diff --git a/book/The-Incompleteness-of-Observation-FULL.md b/book/The-Incompleteness-of-Observation-FULL.md index 676bd86c4..1e3801e0c 100644 --- a/book/The-Incompleteness-of-Observation-FULL.md +++ b/book/The-Incompleteness-of-Observation-FULL.md @@ -941,7 +941,7 @@ The framework's commitment at Level B is therefore structurally substantive, not The framework's gauge structure runs along an axis orthogonal to the observation axis. Where the observation axis is about scope — which substrata fall within the framework's domain — the gauge axis is about transformations: which substratum-level operations leave all observables unchanged. Like the observation axis, the gauge axis has four levels, with each broader level absorbing more degrees of freedom as gauge. -**Level G1: D-gauge on the emergent Hamiltonian.** The narrowest gauge freedom in the framework is the diagonal-unitary equivalence on the emergent Hamiltonian: $H \to D H D^\dagger$ with $D$ a diagonal unitary in the measurement-outcome basis. This freedom preserves transition probabilities exactly. Energy shifts and the antiunitary reflection also preserve them; completeness of these alternatives under the stated gap and overlap hypotheses is conditional on the sign-aware integer classification H-BG ([GR §3.3]). Two Hamiltonians $H$ and $D H D^\dagger$ produce identical Born-rule probabilities $T_{ij}(t) = |\langle j | e^{-iHt/\hbar} | i \rangle|^2$ in the measurement basis, so they are operationally indistinguishable. The transformation rephases the measurement basis, changing off-diagonal phases of $H$ while leaving its diagonal entries and every transition probability unchanged. +**Level G1: D-gauge on the emergent Hamiltonian.** The narrowest gauge freedom in the framework is the diagonal-unitary equivalence on the emergent Hamiltonian: $H \to D H D^\dagger$ with $D$ a diagonal unitary in the measurement-outcome basis. This freedom preserves transition probabilities exactly. Energy shifts and the antiunitary reflection also preserve them; completeness of these alternatives under the stated gap and overlap hypotheses is conditional on the sign-aware integer classification H-BG ([GR §3.3]; Appendix B.7). Two Hamiltonians $H$ and $D H D^\dagger$ produce identical Born-rule probabilities $T_{ij}(t) = |\langle j | e^{-iHt/\hbar} | i \rangle|^2$ in the measurement basis, so they are operationally indistinguishable. The transformation rephases the measurement basis, changing off-diagonal phases of $H$ while leaving its diagonal entries and every transition probability unchanged. Level G1 is the smallest gauge group in the hierarchy. It operates within a single Hamiltonian realization in a single emergent quantum mechanics — within Level D × Level G1 in the joint hierarchy, the framework's most specific intersection. Its content is the bookkeeping freedom in choosing phase conventions for the emergent basis states, with no implications beyond that. @@ -5806,7 +5806,7 @@ The appendix's content is organized by mathematical structure rather than by cha *B.6 Gauge-theoretic derivations.* Two key gauge-theoretic results from Chapter 5: the conditional H-link link count for $K = 2dm = 6$ internal components per site, and the cubic-rotation-group decomposition of the coupling matrix's eigenvalue multiplicities into $(3, 2, 1)$ — the multiplicities that determine $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$. -*B.7 The lemma chain for the characterization.* The framework's characterization theorem — accessible non-Markovianity ⟺ C1/C3/C4 realization per horizon, with the quantum representation internal and universal ($S \iff D \iff Q_{\mathrm{fb}}$, [Main §3.4]) — relies on a chain of intermediate lemmas: partition-relativity, emergent stochasticity, P-indivisibility from a non-permutation witness, accessible-timescale backflow, and phase-locking. This appendix collects these lemmas with their proofs in compact form, providing the reader a self-contained reference for the framework's logical structure. +*B.7 The lemma chain for the characterization.* The framework's characterization theorem — accessible non-Markovianity ⟺ C1/C3/C4 realization per horizon, with the quantum representation internal and universal ($S \iff D \iff Q_{\mathrm{fb}}$, [Main §3.4]) — relies on a chain of intermediate lemmas: partition-relativity, emergent stochasticity, P-indivisibility from a non-permutation witness, accessible-timescale backflow, and phase-locking, together with the results on which phase-locking rests: Bohr-frequency completeness, the real-to-integer transfer and the two-branch classification conditional on H-BG. This appendix collects these lemmas with their proofs in compact form, providing the reader a self-contained reference for the framework's logical structure. The appendix's content is *complete derivations*, not exposition. Each derivation gives the algebraic steps, the lemmas invoked, and the result with no glosses. For motivation and context, the relevant main-chapter sections are cross-referenced at the start of each derivation. The appendix is designed for readers wanting full technical content; the main chapters give the same results with motivating context but compressed derivations. @@ -6212,10 +6212,32 @@ The framework's characterization theorem — accessible non-Markovianity ⟺ C1/ **Lemma B.7.5 (Phase locking).** *Assume dimension at least three, the global sign-aware classification H-BG of [GR §3.3], and distinct gaps and non-vanishing overlaps for both Hamiltonians being compared. The emergent effective Hamiltonian $\hat{H}_{\text{eff}}$ on the visible sector is then determined by the continuous-time transition data $\{T_{ij}(t)\}$ up to the lemma's full gauge: an overall energy shift, the diagonal-unitary rephasing freedom, and the antiunitary conjugation $\hat{H} \to -\hat{H}^*$ — a genuine twofold ambiguity rather than a convention ([Main §3.1]).* -*Proof.* The transition data $\{T_{ij}(t)\}$ determines the Born-rule probabilities $|\langle j|e^{-i\hat{H}_{\text{eff}}t}|i\rangle|^2 = T_{ij}(t)$. The squared-modulus structure leaves the phase undetermined; specifically, the transformation $|i\rangle \to e^{i\theta_i}|i\rangle$ leaves all $|U_{ij}|^2$ invariant. This is the basis-convention freedom — the choice of phase for each basis state — which is physically trivial. H-BG and the coefficient-labelled exceptional obstruction classify the possible mode correspondences ([GR §3.3]); this step is not supplied by Fourier extraction alone. Conditional on H-BG, the remaining alternatives are this rephasing freedom, the energy shift and the antiunitary conjugation $\hat{H} \to -\hat{H}^*$ stated above. $\square$ +*Proof.* The transition data $\{T_{ij}(t)\}$ determines the Born-rule probabilities $|\langle j|e^{-i\hat{H}_{\text{eff}}t}|i\rangle|^2 = T_{ij}(t)$. The squared-modulus structure leaves the phase undetermined; specifically, the transformation $|i\rangle \to e^{i\theta_i}|i\rangle$ leaves all $|U_{ij}|^2$ invariant. This is the basis-convention freedom — the choice of phase for each basis state — which is physically trivial. H-BG and the coefficient-labelled exceptional obstruction classify the possible mode correspondences (Theorem B.7.5c below; [GR §3.3]); this step is not supplied by Fourier extraction alone. Conditional on H-BG, the remaining alternatives are this rephasing freedom, the energy shift and the antiunitary conjugation $\hat{H} \to -\hat{H}^*$ stated above. $\square$ *Scope.