generator: 4 new speculations across co-emergence and gaussian-gravitational-decoherence

programs/co-emergence/claims/ce-ftoy-attractive-coupling-subcritical.md

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+---
+id: ce-ftoy-attractive-coupling-subcritical
+program: co-emergence
+tier: speculation
+status: live
+statement: >
+  For real gamma > 0 (attractive self-consistency weighting, the sign opposite
+  every instance tested to date, all fixed at gamma = -1 or -2) F_toy's unique
+  fixed point loses stability through a subcritical flip bifurcation rather
+  than the supercritical one established at gamma = -1: onset produces a
+  finite-amplitude jump to a period-2 cycle, with a finite beta-window of
+  bistability (stable fixed point and stable cycle coexisting) below the
+  crossing, rather than a continuous cycle-amplitude bound growing
+  continuously as sqrt(beta - beta_flip) from zero.
+object:
+  name: F_toy
+  definition: "F(psi)_sigma = exp(gamma R_sigma(psi)) / ||exp(gamma R(psi))||_2"
+  space: the probability simplex (real branch, gamma real)
+hypotheses:
+  - gamma real and strictly positive
+  - A symmetric
+  - h in R^N
+  - beta swept across the flip eigenvalue crossing at -1
+depends_on:
+  - id: ce-toy-fixed-point-multiplicity
+    role: context
+    transfers: same-object
+    justification: >
+      Same object F_toy; explicitly not the same question. That dead
+      speculation asked whether a SECOND fixed point of F exists at the
+      cycling parameters (killed: exactly one root found by exhaustive
+      root-finding). This asks whether a second stable ATTRACTOR (a cycle)
+      can coexist with the unique fixed point in a finite parameter window -
+      a bistability question the multiplicity search did not address, since
+      it searched for additional roots of F, not additional attracting sets
+      of the iteration.
+  - id: ce-riem-classical-unique
+    role: context
+    transfers: same-object
+    justification: >
+      Same F_toy object at a different, unexplored sign of gamma; the
+      uniqueness claim there is stated and evidenced only for the tested
+      gamma = -1, -2 branch and does not cover gamma > 0.
+falsifier: >
+  Track the cycle amplitude near the flip crossing for several gamma > 0
+  values; a continuous onset with amplitude growing as sqrt(beta - beta_flip)
+  from exactly zero, matching the gamma = -1 supercritical result, or the
+  absence of any beta-window where both the fixed point and a period-2 cycle
+  are simultaneously stable under forward and backward beta sweeps, falsifies
+  the subcritical claim.
+consequence: >
+  A subcritical transition would mean self-consistency selection is
+  history-dependent (hysteretic) for attractive coupling: which configuration
+  F_toy's iteration lands on would depend on initial conditions near the
+  transition, not only on beta and gamma. That is a genuinely different
+  selection-principle gap from the multiplicity question already closed
+  (a second fixed point), and would matter for any argument that treats "the"
+  self-consistent solution as the map's unique attractor rather than merely
+  its unique fixed point.
+provenance: {born: 2026-08-03, born_by: generator}
+---
+
+Every real-gamma instance in the record — [[ce-toy-fixed-point-multiplicity]],
+[[ce-riem-classical-unique]], [[ce-second-iterate-real-spectrum]] — fixes
+gamma at -1 or -2 (repulsive self-consistency weighting: R appears with a
+negative sign in the exponent). The opposite-sign branch has not been swept in
+any recorded instance. Bistability (coexisting fixed point and cycle) is a
+structurally different failure mode from the multiplicity question the
+generator already asked and lost: that speculation was about a second FIXED
+point of F; this is about a second stable ATTRACTOR of the iteration, which a
+uniqueness-of-fixed-points result does not by itself exclude.
+
+Axiom check: purely an internal statement about the toy model's Riemannian
+branch (gamma real, theta = 0); no background metric, foliation, or time
+parameter is invoked — F_toy is a discrete self-map, and "history-dependence"
+here means dependence on the iteration's starting point, not on any physical
+time evolution. No axiom violation is apparent. Flagged honestly: if
+subcritical, describing the outcome as "selection" already presupposes F_toy's
+iteration is the relevant dynamical process picking out the physical
+configuration, a reading the paper's own framing licenses but this claim does
+not re-derive.

