Em03-H-v8 Exact Fermionic Extraction, Quench Diagnostics, and Stored-Packet Excitation-Channel Recovery Complete-channel proofs, reported computational results, and reproducible certification

We distinguish a fixed microscopic recovery experiment, a restricted continuum excitation-code experiment, and a gravitational interpretation requiring an independent physical identification.

For a quartic Majorana Hamiltonian and an even spectral code, we construct complete accessible and complementary channels, retaining arbitrary references and preparation purifications. We prove exact quench-leakage, best-compression, departure, and interpolation formulas. A resonant controlled extractor has explicit signed fermionic branches and a complete decoder. Ideal full transfer is exact. Its minimal jointly invariant reached space has dimension ${\sum }_{m\mathrm{even}}(m+1)\mathrm{rank}(P_{m}V)$, at most forty for the frozen eight-mode instance P8.

The supplied P8-new execution materials report $E_{\mathrm{code}}\approx 4.318140424446922$, $L_N\approx 1.4473228981681243$, and departures $0.032277775445887835$ and $0.12385356780895762$ at times $1/8$ and $1/4$. The new results summary additionally reports agreement between Volterra and spectral/RK candidate calculations near $q\approx -0.77925+0.07681\mathrm{i}$, with $\eta =|q{|}^2\approx 0.61313$.

For the declared continuum reduction, storage changes the occupied packet while extraction remains coupled to the fixed packet operator. A unique Volterra solution determines the complete signed excitation-qubit channel and its information-bearing complement. The specified complete decoder has exact half-diamond error $1-\eta$. The reported candidate values therefore suggest error about $0.38687$, above one quarter, for this decoder. They do not establish a rigorous numerical exclusion.

The supplied summary reports a patched rational defect-checking run with uniform defect bound about $0.117$, but its resulting transmission enclosure is $[0,1]$. Exact certificate rationals, original candidate arrays, patched-source identifiers, and complete inspection reports are not supplied. Version 8 incorporates these reported results, repairs mixed interval/complex dispatch, and introduces journaled failure recording. The new v8 programs remain unexecuted. P8 rank certification, a nontrivial continuum amplitude enclosure, and the validated quarter-error verdict remain open. No P8 gravitational dictionary is constructed.

The mathematical and physical tasks are separate:

The first two chains are constructed below. The third is conditional.

The supplied analytical sources are -input.tex and 03-H-v7.tex. The supplied P8-new execution materials and the new 03-H-v7-results-summary.txt are documentary numerical inputs, not mathematical substitutes for error bounds.

$\mathsf{A}$: analytical.

A consequence of a stated mathematical specification and proof.

$\mathsf{R}$: reported.

A numerical result or execution claim attributed to supplied records, without independent authentication of its underlying binary artifacts.

$\mathsf{I}$: inspected.

An artifact reloaded and checked in a separately recorded invocation.

$\mathsf{V}$: validated.

A numerical conclusion with rigorous arithmetic, selection, approximation, and applicable model-error allowances.

Physical implementation is a further evidential task.

A statement in a summary that an inspection succeeded is itself reported evidence unless its inspection artifact is available. Likewise, a displayed rational-checker status is not an exact certificate when only rounded summary values are supplied.

{}1.12

Version 8:

1. incorporates the newly reported candidate values and coarse defect-check result;

2. distinguishes exploratory failure of the specified decoder from a validated exclusion and from optimal-recovery impossibility;

3. corrects the interval/complex multiplication dispatch defect in the printed v7 checker;

4. replaces repeated fixed-name checkpoint writes with a journal and unique temporary snapshot files;

5. retains exact JSON/NPZ node-link checks and complete saved spectral/RK states;

6. states an explicit numerical accuracy target for the next candidate certificate.

The primary controls, coupling packet, storage interval, microscopic coefficients, and code-selection rules are unchanged. No new physical identification is inferred from solver agreement.