* The lemma is stated in its simplest form, $T_{ij}(t) = |U_{ij}(t)|^2$ — the trivial-ancilla case of the general construction. In the ancilla-marginal form, local generic rigidity is checked at the certified block structures; general completeness is open. The visible data constrain the *dilated* Hamiltonian and configuration projectors, with the energy shift and diagonal-phase freedoms exactly invisible; the extension, its Fourier structure, and the constraint analysis of the larger pair-phase gauge are given in the companion paper ([Main §3.4]) — where the pair-phase gauge reduces to the diagonal one for generic families at each fixed dimension — together with the antiunitary conjugate $H \to -H^*$, retained there as an intrinsic ambiguity of transition data. +**Bohr frequencies and the D-gauge classification.** The proof of Lemma B.7.5 rests on the results collected here, each stated with its hypotheses ([GR §3.3]). Throughout, $U(t) = e^{-iHt}$ with $H$ in frequency units and $H = V\,\mathrm{diag}(E)\,V^\dagger$ on a finite configuration space, so that $V_{ia}$ is the overlap of configuration $i$ with eigenvector $a$, and likewise $H' = W\,\mathrm{diag}(E')\,W^\dagger$; the antiunitary reflection $-\bar{H}$ is the $-\hat{H}^*$ of Lemma B.7.5. + +**Theorem B.7.5a (Bohr-frequency completeness).** *Let $U(t) = e^{-iHt}$ with non-degenerate eigenvalues and non-vanishing configuration-basis overlaps. Then the transition probabilities determine the Bohr-frequency set $\{E_a - E_b\}$: each return probability expands as $|U_{ii}(t)|^2 = \sum_{a,b} |V_{ia}|^2 |V_{ib}|^2\, e^{-i(E_a - E_b)t}$, in which every gap value carries a strictly positive coefficient; every $|U_{ij}(t)|^2$ is a combination of the same frequencies; and consequently any $U'$ of the same form with $|U'_{ij}(t)|^2 = |U_{ij}(t)|^2$ for all $i, j, t$ has $\{E'_a - E'_b\} = \{E_a - E_b\}$.* + +*Proof.* Expanding $U_{ij}(t) = \sum_a V_{ia}\, e^{-iE_a t}\, V_{ja}^*$ gives $|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a - E_b)t}$; at $i = j$ the coefficients are $|V_{ia}|^2 |V_{ib}|^2 > 0$, and grouping terms by frequency value only adds positive quantities — degenerate gaps merge coefficients but cannot cancel them. Distinct characters $t \mapsto e^{-i\omega t}$ of $(\mathbb{R}, +)$ are linearly independent (Dedekind), so equality of two probability families forces equality of the per-frequency coefficients over the union of the two gap sets, and positivity makes the supports — the gap sets — coincide. $\square$ + +**The dimensional obstruction.** Theorem B.7.5a is what the gap equation of §B.4 consumes, and it is less than a reconstruction of $H$. The unitary $U(t)$ is dimensionless, and $\hbar$ enters only through $\hat{H} = i\hbar\,\partial_t U\, U^\dagger$. The transition probabilities are dimensionless functions of $t$ and determine temporal frequencies; converting a frequency to an energy requires the action scale $\hbar$ itself, so no amount of transition data fixes it. The residual ambiguities in passing from frequencies to a Hamiltonian carry no scale either: diagonal rephasing $H \mapsto DHD^\dagger$ moves no modulus, the energy origin $H \mapsto H + E_0$ contributes only the global phase $e^{-iE_0 t}$, the antiunitary reflection $H \mapsto -\bar{H}$ satisfies $e^{-i(-\bar{H})t} = \overline{e^{-iHt}}$ entrywise and so preserves every $|U_{ij}(t)|^2$ exactly, and Theorem B.7.5c below excludes further ambiguities conditional on its stated classification hypothesis. The thermal self-consistency of §B.4, Step 4, is therefore the mathematically obligatory entry point for the dimensionful input. + +**Lemma B.7.5b (Real-to-integer transfer).** *Let $E, E'$ be real spectra on $n$ modes, each with distinct directed gaps — $E_b - E_a = E_d - E_c$ with $a \neq b$ and $c \neq d$ only when $(a, b) = (c, d)$ — and with equal gap sets. Then $E'$ is, after relabeling its modes, a translate $E + E_0$ of $E$ or a translate $-E + E_0$ of its reflection; or else there are relabelings of the two spectra and strictly increasing integer rulers $F, F'$ on $n$ marks, each with distinct positive differences, with equal difference sets, and neither a translate nor a reflection of the other, such that every integer gap identity $F'_d - F'_c = F_b - F_a$ with $a \neq b$ holds for the relabeled real spectra as well.* + +The lemma assumes no classification hypothesis. It is proved in the framework's Lean formalization (kernel: `real_to_integer_surrogate` in `OIBridge/TurnpikeScopeTransfer.lean`), and it is the step through which a classification of integer rulers, such as H-BG below, applies to real spectra. + +**Theorem B.7.5c (D-gauge completeness, two-branch form).** *Assume H-BG: every non-congruent pair of strictly ordered integer rulers with distinct positive differences and equal difference sets has six marks and, after relabeling, realizes either $\mu$ or its global reversal, where $\mu$ sends each ascending mark pair of $\{0,1,4,10,12,17\}$ to the ascending mark pair of $\{0,1,8,11,13,17\}$ with the same difference. The orientation is chosen once for the whole pair. Let $H,H'$ act on the same finite configuration space of dimension $m\ge3$, with distinct nonzero energy gaps and non-vanishing configuration-basis overlaps for both Hamiltonians. If $|U'_{ij}(t)|^2 = |U_{ij}(t)|^2$ for all $i,j,t$, then $H' = DHD^\dagger + E_0$ or $H' = -D\bar{H}D^\dagger + E_0$, for a time-independent diagonal unitary $D$ and a real $E_0$.* + +*Conditional proof.* Equality of the probability families forces equality of the per-frequency coefficients of $|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a-E_b)t}$. Under the stated overlap and gap hypotheses, the spectra have equal difference sets and the coefficient matching induces a bijection of ordered mode pairs preserving directed gaps. A fixed correspondence can be transferred from real to integer spectra while preserving its gap identities, strict order, distinct differences and non-congruence (Lemma B.7.5b): its realization constraints are homogeneous rational-linear equalities, strict inequalities and inequations, and rational approximation in the equality subspace followed by denominator clearing preserves the finite list of strict conditions. H-BG then supplies a congruent pair or one of the two globally oriented exceptional correspondences. For congruent spectra, diagonal-line rigidity fixes the moduli and the phase-coboundary relation fixes the phases, giving $W_{i,\tau a}=d_i\beta_a V_{ia}$ with $d,\beta$ unimodular. The column phases cancel in the reconstructed Hamiltonian. Reflection congruence applies this argument to $(\bar V,-E)$ and gives the second branch. For the positive exceptional correspondence $\mu$, the coefficient identities force flat moduli and constrain the phase data to a graph of maximum clique three, excluding the six orthogonal rows required by unitarity. For the reversed correspondence, reversing the source mode pair conjugates its coefficient tensor; applying the same obstruction to the conjugated source eigenbasis excludes that orientation as well. Thus the two Hamiltonian branches follow conditional on H-BG. $\square$ + +*Scope.