programs/co-emergence/claims/ce-mass-phase-residual-in-ggd-kernel.md

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+---
+id: ce-mass-phase-residual-in-ggd-kernel
+program: co-emergence
+tier: speculation
+status: live
+statement: >
+  Because mass requires Lorentzian signature (ce-mass-signature) and Lorentzian
+  signature carries the complex self-consistency phase theta (the CE-10
+  theta-signature identification), a genuinely massive two-branch superposition
+  cannot supply the gaussian-gravitational-decoherence noise kernel
+  N_0000(x, x') exactly as the real bilinear (c^4/4) Delta_rho(x) Delta_rho(x')
+  that ggd-noise-kernel derives: writing each branch's expected stress-energy
+  with its self-consistency-weight phase factor explicit produces a residual
+  term of order sin(theta) that survives in N_0000 for any pair of branches
+  whose co-emergence phases differ, vanishing only in the theta -> 0
+  (Riemannian, hence per ce-mass-signature formally massless) limit.
+object:
+  name: N_0000(x, x'), decomposed via each branch's self-consistency phase theta
+  space: expectation values of the stress-energy operator T00 over the two branches entering the Einstein-Langevin analysis
+hypotheses:
+  - each branch is a genuinely massive configuration (Lorentzian per ce-mass-signature)
+  - a curved, dynamical, generically symmetry-free 4-manifold
+  - each branch's co-emergence phase theta is well defined and, in general, differs between the two branches
+  - non-relativistic stress-energy operator T00 = c^2 rho(x - X_CM)
+  - orthogonal branches
+  - each branch has definite centre-of-mass position
+depends_on:
+  - id: ce-mass-signature
+    role: load-bearing
+    transfers: cross-object
+    justification: >
+      Different object (a particle-classification argument on a curved
+      manifold vs. an expectation-value calculation on branch densities); the
+      claim borrowed is the identification mass <-> Lorentzian signature, not
+      any structure of the mass-shell argument itself. Note this dependency
+      currently carries an audit verdict of "demote" (Wigner's flat-space
+      classification applied to a curved, generically symmetry-free
+      manifold); this speculation is exposed to that same gap and does not
+      independently repair it.
+  - id: ggd-noise-kernel
+    role: load-bearing
+    transfers: cross-object
+    justification: >
+      Different object (a semiclassical stress-energy expectation-value
+      identity vs. this speculation's proposed phase-dependent correction to
+      it); the two formalisms have never been checked for mutual consistency
+      across the program boundary, which is exactly what this speculation
+      tests.
+falsifier: >
+  Recompute N_0000 keeping the self-consistency phase explicit in each
+  branch's T00 expectation value (writing the branch amplitude with its
+  e^{i theta} weight factor rather than treating Delta_rho as a real classical
+  density difference from the outset). Expectation values of a Hermitian
+  operator in any physical state are real by construction; if the phase
+  factors cancel identically for exactly this reason, with no residual
+  sin(theta) surviving in N_0000 to any order, the speculation is falsified
+  outright.
+consequence: >
+  If a real residual term survives, GGD's noise kernel is incomplete for any
+  branch pair with differing co-emergence phase, predicting an additional
+  decoherence channel set by theta rather than by Delta_rho alone -
+  potentially discriminating from Delta_rho-only. If the residual does not
+  survive (the more likely outcome a priori, given that T00 expectation
+  values are manifestly real), this would be the first explicit check that
+  co-emergence's mass/signature construction and GGD's stress-energy
+  construction are mutually consistent where they overlap - a cross-program
+  consistency question neither program's record currently addresses.
+provenance: {born: 2026-08-03, born_by: generator}
+---
+
+A cross-program consequence: no existing claim connects co-emergence's
+mass-signature identification to gaussian-gravitational-decoherence's noise
+kernel, and the two frameworks have never been checked against each other at
+the point where they should overlap - both concern the stress-energy content
+of a massive superposition.
+
+Axiom check, stated honestly rather than resolved: combining the two
+requires specifying at what point in "time" theta is evaluated, since GGD's
+T00 expectation value is defined at a Schrödinger-picture instant on a fixed
+background time slice, while co-emergence's theta is a property of a
+self-consistent FIXED POINT of a timeless map, not of a state's evolution.
+Treating "the branch's theta at the moment N_0000 is evaluated" as
+well-defined risks smuggling a preferred-time identification between GGD's t
+and co-emergence's fixed-point locus that neither program's axioms license.
+This is flagged as a genuine hole in how the two objects would even be
+compared, not resolved here - a speculation that violates an axiom in its own
+construction is still worth proposing, per the generator's own criteria,
+provided the violation is named rather than hidden.

