A quantum memory that accumulates k successive records of a single system can audit itself: its memory-side statistics obey polynomial identities, and its physical realizability is a positivity question. The companion paper (Gleason from Inside, doi:10.5281/zenodo.22014602) proved the two-record certificate (identity ideal + one back-action-corrected moment inequality M*) and left the k >= 3 ladder open. This note closes it for the single-collision comb architecture at every k. The entire identity structure is generated by the transverse Bloch block of the per-record collision channel, read in its two operator orders: a route-damping chain law (every skip-pair correlation as a sum over sine-pair routes, each damped by the visibility of every record strictly between its sine vertices) and a response transfer law carrying odd higher-sine routes; both are census-exact at machine precision through five records at every span and start. The identity variety's free chart has dimension exactly 2k+1, one transport identity per added record. The completeness theorem: a chart point is realizable if and only if its outcome probabilities are valid and the FIRST pair's M* is positive semidefinite; every later adjacent certificate is a consequence (the transported effective state inherits the Bloch ball). The proof is a constructive, optimization-free rung inversion; machine confirmations include 200/200 exact round trips, a 1703-point redundancy test with zero violations, and exact inversion of all 1094 points of a persisted certification sample. The comb moment hierarchy collapses at level one: identities do all equality work (linear in k) and a single two-record inequality does all positivity work forever. The frontier is named: multi-collision cells, reused cells, and non-commuting rotation axes are outside the theorem. The ledger of failed predictions, including two wrong transfer compositions killed by census gates, is an appendix.
A quantum memory that accumulates k successive records of a single system can audit itself: its memory-side statistics obey polynomial identities, and its physical realizability is a positivity question. The companion paper (Gleason from Inside, doi:10.5281/zenodo.22014602) proved the two-record certificate (identity ideal + one back-action-corrected moment inequality M*) and left the k >= 3 ladder open. This note closes it for the single-collision comb architecture at every k. The entire identity structure is generated by the transverse Bloch block of the per-record collision channel, read in its two operator orders: a route-damping chain law (every skip-pair correlation as a sum over sine-pair routes, each damped by the visibility of every record strictly between its sine vertices) and a response transfer law carrying odd higher-sine routes; both are census-exact at machine precision through five records at every span and start. The identity variety's free chart has dimension exactly 2k+1, one transport identity per added record. The completeness theorem: a chart point is realizable if and only if its outcome probabilities are valid and the FIRST pair's M* is positive semidefinite; every later adjacent certificate is a consequence (the transported effective state inherits the Bloch ball). The proof is a constructive, optimization-free rung inversion; machine confirmations include 200/200 exact round trips, a 1703-point redundancy test with zero violations, and exact inversion of all 1094 points of a persisted certification sample. The comb moment hierarchy collapses at level one: identities do all equality work (linear in k) and a single two-record inequality does all positivity work forever. The frontier is named: multi-collision cells, reused cells, and non-commuting rotation axes are outside the theorem. The ledger of failed predictions, including two wrong transfer compositions killed by census gates, is an appendix.
An interior observer cannot step outside its world to test the quantum probability rule; everything it knows arrives through records written into its own memory. This paper asks what such an observer can certify about the rule from those records alone, within a minimal two-record collision register. Results: the register's complete memory-side statistics (census) in closed form, making it a self-calibrating interior tomograph; an interior Gleason theorem at first order (the audit's response matrix has full rank with kernel exactly the Born-model tangent; the threshold is two records); the global forgery geography (a bounded shell and a lattice stratum pass every identity) with an identity-futility theorem proving this necessary and an exact repair (one PSD condition on a back-action-corrected moment matrix, complete at two records with gauge classified); the completed arena (det M_c = sin^2 phi (1 - |b|^2): one complex PSD condition is physicality plus tomography, its imaginary part the register's measurable complexness); the real-amplitude strike (complexness is credential-relative: cell dimension and readout sharpness; with both enforced, real registers are convicted by the census's zero channels at a dial rate of 0.40 census gap per unit certified sharpness); and a hardware campaign on ibm_kingston in which the audit detected, localized, mechanism-confirmed, and time-tracked a device systematic. Pre-registered throughout; failed predictions ledgered in an appendix.
A proof assistant's kernel evaluates by reducing terms symbolically—and that is slow. Asking the kernel to sum a stabilizer code's weight enumerator over its 224 elements is hours of reduction; the same sum, compiled, is milliseconds. This gap is not a footnote. It means a machine-checked certificate is not one thing: for an object too large for the kernel to reduce, certification splits into a ladder. On the first rung the kernel proves the method exact—that bucket elimination equals the flat sum, a small general theorem with no compiler trust. On the second the compiler executes the instance—the actual census over a real code's generators, carrying only the compiler-correctness axiom. On the third the consequences are axiom-free: the quantum MacWilliams identity forces the normalizer spectrum, the code distance, and the sub-threshold error-rate prefactor from the executed spectrum, with an empty footprint. We make this ladder the object of study, draw its trust boundary exactly, and certify the rotated surface code [[25, 1, 5]]—distance 5, prefactor 160—past a census the kernel cannot touch. The ladder has an outer wall that is a theorem, not a limitation: computing an arbitrary code's exact spectrum is #P-hard, so the bounded-treewidth codes—the surface and toric family that hardware actually deploys—are the tractable island by complexity theory, and the determinant shortcut is provably blocked on the coupled CSS enumerator by a local free-fermion obstruction. We also state the discipline the ladder demands: a certificate that reads back a constant it was handed certifies nothing; certification must be tied to the actual object, and we show the difference is a theorem that fails when the tie is cut. Published by The Institute for Applied Ontological Mathematics.
