# RRLS v0.30 — Dual-Phase Expansion/Collapse Probe ## Status **P = NP is not proved.** **No claim is made that the physical universe literally expands/collapses according to this algebra.** v0.30 formalizes the proposed simultaneous opposite-phase experiment as an exact involution on Boolean/dimensional coordinates and asks three separate questions: 1. does observing the opposite phase add independent information? 2. can phase-equivalent descriptions be quotiented exactly? 3. do phase crossings expose semantic structure useful to RRLS? The answer is: **yes for nonlinear/parity structure and representation canonicalization; no for merely duplicating a linear probe.** --- ## 1. Opposite phase Use signed Boolean coordinates z_i = 2 x_i - 1 in {-1,+1}. The full antipodal inversion is J(z) = -z, or in Boolean form J(x) = x XOR 1^n. For CNF F, define J(F)(x) = F(NOT x). In CNF syntax this simply flips every literal sign. Because x -> NOT x is a bijection, SAT(F) = SAT(J(F)) and #SAT(F) = #SAT(J(F)). This is exact. The v0.30 differential tested 120 random small CNFs against an independent Boolean enumerator and the residual engine. Formula, inverted formula, baseline #SAT and antipodal-residual #SAT all agreed. --- ## 2. Expansion/collapse coordinate For signed response s(z) in {-1,+1}, define E(z) = [s(z) + s(-z)] / 2 O(z) = [s(z) - s(-z)] / 2. E is phase-even: E(-z)=E(z). O is phase-odd: O(-z)=-O(z). Define the exact rational interpolation R_lambda(z) = E(z) + lambda O(z), lambda in [-1,+1]. Then R_(+1)(z) = s(z) R_(-1)(z) = s(-z) R_0(z) = E(z). Computationally, lambda=0 is a phase-collapse point: the orientation-dependent component is removed and only the inversion-invariant core remains. This is an algebraic interpolation in response space. Intermediate lambda values are not asserted to be physical Boolean realities. --- ## 3. Where opposite trajectories cross Compare the two branches R_+(lambda)=E+lambda O R_-(lambda)=E-lambda O. They always meet at lambda=0. For lambda != 0, they coincide iff O=0. Therefore nontrivial/persistent opposite-phase crossings identify **phase-invariant coordinates**. For one antipodal Boolean pair there are only four possibilities: 00 -> E=-1, O=0 11 -> E=+1, O=0 01 -> E=0, O=-1 10 -> E=0, O=+1. The sign of O records which oriented phase supplied the TRUE side, relative to the canonical pair orientation. If O=0, the starting phase is fundamentally unidentifiable from that response because the two phases are observationally identical. --- ## 4. Exact Walsh/Fourier consequence For the Walsh character chi_S(z), chi_S(-z) = (-1)^|S| chi_S(z). Thus global phase inversion leaves every even-degree Walsh coefficient unchanged and flips every odd-degree coefficient. The two-phase measurement therefore separates the spectrum exactly: [f(z)+f(-z)]/2 -> even-degree channel [f(z)-f(-z)]/2 -> odd-degree channel. 100/100 random finite controls passed this inversion theorem exactly with integer Walsh transforms. Family controls: - NAE-3: zero odd-channel energy under full global inversion. - even-dimensional parity: entirely phase-even. - odd-dimensional parity: entirely phase-odd. - generic random 3-SAT: mixed even/odd content. So opposite phase is useful as a **parity/nonlinearity separator**. --- ## 5. Crucial negative result: opposite linear force does not double information If a linear measurement is y = A z, then the opposite phase gives y^- = A(-z) = -Az = -y. The second observation is algebraically dependent on the first. Over GF(257), v0.30 tested 17 measurement matrices. Adding every opposite row -A produced rank([A;-A]) = rank(A) in 17/17 cases. This is a theorem for linear algebra, not merely a benchmark observation. Therefore simply pushing equally left/right or inward/outward in a linear symmetric system cannot create independent information. The new information must come from **nonlinear response, collision/contact, parity asymmetry, projection, or another symmetry-breaking operator**. --- ## 6. Global phase quotient A residual CNF R and its complete inversion J(R) have identical remaining model counts. For #SAT recursion, inversion also swaps the two children of every variable: J(R)|_(x_v=0) ~ J(R|_(x_v=1)) J(R)|_(x_v=1) ~ J(R|_(x_v=0)). So the unordered child multiset is invariant. v0.30 implements an exact residual quotient using the canonical pair C(R)=min(R,J(R)). 