programs/co-emergence/claims/ce-second-iterate-radius-power-law.md

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+---
+id: ce-second-iterate-radius-power-law
+program: co-emergence
+tier: speculation
+status: live
+statement: >
+  The second-iterate spectral radius rho_2(beta) of D(G o G) at the F_toy
+  attracting fixed point decays as a power law rho_2(beta) ~ C beta^{-1} for
+  beta well past the flip onset, with C independent of N, driven by the
+  Jacobian's spectral weight concentrating onto a single direction aligned
+  with the dominant softmax component as the fixed point sharpens toward a
+  simplex vertex.
+object:
+  name: D(G o G)
+  definition: "D(G o G)(a) = S(e) M S(b) M, with S(p) = diag(p) - p p^T, b = G(a), e = G(b)"
+  space: tangent space of the probability simplex
+hypotheses:
+  - gamma real
+  - gamma = -1
+  - the marginal-coupling matrix A symmetric
+  - the softmax form induced by the self-consistency weight
+  - beta swept at least two orders of magnitude past the flip onset
+  - h in R^N
+depends_on:
+  - id: ce-second-iterate-real-spectrum
+    role: load-bearing
+    transfers: same-object
+    justification: >
+      Same object, D(G o G); this speculation proposes a specific rate for the
+      decay that claim's own gaps field records as unexplained ("does not
+      explain WHY the second-iterate spectral radius decays to ~0.037 rather
+      than merely staying below 1").
+falsifier: >
+  Fit log(rho_2(beta)) against log(beta) across at least two decades of beta at
+  several N; a fitted exponent that is not close to -1 (say outside
+  [-1.3, -0.7]), or that drifts systematically with N, falsifies the power law
+  as stated. The recorded data point rho_2 ~ 0.037 near beta ~ 50 is a single
+  sample and does not by itself confirm or refute a specific exponent.
+consequence: >
+  A clean inverse-beta law would give the missing mechanism the parent claim
+  flags as absent (the failed Rayleigh-quotient bound diverges instead), and
+  would let ce-riem-classical-unique's bound on the exact-to-bound contraction
+  ratio be re-derived analytically rather than only observed to fall to
+  1e-310 at large beta. A non-power-law or N-dependent exponent would instead
+  show the decay is not a single universal mechanism, narrowing what
+  ce-second-iterate-real-spectrum's palindrome argument can be said to explain.
+provenance: {born: 2026-08-03, born_by: generator}
+---
+
+Mined from [[ce-second-iterate-real-spectrum]]'s own gaps field and the
+Rayleigh-quotient failure recorded on [[ce-riem-classical-unique]] ("the bound
+diverges... because the smallest eigenvalue of S(e) tends to zero as the cycle
+concentrates near a simplex vertex"). That failure is itself evidence for the
+concentration mechanism proposed here, but a divergent bound is not a
+convergent rate, and no exponent has been fit.
+
+Axiom check: this is a purely internal statement about the toy model's
+Jacobian at gamma real (Riemannian branch, theta = 0); it does not invoke a
+background metric, a preferred foliation, or unitary time evolution — F_toy is
+iterated as a discrete self-map with no time parameter involved. No axiom
+appears to be smuggled. The one honesty flag: "concentrating near a simplex
+vertex" describes the *attracting* orbit's location, which is itself an
+empirical fact about this map (not assumed here), carried over unchanged from
+the parent claim's evidence record.

programs/gaussian-gravitational-decoherence/claims/ggd-cpmg-odd-even-colored-asymmetry.md