The new v8 programs are not the patched v7 source reportedly used. That patched source and its hash must still be supplied to authenticate the reported invocation. Historical sources and manifests are not overwritten.

The message is $M$, reference $Q$, boundary $B$, and bath $R$. Accessible bath modes form $A$. A non-message factor $K$ may have an inaccessible purifying register $G$.

Unsubscripted norms are operator norms. Subscripts $1$ and $\mathrm{F}$ denote trace and Frobenius norms. Set

All channel distances retain arbitrary references. For an isometry $S$, ${\mathrm{Ad}}_S(X)=SX{S}^{†}$, and ${\mathcal{R}}_{\tau }(X)=\mathrm{tr}(X)\tau$. Matrix units are $E_{\alpha \beta }=|\alpha ⟩⟨\beta |$.

For a qubit channel,

Let $|\Phi ⟩=(|00⟩+|11⟩)/\sqrt{2}$. Define squared target-overlap fidelities

Root fidelity $f(\rho ,\sigma )={\Vert \sqrt{\rho }\sqrt{\sigma }\Vert }_1$ satisfies $f\ge 1-d_{\mathrm{tr}}(\rho ,\sigma )$. Its maximal purification overlap equals $f$, with sufficiently enlarged purifying spaces. The diamond norm of a difference of finite channels can be optimized on referenced density operators.

These conventions and facts are standard Watrous,FuchsVandeGraaf. Recovery and information–disturbance frameworks are established BenyOreshkov,KSW. A channel error at most $e$ against the identity implies both target overlaps are at least $1-e$, by the pure-target projector test.

Use ${\mathcal{H}}_B={\mathcal{F}}_{\mathrm{f}}({\mathbb{C}}^n)$, $n\ge 2$, with

The annihilation rule is

and creation uses the corresponding preceding-occupation sign.

These operators satisfy

For

one has $\{{\chi }_a,{\chi }_b\}={\delta }_{ab}I$ and

Proof. For equal labels the two orders give complementary occupation projections. For unequal labels, exchanging actions reverses the preceding-occupation sign. Expansion gives the Majorana relations. Finally, $\mathrm{i}{\chi }_{2j-1}{\chi }_{2j}={c_j}^{†}{c}_j-I/2$.

Fix real coefficients and set

where $E_{\min }$ is the minimum over both parity sectors. Quartic terms are Hermitian because reversal requires six exchanges. With ${\mathsf{P}}_B=(-1{)}^{N_{\mathrm{B}}}$,

Number need not commute with the quartic Hamiltonian.

SYK motivates the quartic setting but does not identify this fixed code with a black-hole microstate code MaldacenaStanford. No disorder average replaces the fixed coefficient array.

Use an independent initially vacuum bath ${\mathcal{H}}_R={\mathcal{F}}_{\mathrm{f}}({\mathbb{C}}^n)$, with

The one-particle order is $c_1,\dots ,c_n,b_1,\dots ,b_n$. Odd operators in the two blocks anticommute. Pair hopping terms commute and $\Vert G_{\mathrm{hop}}\Vert \le n$.

The signed regrouping ${\mathcal{R}}_A$ places accessible modes first, preserving each declared internal order. For whole-bath access,

Target A stores under $H_{\mathrm{B}}$, replaces it by $\Omega N_{\mathrm{B}}$, and extracts through resonant hopping. Target B retains an interacting boundary generator during transfer. A result for one target does not implement the other.

The exact microscopic construction solves Target A. The continuum experiment below is independently declared and retains its own storage generator.

A general finite path is

with bounded piecewise continuous real controls, fixed independently of the unknown logical input.

Choose $D=2g$ orthonormal even eigenvectors ${\beta }_{ia}$, $i=0,1$, $a=1,\dots ,g$, with fixed spectral, phase, and degeneracy conventions. Set

The full code algebra is $V\mathcal{B}(MK)V^{†}$. The message algebra is $V(\mathcal{B}(M)\otimes I_K)V^{†}$.