* The complete comparison of H-BG with the published integer classification of Bekir and Golomb [A. Bekir and S. W. Golomb, IEEE Trans. Inform. Theory 53 (2007) 2864–2867], including all normalizations and exceptional orientations, is open. The unsigned formal premise `BGIntegerClassification` of the Lean formalization is false: the strictly ordered rulers $(0,1,4,10,12,17)$ and $(0,4,6,9,16,17)$ have distinct positive differences and equal difference sets, are neither translates nor reflections of one another, and satisfy the reversed exceptional correspondence while admitting no relabelings with the required positive orientation. The kernel implication `twoBranch_of_BGClassification` therefore does not establish reconstruction from the physical hypotheses. The example contradicts that formal premise, not the published classification. Bohr-frequency completeness, the real-to-integer transfer, congruent reconstruction and the positive six-mode obstruction are kernel-proved independently of it. The listed ambiguities are exactly invisible: $H+E_0$ contributes only a global phase, while $-\bar H$ conjugates the propagator entrywise. The gap condition is used where coefficient matching induces a bijection of ordered mode pairs: with degenerate gaps one frequency's coefficient is a sum over several mode pairs. The $C_4$ ring of [SM] Proposition 20 has distinct eigenvalues, uniform overlaps and degenerate gaps; its conjugate $\bar H$ reproduces every $|U_{ij}(t)|^2$ and conjugates the loop product $H_{01}H_{12}H_{23}H_{30}$, which no diagonal conjugation changes. The ring therefore does not exhibit a failure, without the gap condition, of the two-branch form conditional on H-BG: $\bar H$ lies in the reflection branch, $\bar H = -D_s\bar H D_s^\dagger$ with $D_s = \mathrm{diag}(1,-1,1,-1)$. The raw frequency set alone cannot distinguish the non-congruent homometric rulers; reconstruction uses coefficient-labelled frequency data. + +**Corollary B.7.5d (Operational antiunitary invariance).** *Transposing the preparation, every operation, and the effect simultaneously — $\rho \mapsto \rho^T$, $\mathcal{G} \mapsto T \circ \mathcal{G} \circ T$, $E \mapsto E^T$ — preserves every finite circuit probability, for arbitrary operations and arbitrary finite sequences, hence for every fixed outcome branch of any adaptive instrument strategy; the transformation carries Kraus operations to Kraus operations with conjugated operators, preserves the trace normalization, and on unitary evolution is exactly the branch map $H \mapsto -\bar{H}$.* + +Enlarging the data from transition probabilities to arbitrary preparations, instruments, adaptive finite circuits, and effects therefore cannot remove the second branch: at the purely operational level the two branches are not two empirically distinguishable theories but two antiunitarily related reconstructions of the same complete circuit statistics, and they become physically distinct only once an oriented structure is fixed ([GR §3.3]). (kernel: `circuit_invariance`, `string_invariance`, `transposeMap_kraus`, `kraus_normalization`, `unitary_channel_transpose` in `OIBridge/AntiunitaryInvariance.lean`) + **Theorem B.7.6 (Characterization theorem).** *In the redefined form of [Main §3.4]: accessible non-Markovianity of the visible process is equivalent, on each finite horizon, to embedded observation under C1, C3, C4 (C2 outside the equivalence, carried by the conditional physical-memory theorem); the unitary quantum representation is internal and universal — $S \iff D \iff Q_{\mathrm{fb}}$, the realization's permutation unitary, the compressed form under (T) — nontrivial at the recurrence scale where the dynamics is P-indivisible, with the fixed-$\hat{H}$ form constructive for realized processes:* $$\text{Accessible non-Markovianity on } V \iff \text{Embedded observation of } (S, \varphi) \text{ with C1, C3, C4, per finite horizon}$$ diff --git a/book/The-Incompleteness-of-Observation-FULL.pdf b/book/The-Incompleteness-of-Observation-FULL.pdf index 975f3aa67..b3887a418 100644 Binary files a/book/The-Incompleteness-of-Observation-FULL.pdf and b/book/The-Incompleteness-of-Observation-FULL.pdf differ diff --git a/book/The-Incompleteness-of-Observation-FULL.tex b/book/The-Incompleteness-of-Observation-FULL.tex index 4e8fd7bf1..eddb0b688 100644 --- a/book/The-Incompleteness-of-Observation-FULL.tex +++ b/book/The-Incompleteness-of-Observation-FULL.tex @@ -4354,11 +4354,11 @@ \subsection{3.4 The gauge axis}\label{the-gauge-axis}} transition probabilities exactly. Energy shifts and the antiunitary reflection also preserve them; completeness of these alternatives under the stated gap and overlap hypotheses is conditional on the sign-aware -integer classification H-BG ({[}GR §3.3{]}). Two Hamiltonians \(H\) and -\(D H D^\dagger\) produce identical Born-rule probabilities -\(T_{ij}(t) = |\langle j | e^{-iHt/\hbar} | i \rangle|^2\) in the -measurement basis, so they are operationally indistinguishable. The -transformation rephases the measurement basis, changing off-diagonal +integer classification H-BG ({[}GR §3.3{]}; Appendix B.7). Two +Hamiltonians \(H\) and \(D H D^\dagger\) produce identical Born-rule +probabilities \(T_{ij}(t) = |\langle j | e^{-iHt/\hbar} | i \rangle|^2\) +in the measurement basis, so they are operationally indistinguishable. +The transformation rephases the measurement basis, changing off-diagonal phases of \(H\) while leaving its diagonal entries and every transition probability unchanged. @@ -25941,9 +25941,12 @@ \subsection{B.1 What this appendix universal (\(S \iff D \iff Q_{\mathrm{fb}}\), {[}Main §3.4{]}) --- relies on a chain of intermediate lemmas: partition-relativity, emergent stochasticity, P-indivisibility from a non-permutation witness, -accessible-timescale backflow, and phase-locking. This appendix collects -these lemmas with their proofs in compact form, providing the reader a -self-contained reference for the framework's logical structure. +accessible-timescale backflow, and phase-locking, together with the +results on which phase-locking rests: Bohr-frequency completeness, the +real-to-integer transfer and the two-branch classification conditional +on H-BG. This appendix collects these lemmas with their proofs in +compact form, providing the reader a self-contained reference for the +framework's logical structure. The appendix's content is \emph{complete derivations}, not exposition. Each derivation gives the algebraic steps, the lemmas invoked, and the @@ -27316,11 +27319,11 @@ \subsection{B.7 The lemma chain for emergent quantum \(|U_{ij}|^2\) invariant. This is the basis-convention freedom --- the choice of phase for each basis state --- which is physically trivial. H-BG and the coefficient-labelled exceptional obstruction classify the -possible mode correspondences ({[}GR §3.3{]}); this step is not supplied -by Fourier extraction alone. Conditional on H-BG, the remaining -alternatives are this rephasing freedom, the energy shift and the -antiunitary conjugation \(\hat{H} \to -\hat{H}^*\) stated above. -\(\square\) +possible mode correspondences (Theorem B.7.5c below; {[}GR §3.3{]}); +this step is not supplied by Fourier extraction alone. Conditional on +H-BG, the remaining alternatives are this rephasing freedom, the energy +shift and the antiunitary conjugation \(\hat{H} \to -\hat{H}^*\) stated +above. \(\square\) \emph{Scope.} The lemma is stated in its simplest form, \(T_{ij}(t) = |U_{ij}(t)|^2\) --- the trivial-ancilla case of the @@ -27336,6 +27339,171 @@ \subsection{B.7 The lemma chain for emergent quantum \(H \to -H^*\), retained there as an intrinsic ambiguity of transition data. +\textbf{Bohr frequencies and the D-gauge classification.} The proof of +Lemma B.7.5 rests on the results collected here, each stated with its +hypotheses ({[}GR §3.3{]}). Throughout, \(U(t) = e^{-iHt}\) with \(H\) +in frequency units and \(H = V\,\mathrm{diag}(E)\,V^\dagger\) on a +finite configuration space, so that \(V_{ia}\) is the overlap of +configuration \(i\) with eigenvector \(a\), and likewise +\(H' = W\,\mathrm{diag}(E')\,W^\dagger\); the antiunitary reflection +\(-\bar{H}\) is the \(-\hat{H}^*\) of Lemma B.7.5. + +\textbf{Theorem B.7.5a (Bohr-frequency completeness).