deltaPhi
plain-language theorem explainer
Defines the BMV entangling phase invariant at the named SI lab geometry: the weak-field combination (G m1 m2 T / ℏ)(1/r_LL + 1/r_RR − 1/r_LR − 1/r_RL). Anyone citing the certified falsifier band or the exact rational value 26696/47475 starts here. The body is a direct specialization of the upstream weak-field branch invariant to the module's fixed masses, time, and four branch separations.
Claim. Let $\Delta\Phi$ be the real number obtained by evaluating the weak-field BMV branch invariant at the named SI inputs $G$, $\hbar$, masses $m_1,m_2$, interaction time $T$, and the four branch separations $r_{LL},r_{LR},r_{RL},r_{RR}$. Explicitly, $\Delta\Phi = \frac{G m_1 m_2 T}{\hbar}\bigl(\frac{1}{r_{LL}}+\frac{1}{r_{RR}}-\frac{1}{r_{LR}}-\frac{1}{r_{RL}}\bigr)$.
background
This module is the BMV falsifier floor, not a pillar-3 discriminator. Any quantum mediator predicts BMV entanglement, so a positive signal cannot separate Recognition Science from GR+QFT. What is formalized is narrower: under the Newtonian weak-field phase model plus a fixed named geometry, the two-qubit branch state is non-product, with the entangling invariant certified in the band $[1/2, 7/10]$ and bounded away from $0 \bmod 2\pi$.
The invariant itself is the standard Bose–Marletto–Vedral combination of four gravitational phases. Upstream, BMVPositive supplies the weak-field formula that multiplies the prefactor $G m_1 m_2 T/\hbar$ by the geometric bracket of inverse separations. The SI anchors $G$ and $\hbar$ (and the local mass, time, and length literals) are measured inputs rounded to exact rationals for kernel-checked arithmetic; they are not RS-derived constants.
The module doc is explicit that the magnitude of the branch phases relative to the RS amplitude channel remains MODEL/OPEN: Track 2 forcing uniqueness of the channel does not yet pin the numerical size of $\Delta\Phi$.
proof idea
Pure definition, not a proof. The body is one application of the upstream weak-field branch invariant to the eight named SI parameters of this file (two constants, two masses, one time, four separations). No tactics, no lemmas beyond that specialization. Downstream equalities such as the definitional match to the four-phase branch invariant are then rfl.
why it matters
This is the single numeric object on which the whole certified band rests. Downstream, the definitional equality to the four named phases, the exact rational evaluation $\Delta\Phi = 26696/47475$, the band edges $1/2 \le \Delta\Phi \le 7/10$, strict positivity, $\Delta\Phi < 2\pi$, and non-congruence to $0 \bmod 2\pi$ all unfold or rewrite through it; the witness-band theorem packages those facts for the entanglement claim.
In framework terms it is the concrete falsifier floor quantity: a clean null at this geometry contradicts the package {weak-field phase model + named geometry}. It does not by itself refute the RS framework, because the premise that the RS gravitational channel produces Newtonian weak-field phases of this magnitude is still MODEL/OPEN. The declaration therefore sits at the boundary between fully certified algebra and the unformalized magnitude link.
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