push decreases the projected error can fail on a satisfiable formula. A larger move succeeds for this special family; this observation proves neither a lower bound against all algorithms nor a general efficient SAT method. In general, lifting a formula to include its assignment coordinates makes verification explicit. It does not by itself supply a polynomial method to find the right assignment across all input lengths. That uniform requirement is part of [Cook's P versus NP statement](https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf). ## 6. Correct framing and proof obligations | Proposal | What the mathematics supports | Missing condition for a stronger conclusion | |---|---|---| | Push across zero | Exact signed recovery for the squared-distance sensor | A justified probe law and enough informative directions | | Lift the coordinates | A state space with more distinctions | A sound relationship to the original problem and its full domain | | Read effects instead of causes | A compatible-state fiber | Injectivity or an adequate identification experiment | | A higher law determines lower laws | A law constant on each relevant observation fiber | Proof that the chosen projection preserves sufficient information | | A finite reality grid settles the targets | Finite instances may become exhaustible | Proof that the finite grid preserves the original unlimited quantifiers | Consistency of a proposed model is not a theorem that every question about it is decidable, efficiently computable, or provable from its assumptions. The prize problems ask precise mathematical questions under fixed definitions. Replacing those definitions with a finite grid would require an equivalence theorem before the new calculation could settle the original question. Physical plausibility alone is not that theorem. ## 7. Executed records and PC commands The default run produces 23 independently checked certificates: ten main observability, radial-fiber, and shear examples, plus 13 shear-family orders. The finite-grid operator interpretation is recorded explicitly. The RH and SAT obstruction records have exact witnesses and separate tests; they are not accepted as Millennium certificates by any checker. The release passes 18 targeted test methods, including 80 seeded action models compared with explicit words, 64 exact full-rank radial recoveries, all four radial fiber types, noncommuting action order, the seven-coordinate delay, the sharp reading-error bound, full Fourier verification, the parameterized SAT cases, and isolated/optimized Python runs. Four positive controls pass and all 31 deliberate certificate corruptions are rejected. The standalone checker verifies exact algebra and the declared coverage. The analytic interpretation and all-m interpolation argument have written proofs here. No formal proof-assistant kernel has verified these arguments. Earlier analytic sources and their frozen results remain unchanged. After extracting the project on Windows: ```powershell py -3 tools/null_axis_lab.py py -3 tools/verify_null_axis.py results/null-axis-v020/mixing-probe.json py -3 tools/verify_null_axis.py results/null-axis-v020/shear-order-two.json py -3 tools/challenge_null_axis.py py -3 -m unittest discover -s tests -p "test_null_axis.py" -v ``` Use `python3` on Linux or macOS. For another supported exact model use `--model model.json`; for squared-norm readings use `--radial-data data.json`. Both accept an output directory. `verify_null_axis.py --expected input.json` binds acceptance to that exact model or probe dataset. Input rejection replaces the requested generic output with an explicit failure record. All new modules use the Python standard library; the C++ numerical code is unchanged. No universal prize resolution is claimed.