} \emph{Let +\(U(t) = e^{-iHt}\) with non-degenerate eigenvalues and non-vanishing +configuration-basis overlaps. Then the transition probabilities +determine the Bohr-frequency set \(\{E_a - E_b\}\): each return +probability expands as +\(|U_{ii}(t)|^2 = \sum_{a,b} |V_{ia}|^2 |V_{ib}|^2\, e^{-i(E_a - E_b)t}\), +in which every gap value carries a strictly positive coefficient; every +\(|U_{ij}(t)|^2\) is a combination of the same frequencies; and +consequently any \(U'\) of the same form with +\(|U'_{ij}(t)|^2 = |U_{ij}(t)|^2\) for all \(i, j, t\) has +\(\{E'_a - E'_b\} = \{E_a - E_b\}\).} + +\emph{Proof.} Expanding +\(U_{ij}(t) = \sum_a V_{ia}\, e^{-iE_a t}\, V_{ja}^*\) gives +\(|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a - E_b)t}\); +at \(i = j\) the coefficients are \(|V_{ia}|^2 |V_{ib}|^2 > 0\), and +grouping terms by frequency value only adds positive quantities --- +degenerate gaps merge coefficients but cannot cancel them. Distinct +characters \(t \mapsto e^{-i\omega t}\) of \((\mathbb{R}, +)\) are +linearly independent (Dedekind), so equality of two probability families +forces equality of the per-frequency coefficients over the union of the +two gap sets, and positivity makes the supports --- the gap sets --- +coincide. \(\square\) + +\textbf{The dimensional obstruction.} Theorem B.7.5a is what the gap +equation of §B.4 consumes, and it is less than a reconstruction of +\(H\). The unitary \(U(t)\) is dimensionless, and \(\hbar\) enters only +through \(\hat{H} = i\hbar\,\partial_t U\, U^\dagger\). The transition +probabilities are dimensionless functions of \(t\) and determine +temporal frequencies; converting a frequency to an energy requires the +action scale \(\hbar\) itself, so no amount of transition data fixes it. +The residual ambiguities in passing from frequencies to a Hamiltonian +carry no scale either: diagonal rephasing \(H \mapsto DHD^\dagger\) +moves no modulus, the energy origin \(H \mapsto H + E_0\) contributes +only the global phase \(e^{-iE_0 t}\), the antiunitary reflection +\(H \mapsto -\bar{H}\) satisfies +\(e^{-i(-\bar{H})t} = \overline{e^{-iHt}}\) entrywise and so preserves +every \(|U_{ij}(t)|^2\) exactly, and Theorem B.7.5c below excludes +further ambiguities conditional on its stated classification hypothesis. +The thermal self-consistency of §B.4, Step 4, is therefore the +mathematically obligatory entry point for the dimensionful input. + +\textbf{Lemma B.7.5b (Real-to-integer transfer).} \emph{Let \(E, E'\) be +real spectra on \(n\) modes, each with distinct directed gaps --- +\(E_b - E_a = E_d - E_c\) with \(a \neq b\) and \(c \neq d\) only when +\((a, b) = (c, d)\) --- and with equal gap sets. Then \(E'\) is, after +relabeling its modes, a translate \(E + E_0\) of \(E\) or a translate +\(-E + E_0\) of its reflection; or else there are relabelings of the two +spectra and strictly increasing integer rulers \(F, F'\) on \(n\) marks, +each with distinct positive differences, with equal difference sets, and +neither a translate nor a reflection of the other, such that every +integer gap identity \(F'_d - F'_c = F_b - F_a\) with \(a \neq b\) holds +for the relabeled real spectra as well.} + +The lemma assumes no classification hypothesis. It is proved in the +framework's Lean formalization (kernel: +\texttt{real\_to\_integer\_surrogate} in +\texttt{OIBridge/TurnpikeScopeTransfer.lean}), and it is the step +through which a classification of integer rulers, such as H-BG below, +applies to real spectra. + +\textbf{Theorem B.7.5c (D-gauge completeness, two-branch form).} +\emph{Assume H-BG: every non-congruent pair of strictly ordered integer +rulers with distinct positive differences and equal difference sets has +six marks and, after relabeling, realizes either \(\mu\) or its global +reversal, where \(\mu\) sends each ascending mark pair of +\(\{0,1,4,10,12,17\}\) to the ascending mark pair of +\(\{0,1,8,11,13,17\}\) with the same difference. The orientation is +chosen once for the whole pair. Let \(H,H'\) act on the same finite +configuration space of dimension \(m\ge3\), with distinct nonzero energy +gaps and non-vanishing configuration-basis overlaps for both +Hamiltonians. If \(|U'_{ij}(t)|^2 = |U_{ij}(t)|^2\) for all \(i,j,t\), +then \(H' = DHD^\dagger + E_0\) or \(H' = -D\bar{H}D^\dagger + E_0\), +for a time-independent diagonal unitary \(D\) and a real \(E_0\).} + +\emph{Conditional proof.} Equality of the probability families forces +equality of the per-frequency coefficients of +\(|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a-E_b)t}\). +Under the stated overlap and gap hypotheses, the spectra have equal +difference sets and the coefficient matching induces a bijection of +ordered mode pairs preserving directed gaps. A fixed correspondence can +be transferred from real to integer spectra while preserving its gap +identities, strict order, distinct differences and non-congruence (Lemma +B.7.5b): its realization constraints are homogeneous rational-linear +equalities, strict inequalities and inequations, and rational +approximation in the equality subspace followed by denominator clearing +preserves the finite list of strict conditions. H-BG then supplies a +congruent pair or one of the two globally oriented exceptional +correspondences. For congruent spectra, diagonal-line rigidity fixes the +moduli and the phase-coboundary relation fixes the phases, giving +\(W_{i,\tau a}=d_i\beta_a V_{ia}\) with \(d,\beta\) unimodular. The +column phases cancel in the reconstructed Hamiltonian. Reflection +congruence applies this argument to \((\bar V,-E)\) and gives the second +branch. For the positive exceptional correspondence \(\mu\), the +coefficient identities force flat moduli and constrain the phase data to +a graph of maximum clique three, excluding the six orthogonal rows +required by unitarity. For the reversed correspondence, reversing the +source mode pair conjugates its coefficient tensor; applying the same +obstruction to the conjugated source eigenbasis excludes that +orientation as well. Thus the two Hamiltonian branches follow +conditional on H-BG. \(\square\) + +\emph{Scope.} The complete comparison of H-BG with the published integer +classification of Bekir and Golomb {[}A. Bekir and S. W. Golomb, IEEE +Trans. Inform. Theory 53 (2007) 2864--2867{]}, including all +normalizations and exceptional orientations, is open. The unsigned +formal premise \texttt{BGIntegerClassification} of the Lean +formalization is false: the strictly ordered rulers \((0,1,4,10,12,17)\) +and \((0,4,6,9,16,17)\) have distinct positive differences and equal +difference sets, are neither translates nor reflections of one another, +and satisfy the reversed exceptional correspondence while admitting no +relabelings with the required positive orientation. The kernel +implication \texttt{twoBranch\_of\_BGClassification} therefore does not +establish reconstruction from the physical hypotheses. The example +contradicts that formal premise, not the published classification. +Bohr-frequency completeness, the real-to-integer transfer, congruent +reconstruction and the positive six-mode obstruction are kernel-proved +independently of it. The listed ambiguities are exactly invisible: +\(H+E_0\) contributes only a global phase, while \(-\bar H\) conjugates +the propagator entrywise. The gap condition is used where coefficient +matching induces a bijection of ordered mode pairs: with degenerate gaps +one frequency's coefficient is a sum over several mode pairs. The +\(C_4\) ring of {[}SM{]} Proposition 20 has distinct eigenvalues, +uniform overlaps and degenerate gaps; its conjugate \(\bar H\) +reproduces every \(|U_{ij}(t)|^2\) and conjugates the loop product +\(H_{01}H_{12}H_{23}H_{30}\), which no diagonal conjugation changes. The +ring therefore does not exhibit a failure, without the gap condition, of +the two-branch form conditional on H-BG: \(\bar H\) lies in the +reflection branch, \(\bar H = -D_s\bar H D_s^\dagger\) with +\(D_s = \mathrm{diag}(1,-1,1,-1)\). The raw frequency set alone cannot +distinguish the non-congruent homometric rulers; reconstruction uses +coefficient-labelled frequency data. + +\textbf{Corollary B.7.5d (Operational antiunitary invariance).