For a message channel prepare ${\rho }_{QM}\otimes {\sigma }_K$, with independent ${\sigma }_K$. Uniform subsystem recovery instead allows arbitrary ${\rho }_{QMK}$.

The finite path has a unique unitary propagator. For

the accessible channel and complete complement are the two partial traces of ${\mathrm{Ad}}_{Z_{A,t}}$.

If ${\sigma }_K={\sum }_a{p}_a|a_{\sigma }⟩⟨a_{\sigma }|$, define

With $E=B(R\setminus A)G$,

These formulas determine every referenced input.

Proof. Finite-dimensional integral evolution gives existence and uniqueness; self-adjointness gives unitarity. Preparation, evolution, regrouping, and purification form an isometry. Its partial traces are CPTP. Matrix-unit expansion gives the formulas and their reference extensions.

Total-even evolution makes accessible referenced outputs block diagonal in accessible parity. Total parity factors in access order as accessible times environmental parity, and the environmental partial trace gives the assertion. Accessible parity pinching may therefore precede decoding.

Set $n=8$, $g=1$. Enumerate the $1820$ increasing quartets of $\{1,\dots ,16\}$ lexicographically and define

Order exact even-sector eigenspaces by energy. Within each exact degenerate eigenspace, project lexicographic occupation words and apply Gram–Schmidt, discarding zero projections. Give each vector a first nonzero occupation coefficient that is positive real. Select the first two vectors.

Retain the global energy shift over both parities, spectral window $[0,2h_4]$, $t_0=\Omega =T=1$, bath vacuum, and signed whole-bath access.

The coefficient generator is deterministic; typicality is not asserted. The broad spectral window is not a certified gravitational low-energy window.

Define

The pulse is smooth, nonnegative, and has area $\pi /2$. At $u/T=0,1/4,1/2,3/4,1$, its cumulative areas are $0,0,\pi /4,\pi /2,\pi /2$, by support and reflection symmetry.

With ${\Gamma }_a=\sqrt{2}{\chi }_a$, the matrices are signed Gaussian-integer permutations. Hence

The coefficient denominator $4193344$ is not the Hamiltonian denominator.

The exact matrices satisfy

zero even–odd blocks, and

Equivalently,

Proof. Hermiticity and parity give the matrix symmetries and blocks. Quartic Clifford monomials are trace orthogonal and square to $I/16$. The boundary dimension is $256$. Substitute the rational denominators to obtain the integer identity.

The entrywise assembly bound is

Assembly and trace sums fit signed 64-bit integers. The complete square sum is evaluated with arbitrary-size Python integers.

The v8 inspector constructs the same Majoranas independently by

Their monomial entries are exactly representable Gaussian units. This is a distinct construction from the historical occupation-action assembly, not a second call to its assembly routine.

Storage gives $V_{\mathrm{st}}=e^{-\mathrm{i}{t}_0{H}_{\mathrm{B}}}V$, with unchanged range. The actual quench changes the generator by

Define

Furthermore,

attained at $A=\Omega \nu$.

Proof. Insert $P+(I-P)$ between number operators. The retained term is ${\nu }^2$, and the remainder is the Gram matrix of $(I-P)N_{\mathrm{B}}V$. Projection onto $I-P$ gives the compression lower bound; choosing $A=\Omega \nu$ removes the retained component. Rayleigh optimization gives the scalar quantities.

Storage conjugates the compressed matrices by a common code unitary, preserving their spectra.

For $\Omega =1$, set $F(u)=V^{†}{e}^{-\mathrm{i}u{N}_{\mathrm{B}}}V$.

For $u\ge 0$,

For general $\Omega$, replace $u$ by $\Omega u$.