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+---
+id: ggd-cpmg-odd-even-colored-asymmetry
+program: gaussian-gravitational-decoherence
+tier: speculation
+status: live
+statement: >
+  Against a genuinely colored (non-static, non-delta-PSD) gravitational noise
+  spectrum with a finite correlation time - unlike the exactly-static kernel
+  that made ggd-cpmg-degeneracy vacuous - CPMG-n coherence recovery shows a
+  measurable odd/even asymmetry in the pulse count n: odd-n filter functions
+  retain a residual x^2 factor (F ~ x^4 / (64 n^4)) that even-n sequences
+  algebraically cancel (F ~ x^6 / (256 n^4)), so two CPMG sequences with
+  matched total duration and pulse spacing but n and n+1 pulses would show
+  measurably different coherence-recovery magnitudes, providing a diagnostic
+  of the noise spectrum's shape that does not require resolving which
+  mechanism sets its correlation time.
+object:
+  name: filter-function overlap integral for a CPMG-n sequence against a colored noise power spectral density
+  space: single-qubit (or matter-wave interferometer) coherence under the Einstein-Langevin Gaussian-noise model already adopted by this program
+hypotheses:
+  - the noise kernel has some genuinely finite correlation time tau_c (i.e. is not the exactly-static single-realization kernel of ggd-single-realization)
+  - CPMG pulse sequences with n and n+1 pulses at matched total duration T and matched inter-pulse spacing T/n vs T/(n+1)
+  - the power spectral density is colored (not white, not an exact Lorentzian where the asymmetry may wash out - see falsifier)
+depends_on:
+  - id: ggd-cpmg-degeneracy
+    role: context
+    transfers: same-object
+    justification: >
+      Same object (CPMG filter functions against this program's noise
+      kernel), explicitly a different regime. The dead claim's own defense
+      pass derived the odd/even structural finding used here (odd n gives
+      F ~ x^4/(64 n^4), even n gives x^6/(256 n^4)) but only evaluated it
+      against the exactly-static kernel, where the PSD is a delta function
+      at zero frequency and every sequence (odd or even, any n) gives
+      identically zero filter response - the degeneracy that killed the
+      claim. This speculation asks whether the same structural asymmetry
+      survives once the kernel is genuinely colored, a regime the dead claim
+      never reached.
+falsifier: >
+  Compute the filter-function overlap integral for a specific colored PSD
+  (the Lorentzian crossover form entering ggd-material-crossover, or a
+  generic 1/f^alpha form) at matched n and n+1. If the ratio
+  F_odd(n) / F_even(n) tends to 1 as the spectral content moves away from the
+  IR-cutoff-dominated regime where the dead claim's law was derived - i.e.
+  the odd/even structural difference is subleading to the overall n^-4 decay
+  common to both parities for realistic tau_c - the diagnostic is falsified
+  as practically useless, independent of whether the asymmetry exists in
+  principle.
+consequence: >
+  If the asymmetry survives at a computable magnitude, it gives an
+  experimentally accessible pulse-count-parity diagnostic for the shape of
+  the gravitational noise spectrum that is orthogonal to, and available
+  before, the open GGD-1 material-crossover question - it needs only that
+  some finite tau_c exists, not which phonon (or other) mechanism sets it.
+  If it washes out, that sharpens ggd-cpmg-degeneracy's negative result: the
+  odd/even structure found there would then be an artifact of the IR-cutoff
+  convention specifically, not a generic feature of pulse-sequence filtering
+  against colored gravitational noise.
+provenance: {born: 2026-08-03, born_by: generator}
+---
+
+Mined directly from [[ggd-cpmg-degeneracy]]'s tombstone: its defense pass
+derived the odd/even structural finding as a correction to the attacking
+pass's exponents, in service of showing the degeneracy claim was vacuous under
+a static kernel - but the structural finding itself was never evaluated
+against any colored spectrum, only flagged as mattering "for 1/f-type peaked
+spectra" and washing out "against a Lorentzian" in the one comparison made.
+That comparison was for a single fixed n, not the n-vs-(n+1) asymmetry
+proposed here.
+
+Axiom check: this stays entirely within the program's existing
+Einstein-Langevin / Gaussian-noise Assumption 1 and the Diosi-Penrose-style
+kernel already adopted; it introduces no new background structure, preferred
+frame, or foliation. It is conditional on the currently open question of
+whether tau_c is finite at all (GGD-1, halted under needs-human) but is
+explicitly agnostic to which mechanism sets tau_c - phonon, amplitude
+suppression, or otherwise - unlike that milestone, which turns on identifying
+the mechanism. This claim would still need that some finite-tau_c regime
+exists to have any content; if GGD-1 resolves toward "no accessible platform
+reaches the crossover," this speculation would be moot for realistic
+platforms even if its filter-function mathematics is correct in principle.