} +\emph{Transposing the preparation, every operation, and the effect +simultaneously --- \(\rho \mapsto \rho^T\), +\(\mathcal{G} \mapsto T \circ \mathcal{G} \circ T\), \(E \mapsto E^T\) +--- preserves every finite circuit probability, for arbitrary operations +and arbitrary finite sequences, hence for every fixed outcome branch of +any adaptive instrument strategy; the transformation carries Kraus +operations to Kraus operations with conjugated operators, preserves the +trace normalization, and on unitary evolution is exactly the branch map +\(H \mapsto -\bar{H}\).} + +Enlarging the data from transition probabilities to arbitrary +preparations, instruments, adaptive finite circuits, and effects +therefore cannot remove the second branch: at the purely operational +level the two branches are not two empirically distinguishable theories +but two antiunitarily related reconstructions of the same complete +circuit statistics, and they become physically distinct only once an +oriented structure is fixed ({[}GR §3.3{]}). (kernel: +\texttt{circuit\_invariance}, \texttt{string\_invariance}, +\texttt{transposeMap\_kraus}, \texttt{kraus\_normalization}, +\texttt{unitary\_channel\_transpose} in +\texttt{OIBridge/AntiunitaryInvariance.lean}) + \textbf{Theorem B.7.6 (Characterization theorem).} \emph{In the redefined form of {[}Main §3.4{]}: accessible non-Markovianity of the visible process is equivalent, on each finite horizon, to embedded @@ -30318,4 +30486,4 @@ \subsection{Historical and philosophical {[}Origin of the molecular clock hypothesis.{]} \end{document} -% source-sha256: bd20a09a3a84bbe2e11861626356397de14192a532cfa6c39be36214ce309cae +% source-sha256: a17d8302b985c545b3a7060941c59198609c64600e6ab023a6d532db4cec6d6f diff --git a/book/appendix-b-derivations.md b/book/appendix-b-derivations.md index e64b3744c..7765fb998 100644 --- a/book/appendix-b-derivations.md +++ b/book/appendix-b-derivations.md @@ -19,7 +19,7 @@ The appendix's content is organized by mathematical structure rather than by cha *B.6 Gauge-theoretic derivations.* Two key gauge-theoretic results from Chapter 5: the conditional H-link link count for $K = 2dm = 6$ internal components per site, and the cubic-rotation-group decomposition of the coupling matrix's eigenvalue multiplicities into $(3, 2, 1)$ — the multiplicities that determine $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$. -*B.7 The lemma chain for the characterization.* The framework's characterization theorem — accessible non-Markovianity ⟺ C1/C3/C4 realization per horizon, with the quantum representation internal and universal ($S \iff D \iff Q_{\mathrm{fb}}$, [Main §3.4]) — relies on a chain of intermediate lemmas: partition-relativity, emergent stochasticity, P-indivisibility from a non-permutation witness, accessible-timescale backflow, and phase-locking. This appendix collects these lemmas with their proofs in compact form, providing the reader a self-contained reference for the framework's logical structure. +*B.7 The lemma chain for the characterization.* The framework's characterization theorem — accessible non-Markovianity ⟺ C1/C3/C4 realization per horizon, with the quantum representation internal and universal ($S \iff D \iff Q_{\mathrm{fb}}$, [Main §3.4]) — relies on a chain of intermediate lemmas: partition-relativity, emergent stochasticity, P-indivisibility from a non-permutation witness, accessible-timescale backflow, and phase-locking, together with the results on which phase-locking rests: Bohr-frequency completeness, the real-to-integer transfer and the two-branch classification conditional on H-BG. This appendix collects these lemmas with their proofs in compact form, providing the reader a self-contained reference for the framework's logical structure. The appendix's content is *complete derivations*, not exposition. Each derivation gives the algebraic steps, the lemmas invoked, and the result with no glosses. For motivation and context, the relevant main-chapter sections are cross-referenced at the start of each derivation. The appendix is designed for readers wanting full technical content; the main chapters give the same results with motivating context but compressed derivations. @@ -425,10 +425,32 @@ The framework's characterization theorem — accessible non-Markovianity ⟺ C1/ **Lemma B.7.5 (Phase locking).** *Assume dimension at least three, the global sign-aware classification H-BG of [GR §3.3], and distinct gaps and non-vanishing overlaps for both Hamiltonians being compared. The emergent effective Hamiltonian $\hat{H}_{\text{eff}}$ on the visible sector is then determined by the continuous-time transition data $\{T_{ij}(t)\}$ up to the lemma's full gauge: an overall energy shift, the diagonal-unitary rephasing freedom, and the antiunitary conjugation $\hat{H} \to -\hat{H}^*$ — a genuine twofold ambiguity rather than a convention ([Main §3.1]).* -*Proof.* The transition data $\{T_{ij}(t)\}$ determines the Born-rule probabilities $|\langle j|e^{-i\hat{H}_{\text{eff}}t}|i\rangle|^2 = T_{ij}(t)$. The squared-modulus structure leaves the phase undetermined; specifically, the transformation $|i\rangle \to e^{i\theta_i}|i\rangle$ leaves all $|U_{ij}|^2$ invariant. This is the basis-convention freedom — the choice of phase for each basis state — which is physically trivial. H-BG and the coefficient-labelled exceptional obstruction classify the possible mode correspondences ([GR §3.3]); this step is not supplied by Fourier extraction alone. Conditional on H-BG, the remaining alternatives are this rephasing freedom, the energy shift and the antiunitary conjugation $\hat{H} \to -\hat{H}^*$ stated above. $\square$ +*Proof.* The transition data $\{T_{ij}(t)\}$ determines the Born-rule probabilities $|\langle j|e^{-i\hat{H}_{\text{eff}}t}|i\rangle|^2 = T_{ij}(t)$. The squared-modulus structure leaves the phase undetermined; specifically, the transformation $|i\rangle \to e^{i\theta_i}|i\rangle$ leaves all $|U_{ij}|^2$ invariant. This is the basis-convention freedom — the choice of phase for each basis state — which is physically trivial. H-BG and the coefficient-labelled exceptional obstruction classify the possible mode correspondences (Theorem B.7.5c below; [GR §3.3]); this step is not supplied by Fourier extraction alone. Conditional on H-BG, the remaining alternatives are this rephasing freedom, the energy shift and the antiunitary conjugation $\hat{H} \to -\hat{H}^*$ stated above. $\square$ *Scope.* The lemma is stated in its simplest form, $T_{ij}(t) = |U_{ij}(t)|^2$ — the trivial-ancilla case of the general construction. In the ancilla-marginal form, local generic rigidity is checked at the certified block structures; general completeness is open. The visible data constrain the *dilated* Hamiltonian and configuration projectors, with the energy shift and diagonal-phase freedoms exactly invisible; the extension, its Fourier structure, and the constraint analysis of the larger pair-phase gauge are given in the companion paper ([Main §3.4]) — where the pair-phase gauge reduces to the diagonal one for generic families at each fixed dimension — together with the antiunitary conjugate $H \to -H^*$, retained there as an intrinsic ambiguity of transition data. +**Bohr frequencies and the D-gauge classification.