Proof. Retained probability is ${\Vert F(u)x\Vert }^2$; optimize its complement. Center at $A_0=N_{\mathrm{B}}-nI/2$, with norm $n/2$. The scalar phase does not affect departure, and the integral Taylor remainder gives

The off-code amplitude is $-\mathrm{i}u(I-P)N_{\mathrm{B}}V+R_u$, with the displayed remainder bound. Triangle inequalities give the first bounds; the squared-norm difference is at most $2u{L}_N\Vert R_u\Vert +{\Vert R_u\Vert }^2$.

A validated grid of covering radius $\delta$ requires an additional allowance $n\delta$ for a uniform-time claim.

Proof. For $F_0=V^{†}{e}^{-\mathrm{i}u{A}_0}V$, $\Vert F_0\Vert \le 1$, $\Vert {\dot{F}}_0\Vert \le n/2$, and $\Vert \frac{\mathrm{d}}{\mathrm{d}u}({F_0}^{†}{F}_0)\Vert \le n$. Apply the Rayleigh principle.

For every $j$, $V^{†}{c}_{j}V=V^{†}{c_j}^{†}V=0$. Nevertheless even joint hopping can act nontrivially. Also,

A nonzero boundary subspace invariant under all $c_j,{c_j}^{†}$ is the full Fock space.

Proof. Parity makes odd compression its negative. Joint hopping changes both parities together. Insert $P+(I-P)$ for the composite identity. An occupation projector selects a nonzero word in an invariant vector; annihilations give vacuum and creations give every word.

Nonzero $L_N$ excludes exact number dynamics confined to the initial code. It does not exclude an enlarged comparison space or approximate dictionary, and it is not the extraction-loss parameter.

Target A extracts under

Since $[N_{\mathrm{tot}},G_{\mathrm{hop}}]=0$,

For disjoint $L,T$, define $I(L,T)=\#\{(t,l):t\in T,l\in L,t<l\}$.

Writing $c=\cos \Theta (u)$, $s=\sin \Theta (u)$, boundary-first evolution is

If $K_L$ is the bath coefficient, signed bath-first maps are

Consequently,

Zero powers are interpreted as one at endpoints.

Proof. The active creator transformation is

Expand the increasing creator product. Transferred creators cross retained ones, giving the inversion sign. Equation (eq:regroup) supplies the additional access sign. Partial traces give the channels, and unitarity gives completeness.

Set

The full-code channels used below are

For even inputs, an additional branch phase cancels in the accessible channel. It must not be omitted on general parity-coherent inputs.

Let $P_m$ project onto boundary occupation $m$,

For $\varphi ={\sum }_{|P|=m}{\varphi }_P|P⟩$, define

Each $B_{m,k}$ is an isometry. Distinct $(m,k)$ images are orthogonal, and

Absent endpoint terms are zero.

Proof. Each partition has a unique union; distinct partition words are orthogonal, and there are $\left(\genfrac{}{}{0ex}{}{m}{k}\right)$ terms per word. Hopping moves one occupied pair. Counting moves and comparing the binomial normalizations gives the coefficients with the CAR signs already included in $B_{m,k}$.

The smallest subspace containing the prepared stored code and invariant under $N_{\mathrm{tot}},G_{\mathrm{hop}}$ is

Thus

Restricted propagation is exact on the code. For P8, $4\le \dim {\mathcal{K}}_{\mathrm{ext}}\le 40$.

Proof. Number-sector projections are polynomials in $N_{\mathrm{tot}}$ on this finite space. Any invariant subspace containing the initial code contains $B_{m,0}{\mathcal{V}}_m$. The recurrence successively generates every $k$, and proves invariance of the displayed sum. Orthogonality proves the dimension formula.

For P8 the endpoint ranks are at most one and interior ranks at most two, giving $1+6+10+14+9=40$. Also $2=\mathrm{rank}V\le {\sum }_m{r}_m$. If $r_0=1$, a further rank has weight at least three. If $r_0=0$, the total weight is at least six. Hence the unconditional lower bound four.

