Separate classes with separate premises, no shared physical derivation. Exact poset/endofunction censuses through size4 and elementary structural claims checked. Cycles do not select a canonical phase origin.
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Result
Event posets provide ideal states and possible histories but do not generally select growth weights. Supplied finite updates give cycles/transients and closed record cells relative to errors/observation; sink SCCs count maximal separated record cells, not necessarily exhaustive invariant record functions. Transformation monoids allow stable sectors, periodic steps and visible classical noncommutativity together; action and effects remain inputs. Counterworlds show these properties do not follow from composition or information alone.
Required assumptions
Finite event order as starting data
A finite strict partial order is supplied as event/precedence structure without metric, space, rates or probability. Order is an input in this class.
Supplied finite update, observation and errors
Finite X, deterministic step f and observation o, optionally error support T, are inputs. A record-cell system is a family of disjoint nonempty forward-closed observationally distinguishable sets, not automatically an exhaustive invariant partition.
Supplied finite operational action and effects
A transformation monoid acts on finite X; effects define operational distinguishability. Composition, action, effects and any selected time generator are separate starting data.
Complete bounded classification of three binary interfaces; no particle/generation/dimension interpretation. One-shot consistency does not force a fixed order.
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Result
Of 4096 no-self-influence candidates,744 are uniquely consistent under all64 local one-shot contexts.488 are acyclic; each of the remaining256 has an explicit reversible local two-copy control with zero or two fixed points (128 each). All744 process/order entries and256 witnesses were independently reproduced. The488 positive cases analytically work for all replica counts and arbitrary joint deterministic controls.
Required assumptions
Finite deterministic processes with full joint replica control
Supply finite nonempty local input/output sets and total w:product O->product I. Every finite number of coordinatewise independent replicas allows every joint deterministic map I_k^M->O_k^M at corresponding interfaces, including constants/erasure. Require at least one global solution for every such experiment; do not assume a global order.
General elementary finite deterministic theorem, mathematically reviewed twice. Arbitrary encoders, decoders and erasure are used; no general reversible two-copy theorem. Interfaces are not derived events. No novelty priority claim.
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Result
For arbitrary finite alphabets/interfaces, existence under every finite joint local replica control, uniqueness under every such control, acyclic static influence and compatibility with at least one fixed total order are equivalent. A directed cycle of length m allows at most m copies with separately sensitized edges and odd XOR feedback without a solution. Acyclicity gives unique triangular recursion. No unique total order or physical time is selected.
Required assumptions
Finite deterministic processes with full joint replica control
Supply finite nonempty local input/output sets and total w:product O->product I. Every finite number of coordinatewise independent replicas allows every joint deterministic map I_k^M->O_k^M at corresponding interfaces, including constants/erasure. Require at least one global solution for every such experiment; do not assume a global order.
Exact algebraic equivalence of original unmarked pairs and covariance of full passive form transport. No claimed physical P1 symmetry, spatial mirror interpretation of detG, or independently produced source normalization. Original fixed source form and reader still distinguish marked realizations.
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Result
Actual v568 pairs (U0,V0) and (U2,V2) have simultaneous GL3(Z) intertwiner G=[[1,0,0],[0,1,0],[0,2,-1]], G²=I,detG=-1, commuting with Sigma and fixing the specified winding data while sending leading vector (1,2,0) to (1,2,4). All simultaneous intertwiners are tG. The fixed theta(A)=Sigma A^T Sigma is not preserved and its reflected word gives4/8; transporting the entire form gives4/4. Explicit linear intertwiners connect the original three-rate clock to both Qplus/3, so rates alone do not select a twin.
Required assumptions
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Exact finite original case. No nonexistence of B_N, selected continuum state or E8 refutation. Previous energy test uses twist1 without zero modes and is unchanged. Numerical norm diagnostics normal and -OO byte-identical; independent proof review and rerun accepted.
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Result
On unchanged v988 Nx16 Ny6 M1, explicit integral boundary zero vectors, modular ranks over F65537 and exact sign pairing prove ranks190/192/192/192. Periodic counts are95 negative,95 positive,2 zero; other twists96/96/0. Original covariance C=Pminus+Pi0/2 differs from pure occ_frame projector P=Pminus+Q with rankQ1 in the nullspace: norm(C-P)HS²=1/2 and norm(C-C²)HS²=1/8. Both original functions are valid and deliberately distinct. A state-preserving microscopic dictionary cannot silently exchange these periodic states.
Required assumptions
Actual periodic v988 state and occupied frame
Keep unchanged qwz_cylinder, occ_frame and fermi_cov at Nx16 Ny6 M1 and original twists0,1,2,3. Compare their actual finite CAR covariances. Physical selection of either state or equality with the E8 vacuum is not assumed.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only this proxy-fitted constant in this actual finite original case is excluded. The formula is exact; numerical values are checked to1e-9 tolerance. Larger constants, separate historical value2.86 and general continuum estimates are not excluded.
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Result
On unchanged v988 data Nx16,Ny6,M1 the twist1 Slater state against h0 has exact variance formula Tr(P h0(1-P)h0P)=||(1-P)h0P||HS^2. Numerically it gives7.0875042817 and first excitation norm2.6622367065. The original first central sum proxy is identically0; its fit C=1.2040009128 gives C^2=1.4496181980 and already fails to bound the actual norm here.
Required assumptions
Literal charged scaling contract and original Slater case
Audit actual H_N, Psi_lambda,N and bosonic refinement definitions in the original charged scaling contract and unchanged v988 qwz_cylinder/occ_frame functions at Nx16,Ny6,M1, h0 at twist0 and Slater state at twist1. This audits that source path and finite energy case, without assuming microscopic bosonization or a uniform continuum bound.