** The proof of Lemma B.7.5 rests on the results collected here, each stated with its hypotheses ([GR §3.3]). Throughout, $U(t) = e^{-iHt}$ with $H$ in frequency units and $H = V\,\mathrm{diag}(E)\,V^\dagger$ on a finite configuration space, so that $V_{ia}$ is the overlap of configuration $i$ with eigenvector $a$, and likewise $H' = W\,\mathrm{diag}(E')\,W^\dagger$; the antiunitary reflection $-\bar{H}$ is the $-\hat{H}^*$ of Lemma B.7.5. + +**Theorem B.7.5a (Bohr-frequency completeness).** *Let $U(t) = e^{-iHt}$ with non-degenerate eigenvalues and non-vanishing configuration-basis overlaps. Then the transition probabilities determine the Bohr-frequency set $\{E_a - E_b\}$: each return probability expands as $|U_{ii}(t)|^2 = \sum_{a,b} |V_{ia}|^2 |V_{ib}|^2\, e^{-i(E_a - E_b)t}$, in which every gap value carries a strictly positive coefficient; every $|U_{ij}(t)|^2$ is a combination of the same frequencies; and consequently any $U'$ of the same form with $|U'_{ij}(t)|^2 = |U_{ij}(t)|^2$ for all $i, j, t$ has $\{E'_a - E'_b\} = \{E_a - E_b\}$.* + +*Proof.* Expanding $U_{ij}(t) = \sum_a V_{ia}\, e^{-iE_a t}\, V_{ja}^*$ gives $|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a - E_b)t}$; at $i = j$ the coefficients are $|V_{ia}|^2 |V_{ib}|^2 > 0$, and grouping terms by frequency value only adds positive quantities — degenerate gaps merge coefficients but cannot cancel them. Distinct characters $t \mapsto e^{-i\omega t}$ of $(\mathbb{R}, +)$ are linearly independent (Dedekind), so equality of two probability families forces equality of the per-frequency coefficients over the union of the two gap sets, and positivity makes the supports — the gap sets — coincide. $\square$ + +**The dimensional obstruction.** Theorem B.7.5a is what the gap equation of §B.4 consumes, and it is less than a reconstruction of $H$. The unitary $U(t)$ is dimensionless, and $\hbar$ enters only through $\hat{H} = i\hbar\,\partial_t U\, U^\dagger$. The transition probabilities are dimensionless functions of $t$ and determine temporal frequencies; converting a frequency to an energy requires the action scale $\hbar$ itself, so no amount of transition data fixes it. The residual ambiguities in passing from frequencies to a Hamiltonian carry no scale either: diagonal rephasing $H \mapsto DHD^\dagger$ moves no modulus, the energy origin $H \mapsto H + E_0$ contributes only the global phase $e^{-iE_0 t}$, the antiunitary reflection $H \mapsto -\bar{H}$ satisfies $e^{-i(-\bar{H})t} = \overline{e^{-iHt}}$ entrywise and so preserves every $|U_{ij}(t)|^2$ exactly, and Theorem B.7.5c below excludes further ambiguities conditional on its stated classification hypothesis. The thermal self-consistency of §B.4, Step 4, is therefore the mathematically obligatory entry point for the dimensionful input. + +**Lemma B.7.5b (Real-to-integer transfer).** *Let $E, E'$ be real spectra on $n$ modes, each with distinct directed gaps — $E_b - E_a = E_d - E_c$ with $a \neq b$ and $c \neq d$ only when $(a, b) = (c, d)$ — and with equal gap sets. Then $E'$ is, after relabeling its modes, a translate $E + E_0$ of $E$ or a translate $-E + E_0$ of its reflection; or else there are relabelings of the two spectra and strictly increasing integer rulers $F, F'$ on $n$ marks, each with distinct positive differences, with equal difference sets, and neither a translate nor a reflection of the other, such that every integer gap identity $F'_d - F'_c = F_b - F_a$ with $a \neq b$ holds for the relabeled real spectra as well.* + +The lemma assumes no classification hypothesis. It is proved in the framework's Lean formalization (kernel: `real_to_integer_surrogate` in `OIBridge/TurnpikeScopeTransfer.lean`), and it is the step through which a classification of integer rulers, such as H-BG below, applies to real spectra. + +**Theorem B.7.5c (D-gauge completeness, two-branch form).** *Assume H-BG: every non-congruent pair of strictly ordered integer rulers with distinct positive differences and equal difference sets has six marks and, after relabeling, realizes either $\mu$ or its global reversal, where $\mu$ sends each ascending mark pair of $\{0,1,4,10,12,17\}$ to the ascending mark pair of $\{0,1,8,11,13,17\}$ with the same difference. The orientation is chosen once for the whole pair. Let $H,H'$ act on the same finite configuration space of dimension $m\ge3$, with distinct nonzero energy gaps and non-vanishing configuration-basis overlaps for both Hamiltonians. If $|U'_{ij}(t)|^2 = |U_{ij}(t)|^2$ for all $i,j,t$, then $H' = DHD^\dagger + E_0$ or $H' = -D\bar{H}D^\dagger + E_0$, for a time-independent diagonal unitary $D$ and a real $E_0$.* + +*Conditional proof.* Equality of the probability families forces equality of the per-frequency coefficients of $|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a-E_b)t}$. Under the stated overlap and gap hypotheses, the spectra have equal difference sets and the coefficient matching induces a bijection of ordered mode pairs preserving directed gaps. A fixed correspondence can be transferred from real to integer spectra while preserving its gap identities, strict order, distinct differences and non-congruence (Lemma B.7.5b): its realization constraints are homogeneous rational-linear equalities, strict inequalities and inequations, and rational approximation in the equality subspace followed by denominator clearing preserves the finite list of strict conditions. H-BG then supplies a congruent pair or one of the two globally oriented exceptional correspondences. For congruent spectra, diagonal-line rigidity fixes the moduli and the phase-coboundary relation fixes the phases, giving $W_{i,\tau a}=d_i\beta_a V_{ia}$ with $d,\beta$ unimodular. The column phases cancel in the reconstructed Hamiltonian. Reflection congruence applies this argument to $(\bar V,-E)$ and gives the second branch. For the positive exceptional correspondence $\mu$, the coefficient identities force flat moduli and constrain the phase data to a graph of maximum clique three, excluding the six orthogonal rows required by unitarity. For the reversed correspondence, reversing the source mode pair conjugates its coefficient tensor; applying the same obstruction to the conjugated source eigenbasis excludes that orientation as well. Thus the two Hamiltonian branches follow conditional on H-BG. $\square$ + +*Scope.* The complete comparison of H-BG with the published integer classification of Bekir and Golomb [A. Bekir and S. W. Golomb, IEEE Trans. Inform. Theory 53 (2007) 2864–2867], including all normalizations and exceptional orientations, is open. The unsigned formal premise `BGIntegerClassification` of the Lean formalization is false: the strictly ordered rulers $(0,1,4,10,12,17)$ and $(0,4,6,9,16,17)$ have distinct positive differences and equal difference sets, are neither translates nor reflections of one another, and satisfy the reversed exceptional correspondence while admitting no relabelings with the required positive orientation. The kernel implication `twoBranch_of_BGClassification` therefore does not establish reconstruction from the physical hypotheses. The example contradicts that formal premise, not the published classification. Bohr-frequency completeness, the real-to-integer transfer, congruent reconstruction and the positive six-mode obstruction are kernel-proved independently of it. The listed ambiguities are exactly invisible: $H+E_0$ contributes only a global phase, while $-\bar H$ conjugates the propagator entrywise. The gap condition is used where coefficient matching induces a bijection of ordered mode pairs: with degenerate gaps one frequency's coefficient is a sum over several mode pairs. The $C_4$ ring of [SM] Proposition 20 has distinct eigenvalues, uniform overlaps and degenerate gaps; its conjugate $\bar H$ reproduces every $|U_{ij}(t)|^2$ and conjugates the loop product $H_{01}H_{12}H_{23}H_{30}$, which no diagonal conjugation changes. The ring therefore does not exhibit a failure, without the gap condition, of the two-branch form conditional on H-BG: $\bar H$ lies in the reflection branch, $\bar H = -D_s\bar H D_s^\dagger$ with $D_s = \mathrm{diag}(1,-1,1,-1)$. The raw frequency set alone cannot distinguish the non-congruent homometric rulers; reconstruction uses coefficient-labelled frequency data. + +**Corollary B.7.5d (Operational antiunitary invariance).