The exterior identifiability theory (papers 1-3 of this program) is turned interior: in the collision-model architecture of quantum Darwinism, the causal structure itself hands out operation credentials, and the question becomes whether the world can certify its own factorization from the records it keeps. Generator level: one-sided sufficiency with a closed-form sine law; the parity-twist theorem (redundancy is structurally powerless against exactly one global ghost); four restoration laws with derived constants (tilted writes (4/3)/sin^2 eta; second readout basis 8/5 exactly; one odd-dimensional cell free, with the interior/exterior quantifier reversal; dissipation (4/15)(5+p)/p: objectivity needs an open world). Statistics level: the chain law (relational timing with gentleness dampers), the complementarity floor with interior optimal write strength, and the dividend: weak records at the interior optimum extract MORE frame information than projective records, superlinearly (~K^1.85, sustained through K = 10, factor 5.1 at K = 10), with mechanism (shared-fragment covariance conditioning), the readout gap split in half (product bases recover half the quantum-Fisher gap at every K), and re-collision potency with an intra-fragment two-time accumulation mechanism confirmed two ways. Pre-registered; deaths ledgered.
Two walls stand at the edge of the two-ring instrument, and this paper is the map of both. Many qubits first. One setting reads two amplitudes and N qubits carry 2^N, and that wall is the only thing between the instrument and many-body quantum mechanics. Part I is the map of the attempt to scale through it, chains of certified rings in lockstep, each qubit a setting with its digit certificate, entanglement between neighbours as the shared-radius coupling of the companion paper, local turn laws carrying the gates, a geometric form of the tensor networks that are classical physics' own escape from the same wall. What carries over is stated exactly and verified. Across every cut of a chain the two sides share one spectrum, so the cut's radius is its certificate. An eight-link chain built by one layer of levers holds at most one bit per cut and is held by a chain of bond two to a fidelity of 1.000, and a random eight-qubit state has rank 16 at the middle cut and is held to 0.162, the wall exhibited. The move past it is the nesting of the companion repair paper, a ring inside every cell, done here as a setting inside every share. A setting whose two shares are themselves settings, applied down to single qubits, holds any pure state of N qubits exactly, 2^N − 1 settings for 2^N amplitudes, the standard's own count, verified on random states at every size to eight qubits and on the same random state the chain held to 0.162, with every setting on the ladder. At a node the entanglement of that qubit with the rest is sin θ √(1 − |⟨ψ₀|ψ₁⟩|²), the Wootters concurrence for a pair, largest at equal shares over mirror branches. A Z turn on qubit k turns the azimuths at depth k and nothing else, and a measurement reads the root's shares and leaves its branch. On ibm_kingston a random three-qubit state was prepared from its seven settings as the machine's own gate angles and its populations read within 0.048 of the tower's shares. The contraction that turns the chain into a reading is classical arithmetic, and the ladder guard of the companion paper stands on it, 112 fixed-point words carrying their base-60 digit pairs, 200 random corruptions of one to six bits, every one caught from the shadow alone and every word recomputed, while the unguarded contraction returned a wrong energy in 155 of the 200. The five problems of the chain program are each settled where they stand: the tower is the canonical form, a gate on any qubit needs no route through its neighbours, the contraction's cost is measured, the classes the tower compresses are counted, and the width a chain needs to follow the dynamics is the entanglement across its cuts, measured. Three levels second. A pure qutrit carries four real parameters and one setting reads two lengths. The populations are read by two levers, one inside the other, the tower of Part I at its smallest: the first splits level 0 from the rest and the second splits the rest between levels 1 and 2, so every split is reachable, a state fully in one level included. Two certificates ride them, the sum to one and the product law, the coherence between two levels at most the product of their two amplitudes and equal to it for a pure state, and on a thousand random qutrits every one-percent misread of the populations is caught by one of the two, 1,000 of 1,000, 992 of them by the sum alone. The threefold turn a qutrit wants is not a single-digit multiplier, and that is a theorem, not bad luck. The Pisano period over the rank of apparition is 1, 2 or 4 (Vinson 1963), confirmed by enumeration to modulus 250. But the sixty-ring carries the threefold turn as a two-term law, F(n + 20) ≡ 5F(n + 1) + F(n) (mod 10) at all sixty positions, and 178 moduli below 250 carry the same law at a third of their period. The two relative phases are read as well, by three two-level settings, one for each pair of levels, whose azimuths obey the addition law exactly, the third certified by the other two, so that three populations and two certified phases rebuild every pure qutrit, verified on a thousand. Every number is verified by the script in the appendix, and the document is not built unless every check prints True.