Source audit of this particular original path. No nonexistence claim for B_N and no rejection of the established P1/P2 compiler or mathematical E8 field extension.
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Result
The original charged scaling contract starts with H_N as the actual QWZ/CAR GNS space and declares Psi_lambda,N by a normal boson exponential. Its concrete refinements I_charge tensor Gamma_s(j) act instead on a separately chosen bosonic lattice Fock space. No microscopic B_N between these systems with B_2N J_N,2N=V_N,2N B_N and state, energy, parity, sector and localization compatibility is constructed there. The exact charge-lattice isometry F(2q)=lambda does not replace this field morphism.
Required assumptions
Literal charged scaling contract and original Slater case
Audit actual H_N, Psi_lambda,N and bosonic refinement definitions in the original charged scaling contract and unchanged v988 qwz_cylinder/occ_frame functions at Nx16,Ny6,M1, h0 at twist0 and Slater state at twist1. This audits that source path and finite energy case, without assuming microscopic bosonization or a uniform continuum bound.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Stipulated E8_1 lattice vacuum representation. Bosonic currents are not identified with elementary4D fermions or native CCR mediators. Internal quarter grading is not physical quarter time. Contact terms, Z[0], absolute phase and original P1 Phi-dependence do not follow from these normalized coefficients.
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Result
The executable E8 dictionary contains240 root vertices and8 Cartan currents with cocycle, adjoint signs and quarter grades60/64/60/64. The vertex formula and Cartan Ward recursion determine all separated current coefficients of the normalized source series. Mixed spinor currents have5/8+3/8=1; the checked connected four-point response is nonzero. Standard lattice theory implemented and independently checked here, not a new physical interaction.
Required assumptions
Reverse selection in the discrete closure framework
The origin paper discrete E8 closure framework with rank, family, spinor and simple-current conditions is given. Its numerical fixed-point identities are not identified with a physical operator realization.
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
Quarter phase is retained
Comparisons retain the original charged quarter phase; switching to another phase branch is not an admissible solution to the selection problem.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Source audit of the specified reverse chain. No retraction of numerical self-consistency and no independent derivation of the framework.
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Result
Origin954–981 and v6/v15/v350–v352 recover g_car=5 and eight within the chosen closure framework; v154 supplies the conditional local extension. The checked v350–v352 functions do not construct Phi-dependent D_rel/D_ref or physical insertion operators. This scopes those calculations, rather than asserting general impossibility.
Required assumptions
Reverse selection in the discrete closure framework
The origin paper discrete E8 closure framework with rank, family, spinor and simple-current conditions is given. Its numerical fixed-point identities are not identified with a physical operator realization.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Two canonical invariants of the given four-mark object. Not a new proof task; evaluate original field/state maps separately. This records an existing result to repair retrieval and continuity, without claiming a new scientific finding.
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Result
Existing v189 and the Origin paper obtain rank H1(P1\D)=3 and h0(P1,O(D))=5 from ONE four-point divisor D=mu4 on P1. This is an established common geometrical/topological origin of both numbers, not a new hypothesis or reopened task. Retain known anchor identities and reverse selection. Every genesis continuation starts from this theorem; open further source maps do not downgrade it.
Required assumptions
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only the specified R096 branch under its jointly required premises. No derived physical Higgs measure, general H(x) field theory or P1 source completion.
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Result
For actual X_H=(2cosh(ell h_H))^-1, g(u)=Tr_Q X_H=6sum_j[2cosh(ell sqrt(p²+y_j²u))]^-1. Under the specified real face frame and positive nonconstant radial law g strictly decreases, f(u)=u/(1+u) strictly increases, and Cov_mu(f,B_Q)<0. The face identifications cannot be independently rotated without the actual transport.
Required assumptions
Same constant Higgs and two-face carrier
R096 compares the same original constant Higgs parameter on the same actual two-face CAR carrier and fixed source caller. Face labels are not internal chiral L/R labels. This is a conditional source comparison, not derivation of the primitive P1 identification.
Real face frame and positive nonconstant radial law
The negative mixed covariance witness uses the original 36-dimensional quark particle module with one common real face frame, ell>0, p>0 and at least one nonzero original Yukawa singular value. Mu is positive normalized and u=H†H is not almost surely constant. Independent face gauge transport is not supplied.
Only the specified R096 branch under its jointly required premises. No derived physical Higgs measure, general H(x) field theory or P1 source completion.
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Result
The conditional identities <O>_H=24m(H)^2 and <N>_H=3m(H) give Z[O]=8Z[mN] with the same integrable unnormalized weights. Positivity is unnecessary for this branch. Actual origin of the mixed insertion in the original solver remains open.
Required assumptions
Same constant Higgs and two-face carrier
R096 compares the same original constant Higgs parameter on the same actual two-face CAR carrier and fixed source caller. Face labels are not internal chiral L/R labels. This is a conditional source comparison, not derivation of the primitive P1 identification.
Actual R054 color-symmetric quasifree fibers
The four-word connection uses the actual constant color-symmetric conditionally quasifree R054 fermion fibers on the same carrier: <O>_H=24m(H)^2 and <N>_H=3m(H). The averaged source is not assumed quasifree.
Identical integrable unnormalized weights and insertions
The linear identity Z[O]=8Z[mN] uses identical integrable unnormalized weights and conditional fiber values on both sides. Positivity or finite complex normalization is not assumed for this linear statement. Physical use still requires deriving the mixed insertion from the original alphabet.