** *Transposing the preparation, every operation, and the effect simultaneously — $\rho \mapsto \rho^T$, $\mathcal{G} \mapsto T \circ \mathcal{G} \circ T$, $E \mapsto E^T$ — preserves every finite circuit probability, for arbitrary operations and arbitrary finite sequences, hence for every fixed outcome branch of any adaptive instrument strategy; the transformation carries Kraus operations to Kraus operations with conjugated operators, preserves the trace normalization, and on unitary evolution is exactly the branch map $H \mapsto -\bar{H}$.* + +Enlarging the data from transition probabilities to arbitrary preparations, instruments, adaptive finite circuits, and effects therefore cannot remove the second branch: at the purely operational level the two branches are not two empirically distinguishable theories but two antiunitarily related reconstructions of the same complete circuit statistics, and they become physically distinct only once an oriented structure is fixed ([GR §3.3]). (kernel: `circuit_invariance`, `string_invariance`, `transposeMap_kraus`, `kraus_normalization`, `unitary_channel_transpose` in `OIBridge/AntiunitaryInvariance.lean`) + **Theorem B.7.6 (Characterization theorem).** *In the redefined form of [Main §3.4]: accessible non-Markovianity of the visible process is equivalent, on each finite horizon, to embedded observation under C1, C3, C4 (C2 outside the equivalence, carried by the conditional physical-memory theorem); the unitary quantum representation is internal and universal — $S \iff D \iff Q_{\mathrm{fb}}$, the realization's permutation unitary, the compressed form under (T) — nontrivial at the recurrence scale where the dynamics is P-indivisible, with the fixed-$\hat{H}$ form constructive for realized processes:* $$\text{Accessible non-Markovianity on } V \iff \text{Embedded observation of } (S, \varphi) \text{ with C1, C3, C4, per finite horizon}$$ diff --git a/book/ch03-structural-realism.md b/book/ch03-structural-realism.md index 53e2f383a..8e415a544 100644 --- a/book/ch03-structural-realism.md +++ b/book/ch03-structural-realism.md @@ -67,7 +67,7 @@ The framework's commitment at Level B is therefore structurally substantive, not The framework's gauge structure runs along an axis orthogonal to the observation axis. Where the observation axis is about scope — which substrata fall within the framework's domain — the gauge axis is about transformations: which substratum-level operations leave all observables unchanged. Like the observation axis, the gauge axis has four levels, with each broader level absorbing more degrees of freedom as gauge. -**Level G1: D-gauge on the emergent Hamiltonian.** The narrowest gauge freedom in the framework is the diagonal-unitary equivalence on the emergent Hamiltonian: $H \to D H D^\dagger$ with $D$ a diagonal unitary in the measurement-outcome basis. This freedom preserves transition probabilities exactly. Energy shifts and the antiunitary reflection also preserve them; completeness of these alternatives under the stated gap and overlap hypotheses is conditional on the sign-aware integer classification H-BG ([GR §3.3]). Two Hamiltonians $H$ and $D H D^\dagger$ produce identical Born-rule probabilities $T_{ij}(t) = |\langle j | e^{-iHt/\hbar} | i \rangle|^2$ in the measurement basis, so they are operationally indistinguishable. The transformation rephases the measurement basis, changing off-diagonal phases of $H$ while leaving its diagonal entries and every transition probability unchanged. +**Level G1: D-gauge on the emergent Hamiltonian.** The narrowest gauge freedom in the framework is the diagonal-unitary equivalence on the emergent Hamiltonian: $H \to D H D^\dagger$ with $D$ a diagonal unitary in the measurement-outcome basis. This freedom preserves transition probabilities exactly. Energy shifts and the antiunitary reflection also preserve them; completeness of these alternatives under the stated gap and overlap hypotheses is conditional on the sign-aware integer classification H-BG ([GR §3.3]; Appendix B.7). Two Hamiltonians $H$ and $D H D^\dagger$ produce identical Born-rule probabilities $T_{ij}(t) = |\langle j | e^{-iHt/\hbar} | i \rangle|^2$ in the measurement basis, so they are operationally indistinguishable. The transformation rephases the measurement basis, changing off-diagonal phases of $H$ while leaving its diagonal entries and every transition probability unchanged. Level G1 is the smallest gauge group in the hierarchy. It operates within a single Hamiltonian realization in a single emergent quantum mechanics — within Level D × Level G1 in the joint hierarchy, the framework's most specific intersection. Its content is the bookkeeping freedom in choosing phase conventions for the emergent basis states, with no implications beyond that. diff --git a/papers/GR.md b/papers/GR.md index bbce438b8..cb7dfbcc9 100644 --- a/papers/GR.md +++ b/papers/GR.md @@ -177,9 +177,9 @@ Steps 1–2 establish that $\hbar$ depends only on boundary quantities; Step 4 w **Theorem (D-gauge completeness, two-branch form).** *Assume H-BG: every non-congruent pair of strictly ordered integer rulers with distinct positive differences and equal difference sets has six marks and, after relabeling, realizes either $\mu$ or its global reversal, where $\mu$ sends each ascending mark pair of $\{0,1,4,10,12,17\}$ to the ascending mark pair of $\{0,1,8,11,13,17\}$ with the same difference. The orientation is chosen once for the whole pair. Let $H,H'$ act on the same finite configuration space of dimension $m\ge3$, with distinct nonzero energy gaps and non-vanishing configuration-basis overlaps for both Hamiltonians. If $|U'_{ij}(t)|^2 = |U_{ij}(t)|^2$ for all $i,j,t$, then $H' = DHD^\dagger + E_0$ or $H' = -D\bar{H}D^\dagger + E_0$, for a time-independent diagonal unitary $D$ and a real $E_0$.* -*Conditional proof.* Equality of the probability families forces equality of the per-frequency coefficients of $|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_b-E_a)t}$. Under the stated overlap and gap hypotheses, the spectra have equal difference sets and the coefficient matching induces a bijection of ordered mode pairs preserving directed gaps. A fixed correspondence can be transferred from real to integer spectra while preserving its gap identities, strict order, distinct differences and non-congruence: its realization constraints are homogeneous rational-linear equalities, strict inequalities and inequations, and rational approximation in the equality subspace followed by denominator clearing preserves the finite list of strict conditions. H-BG then supplies a congruent pair or one of the two globally oriented exceptional correspondences. For congruent spectra, diagonal-line rigidity fixes the moduli and the phase-coboundary relation fixes the phases, giving $W_{i,\tau a}=d_i\beta_a V_{ia}$ with $d,\beta$ unimodular. The column phases cancel in the reconstructed Hamiltonian. Reflection congruence applies this argument to $(\bar V,-E)$ and gives the second branch. For the positive exceptional correspondence $\mu$, the coefficient identities force flat moduli and constrain the phase data to a graph of maximum clique three, excluding the six orthogonal rows required by unitarity. For the reversed correspondence, reversing the source mode pair conjugates its coefficient tensor; applying the same obstruction to the conjugated source eigenbasis excludes that orientation as well. Thus the two Hamiltonian branches follow conditional on H-BG. $\square$ +*Conditional proof.