120/120 small independent #SAT checks passed. But a global C2 orbit has size at most two, so this operation can never explain an enormous semantic collapse by itself. Hard snapshot results: n=36: 3,258 syntactic residuals -> 3,258 global antipodal classes n=40: 38,191 syntactic residuals -> 37,875 global antipodal classes Only about 1.008x reduction on the n=40 control. --- ## 7. Every coordinate can have its own phase The stronger interpretation of "opposite phase of every coordinate" is the local group G = (Z_2)^n. A phase vector p in {0,1}^n acts by x -> x XOR p. Every member of this orbit has identical SAT status and identical model count. So an exact compiler may store canonical representative + phase mask. The phase mask is precisely the representation-level answer to "which 0/1 phase did this coordinate start in?" v0.30 adds a polynomial **scope-gauge** canonicalizer. Absolute clause scopes are invariant under sign flips. For each variable, the first scope group with nonzero positive-minus-negative occurrence balance fixes that variable's orientation. When every variable is resolved, the result is a full canonical representative under all independent sign flips. Random 3-SAT audit: n=6..15 120/120 formulas fully gauge-resolved 0 observed invariance failures across random phase masks. Repeated-scope exhaustive control: 137 three-variable formulas for which the gauge was complete all 8 possible phase masks produced the same canonical representative. This is a representation/gauge result, not a SAT algorithm. --- ## 8. How much did local phase help the hard residuals? At the v0.28 hard snapshots: n=36: 3,258 syntactic futures 3,170 local-phase classes reduction ~1.028x n=40: 38,191 syntactic futures 24,682 phase classes reduction ~1.547x The phase orbit is therefore real and nontrivial on the n=40 control. But compare the exact semantic audit already known for those same snapshots: n=36: 3,258 syntaxes -> 5 Boolean future functions n=40: 38,191 syntaxes -> 1 Boolean future function. Phase normalization explains only a fraction of the semantic redundancy. This is an important negative result: **the P-vs-NP wall is not merely hidden 0/1 polarity.** --- ## 9. Hard semantic phase signatures For the hard n=40 snapshot, the one semantic future is FALSE. All 1,024 antipodal pairs of its 11 remaining variables are phase-invariant 00 pairs. For the hard n=36 snapshot: - the giant 3,251-syntax UNSAT class is completely phase-invariant; - the small SAT future classes contain 1, 2, or 3 phase-oriented antipodal pairs. Thus the odd channel can expose the orientation of sparse surviving possibilities, while the giant contradiction class collapses entirely into the even channel. --- ## 10. What this means for the user's expansion/collapse analogy A mathematically coherent computational picture is: phase -1 <---- collapse/invariant core ----> phase +1 lambda=0 The endpoints are antipodal descriptions. The center removes orientation-dependent information. Persistent crossings identify inversion-invariant structure. This resembles an expansion/collapse picture geometrically, but no cosmological conclusion follows from the algebra. For physical systems, a mirrored force experiment provides new information only when the response law is not purely odd-linear. Contact, saturation, curvature, torsion, hysteresis and nonlinear coupling are examples that can generate an even channel. --- ## 11. P-vs-NP implication Phase normalization is useful because it quotients a known symmetry group before search. But: - global inversion gives at most a factor-2 orbit; - local sign flips can give a 2^n representation orbit, but they do not change the underlying Boolean function up to input renaming; - phase flips preserve the unsigned incidence/primal graph, so structural width does not disappear; - exact semantic equivalence remains much larger than phase equivalence on the hard controls. Therefore v0.30 does not prove P=NP. The stronger research target is now: semantic canonicalization = phase gauge + algebraic normal forms + structural interfaces + exact future equivalence. Dual phase is one exact symmetry layer inside that larger compiler. --- ## 12. Strongest surviving result The new operator is not "calculate everything twice." It is: evaluate the involution once, separate invariant and oriented components, quotient phase-equivalent descriptions, and spend computation only on structure that survives both directions. That is a genuine compiler optimization when the problem contains phase redundancy.