This reduction uses uniform hopping and common frequency. It does not construct a forty-dimensional boundary CAR representation or reduce the physical bath-output interface to forty dimensions.

For the joint spectral projection ${\Pi }_{m,\lambda }$, where $\lambda =-m,-m+2,\dots ,m$,

Every projected-code rank is $r_m$.

Proof. For a fixed occupied-pair set, each exchange has eigenvalues $\pm 1$; the all-boundary state has overlap squared $1/2$ with each. The sum has the binomial law. Different pair sets are orthogonal and invariant. Every allowed binomial weight is positive.

Let $J$ identify occupation bases and set

Retained Kraus operators are $(I_M\otimes ⟨a{|}_K)W^{†}$. For an off-code basis $z_{\ell }$, add $2^{-1/2}|i{⟩}_M⟨z_{\ell }|$, $i=0,1$. Their effects sum to $I_R$, so the decoder is CPTP on every bath input.

For ${\eta }_{\mathrm{m}}={\sin }^2\Theta (T)$ and $0\le \Theta (T)\le \pi /2$,

Arbitrary correlated $QMK$ inputs are allowed.

At full transfer, with $L_T=J{e}^{-\mathrm{i}(\Omega T+\pi /2)N_{\mathrm{B}}}{V}_{\mathrm{st}}$,

Recovery is exact. For independent gauge preparation, the complete message complement is replacement by $|\varnothing ⟩⟨\varnothing {|}_B\otimes {\sigma }_G$.

Proof. The corrected no-loss branch is $J{{\eta }_{\mathrm{m}}}^{N_{\mathrm{B}}/2}$. For a purified encoded input, overlap with ideal transfer and boundary vacuum is $A=\mathrm{tr}({\rho }_B{{\eta }_{\mathrm{m}}}^{N_{\mathrm{B}}/2})\ge 0$. Since $1-r^k\le k(1-r)$, $1-A\le E_{\mathrm{code}}(1-\sqrt{{\eta }_{\mathrm{m}}})$. Pure-state distance is at most $\sqrt{2(1-A)}$. Decode and contract.

At full transfer only the no-loss branch remains, so the isometry factors as $L_T$ times boundary vacuum. The partial traces give the endpoint identities, and the independent gauge purification gives its fixed complementary factor.

For input dimension $D$, define normalized Choi states

For channels with common input dimension $D$,

If all matrix-unit trace-norm differences are at most $r$, then $d_{\diamond }(\Phi ,\Psi )\le \min (1,Dr/2)$.

Proof. A pure referenced input is $(A\otimes I)|{\Phi }_D⟩$, up to a reference isometry, with ${\Vert A\Vert }^2\le D$. Trace-norm multiplication proves the Choi bound. Alternatively write its vector as ${\sum }_{\alpha }{v}_{\alpha }\otimes |\alpha ⟩$, with ${\sum }_{\alpha }{\Vert v_{\alpha }\Vert }^2=1$. The output difference norm is at most $r({\sum }_{\alpha }\Vert v_{\alpha }\Vert {)}^2\le Dr$. Divide by two and use convexity.

For a finite qubit channel with complete complement $\mathcal{C}$, put

A polar-isometry decoder satisfies

Every deterministic decoder satisfies

Consequently,

Proof. The Bell-input dilation purifies $J_{\mathcal{C}}$. A Bell pair times a purification of ${\tau }_E$ purifies the product center. Enlarging the decoding ancilla, the polar purification isometry attains root overlap $f\ge 1-{\delta }_E$. The decoded Bell distance is at most $\sqrt{1-f^2}\le \sqrt{2{\delta }_E}$. Lemma lem:choi gives the factor two. The maximally mixed constant decoder has error $3/4$, by the four-Pauli decomposition and the Bell test.