Version 4 (2026-08-25). Two additions from the correction campaign, carried as dated in-body correction records. The (2,5) entry, printed as a prediction in v3, is now census-exact -- and it was the wall law's first divergence test. At the constructed maximal-wall stratum point the stiff transverse count is exactly 10, so alpha_stratum(2,5) = 5; a fresh 60k ensemble's calibrated read is consistent with 5, with alpha = 10 (the superseded single-wall count's prediction) excluded at p = 0.000 and demonstrated power ~ 1; and a stratum-conditional paired ladder tracks the exact rank-10 Hessian form to <= 0.002 while a rank-20 comparator separates by >= 2.7 on identical draws. The law m_C(2,d) = 2(d-1) is census-exact at d = 3, 4, 5, now including the first point where it diverges (by 2x) from the superseded count. The (3,4) entry, printed open in v3, is now a PAIR of statements, and neither alone is alpha(3,4). The posted product-wall counting landed census-exact (alpha_stratum(3,4) = 5, so the asymptotic exponent satisfies alpha(3,4) <= 5), while the accessible window at n = 3e4 is steeper (calibrated 90 percent set [5.4, 6.9], with the previously read 4.0-4.1 excluded at power ~ 1): the codim-10 stratum's prefactor has not engaged at reachable depth, and window and asymptote have visibly decoupled. This is the scope lesson of the (d >= 3) x (d' > d) regime -- census plus conditional ladder measure the law, global windows measure the pre-asymptotic shoulder -- and it retro-explains the earlier fresh-seed ladders' instability as shoulder reads. Version 3 (2026-08-20). One SUPERSESSION: the qutrit-qutrit exponent closes. Versions 1 and 2 left alpha(3,3) open, v1 by a small-sample fit reading 3.25 and v2 by the bracket [2.3, 3.0] read off a descending window ladder. Both readings were artifacts of the estimator problem for which the pair-type table was already corrected in v2. The descent is attenuation tracking sample count rather than tail depth: an ensemble drawn from an EXACT alpha = 3.0 law descends by 0.62 across the ladder against the real data's 0.34, so a descending ladder is evidence FOR a constant exponent, not against one. Replacing the window fit with a calibrated Monte-Carlo test and inverting it over candidate exponents gives alpha(3,3) = 3.0 at p = 0.517 with a 90 percent confidence set [2.75, 3.10]; v1's 3.25 is excluded at p = 0.017. Power against alternatives is 0.990 / 0.590 / 0.120 / 0.575 / 0.990 / 1.000 at true alpha = 2.50 / 2.75 / 3.00 / 3.25 / 3.50 / 4.00. An independent re-audit at n = 10^4 reads flat near three on two seeds. Disclosed defect, carried in the paper: the test's own calibration row missed at the low edge (rejection rate 0.207 at a nominal 0.10 when the truth is alpha = 2.50, 4.4 standard errors high), so the lower end of the confidence set is softer than printed. This ensemble separates 3.0 from 3.25 with about 58 percent power and has none to separate 3.0 from 2.95. The (3,3) CARRIER family remains unattributed: the exponent is settled, the stratum that carries the accessible tail there is not. Version 2 (2026-08). Changes from v1: the pair-type exponent table is corrected (v1's alpha(2,4) = 2.08 was a 50-sample order-statistics fluctuation, placed at the 0.15th percentile of its own estimator's sampling distribution; three independent codebases now agree on the wall law alpha(2,d) = d, census-exact at d = 3 and 4); the transverse constant 1/12 is derived in closed form (diagonal cost 1/6, halved by the congruent partner's equal-cost correction); the (2,3) tail's carrier stratum is fully attributed (an isotropic-intertwiner family sharing the exponent 3); the qutrit-qutrit exponent is honestly bracketed open at [2.3, 3.0]; and the dispersion rule (tail-exponent fits print bootstrap dispersion or do not print) is adopted. The v2 ledger appendix carries the correction anatomy. Why is the multipartite certification constant heavy-tailed, where does the tail live, and what kills it? Localization: the credential pencil is exactly block-diagonal over party-subset supports; the tail is the worst pair; geometry statistics reconstruct from pair statistics (14/14 factor-band comparisons). The law: a frame-congruence theorem: whenever the two parties' operation frames subtend equal Bloch angles, the rotation intertwining them is an exact common kernel direction; the zero set is codimension one at generic angles, transverse growth is universally quadratic (mu = (Delta theta)^2/12), and the qubit-pair tail exponent is exactly 1/2 (the previously measured 0.39 is the finite-window slope of this law). The dichotomy: pairs containing a d >= 3 party measure exponents 2.08 to 3.69 because higher-dimensional operation pairs carry many local invariants that must all match: freedom helps the certifier. Engineering: heterogeneous dimension profiles tame the tail; qubit-pair presence splits all measured geometries into two clean classes; screening is a five-minute two-party calculation. Pre-registered; two entry gates that caught wrong conventions and all deaths ledgered.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.