For any decoder, dilate its output and project its pure Bell-input state onto the target Bell vector. Normalizing the environmental projection gives a Bell-product purification at distance $\sqrt{1-F_e}$, when $F_e>0$. After tracing decoder systems, $J_{\mathcal{C}}$ is within that distance of $I_Q/2\otimes \tau$. Since

trace duality gives ${\Vert \mathcal{C}(P)/2\Vert }_1\le 2\sqrt{1-F_e}$. The left side is the complementary pair distance. The Bell-projector test gives error at least $1-F_e$; worst-case overlap cannot exceed Bell overlap. The case $F_e=0$ is immediate.

The complete complement retains all preparation purifications. A discarded environmental marginal cannot silently replace it.

Nearly-$Ad{S}_2$, Schwarzian dynamics, and probe matter motivate the model MSY,EMV,MTV. The following tensor-product reduction is declared, not derived with a controlled full-gravity remainder.

The motivating disk parameters are

A motivating action is

with $h_{\tau \tau }={ϵ}^{-2}$, $\varphi {|}_{\partial M}={\varphi }_{\mathrm{r}}/ϵ$. Eight complex massless Dirac species have action

A disk-extending spin structure and homogeneous reflecting boundary condition may be declared using

with $P_{-}\psi ={\overline{\psi P}}_{-}=0$, including variations. The boundary variation vanishes because $P_{+}{\gamma }_n{P}_{+}=0$.

The exterior metric

motivates the reflected positive-frequency one-body model. The exact analysis begins with the directly declared multiplication operator

and normal-ordered matter Hamiltonian

Declare

The spectral normalization is part of the declaration. Backreaction, fluctuating dressing, and loop corrections are neither included nor bounded. The static Fock vacuum is not assumed to be a thermal or horizon-regular Hartle–Hawking state.

Set

Creation is linear and annihilation antilinear in its packet. These are normalized orthogonal even excitation states, not identified black-hole microstates. Packet operators are bounded CAR operators; they are not unsmeared coincident fields.

Direct integration gives

The code-state storage-energy means are $3/2,5/2$, with variances $1/12,7/12$. They are not energy eigenvectors.

Extraction retains storage and adds

The primary experiment remains

with pulse (eq:pulse). Other six species and bath modes remain vacuum.

Storage changes the occupied packet to $f_{\mathrm{st}}(\omega )=e^{-\mathrm{i}\omega t_0}f(\omega )$. Extraction still couples to the fixed $f$. The common Schwarzian state evolves to $e^{-\mathrm{i}(t_0+T)E}\varphi (E)$. It remains factorized and independent of the logical input, so it is a harmless spectator for the reduced channel, while retained as a pure complementary factor.

On $L^2({\mathbb{R}}_{+})\oplus \mathbb{C}|b⟩$, set

For bounded piecewise continuous real $v,\kappa$, $K(u)$ has a unique interaction-picture unitary propagator.

Proof. Relative to $h\oplus 1$, the perturbation is bounded, self-adjoint, and finite rank. Strong continuity on the finitely many packet vectors gives norm continuity of the corresponding rank-one interaction-picture operators on each control segment. The norm-convergent Dyson series gives existence and uniqueness; self-adjointness gives unitarity.

The exact evolution is equivalent to the unique continuous solution

The matter packet is

For the primary experiment, $A(0)=(4-2\mathrm{i})/5$, $B(0)=0$.

Proof. Variation of constants gives the packet formula. Contracting with $f$ uses $⟨f,e^{-\mathrm{i}th}f⟩=g(t)$. Variation of constants for $\mathrm{i}\dot{B}=B+\kappa A$ gives the second equation. Conversely, the packet formula reconstructs the mild Schrödinger solution; uniqueness proves equivalence.

Let $q=B(1)$, $\eta =|q{|}^2$. Unitarity gives ${\Vert \psi (1)\Vert }^2=1-\eta$. Zero coupling gives $B=0$; if $v=\kappa =0$, $A(u)=g(u+1)$.

For the primary Volterra operator, the componentwise maximum-norm majorant is $1+|\kappa (u)|$. Thus