* Equality of the probability families forces equality of the per-frequency coefficients of $|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a-E_b)t}$. Under the stated overlap and gap hypotheses, the spectra have equal difference sets and the coefficient matching induces a bijection of ordered mode pairs preserving directed gaps. A fixed correspondence can be transferred from real to integer spectra while preserving its gap identities, strict order, distinct differences and non-congruence: its realization constraints are homogeneous rational-linear equalities, strict inequalities and inequations, and rational approximation in the equality subspace followed by denominator clearing preserves the finite list of strict conditions. H-BG then supplies a congruent pair or one of the two globally oriented exceptional correspondences. For congruent spectra, diagonal-line rigidity fixes the moduli and the phase-coboundary relation fixes the phases, giving $W_{i,\tau a}=d_i\beta_a V_{ia}$ with $d,\beta$ unimodular. The column phases cancel in the reconstructed Hamiltonian. Reflection congruence applies this argument to $(\bar V,-E)$ and gives the second branch. For the positive exceptional correspondence $\mu$, the coefficient identities force flat moduli and constrain the phase data to a graph of maximum clique three, excluding the six orthogonal rows required by unitarity. For the reversed correspondence, reversing the source mode pair conjugates its coefficient tensor; applying the same obstruction to the conjugated source eigenbasis excludes that orientation as well. Thus the two Hamiltonian branches follow conditional on H-BG. $\square$ -The complete comparison of H-BG with the published integer classification of Bekir and Golomb [A. Bekir and S. W. Golomb, IEEE Trans. Inform. Theory 53 (2007) 2864–2867], including all normalizations and exceptional orientations, remains open. The unsigned formal premise `BGIntegerClassification` is false: the strictly ordered rulers $(0,1,4,10,12,17)$ and $(0,4,6,9,16,17)$ have distinct positive differences and equal difference sets, are neither translates nor reflections of one another, and satisfy the reversed exceptional correspondence while admitting no relabelings with the required positive orientation. The kernel implication `twoBranch_of_BGClassification` therefore does not establish reconstruction from the physical hypotheses. The example contradicts that formal premise, not the published classification. Bohr-frequency completeness, congruent reconstruction and the positive six-mode obstruction remain kernel-proved independently of it. The listed ambiguities are exactly invisible: $H+E_0$ contributes only a global phase, while $-\bar H$ conjugates the propagator entrywise. Omitting the gap condition fails on the $C_4$ ring of [SM] Proposition 20, whose distinct eigenvalues have degenerate gaps and whose loop phase is not preserved by diagonal conjugation. The raw frequency set alone cannot distinguish the non-congruent homometric rulers; reconstruction uses coefficient-labelled frequency data. +The complete comparison of H-BG with the published integer classification of Bekir and Golomb [A. Bekir and S. W. Golomb, IEEE Trans. Inform. Theory 53 (2007) 2864–2867], including all normalizations and exceptional orientations, remains open. The unsigned formal premise `BGIntegerClassification` is false: the strictly ordered rulers $(0,1,4,10,12,17)$ and $(0,4,6,9,16,17)$ have distinct positive differences and equal difference sets, are neither translates nor reflections of one another, and satisfy the reversed exceptional correspondence while admitting no relabelings with the required positive orientation. The kernel implication `twoBranch_of_BGClassification` therefore does not establish reconstruction from the physical hypotheses. The example contradicts that formal premise, not the published classification. Bohr-frequency completeness, the real-to-integer transfer (kernel: `real_to_integer_surrogate` in `OIBridge/TurnpikeScopeTransfer.lean`), congruent reconstruction and the positive six-mode obstruction remain kernel-proved independently of it. The listed ambiguities are exactly invisible: $H+E_0$ contributes only a global phase, while $-\bar H$ conjugates the propagator entrywise. The gap condition is used where coefficient matching induces a bijection of ordered mode pairs: with degenerate gaps one frequency's coefficient is a sum over several mode pairs. The $C_4$ ring of [SM] Proposition 20 has distinct eigenvalues, uniform overlaps and degenerate gaps; its conjugate $\bar H$ reproduces every $|U_{ij}(t)|^2$ and conjugates the loop product $H_{01}H_{12}H_{23}H_{30}$, which no diagonal conjugation changes. The ring therefore does not exhibit a failure, without the gap condition, of the two-branch form conditional on H-BG: $\bar H$ lies in the reflection branch, $\bar H = -D_s\bar H D_s^\dagger$ with $D_s = \mathrm{diag}(1,-1,1,-1)$. The raw frequency set alone cannot distinguish the non-congruent homometric rulers; reconstruction uses coefficient-labelled frequency data. **Corollary (operational antiunitary invariance).** *Transposing the preparation, every operation, and the effect simultaneously — $\rho \mapsto \rho^T$, $\mathcal{G} \mapsto T \circ \mathcal{G} \circ T$, $E \mapsto E^T$ — preserves every finite circuit probability, for arbitrary operations and arbitrary finite sequences, hence for every fixed outcome branch of any adaptive instrument strategy; the transformation carries Kraus operations to Kraus operations with conjugated operators, preserves the trace normalization, and on unitary evolution is exactly the branch map $H \mapsto -\bar{H}$.* diff --git a/papers/GR.pdf b/papers/GR.pdf index 22f08cb47..1b4178399 100644 Binary files a/papers/GR.pdf and b/papers/GR.pdf differ diff --git a/papers/GR.tex b/papers/GR.tex index 795849eda..286492c86 100644 --- a/papers/GR.tex +++ b/papers/GR.tex @@ -1187,7 +1187,7 @@ \subsubsection{3.3 Gauge-fixing and the dimensional \emph{Conditional proof.} Equality of the probability families forces equality of the per-frequency coefficients of -\(|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_b-E_a)t}\). +\(|U_{ij}(t)|^2 = \sum_{a,b} V_{ia} V_{ja}^* V_{ib}^* V_{jb}\, e^{-i(E_a-E_b)t}\). Under the stated overlap and gap hypotheses, the spectra have equal difference sets and the coefficient matching induces a bijection of ordered mode pairs preserving directed gaps. A fixed correspondence can @@ -1225,15 +1225,25 @@ \subsubsection{3.3 Gauge-fixing and the dimensional \texttt{twoBranch\_of\_BGClassification} therefore does not establish reconstruction from the physical hypotheses. The example contradicts that formal premise, not the published classification. Bohr-frequency -completeness, congruent reconstruction and the positive six-mode -obstruction remain kernel-proved independently of it. The listed -ambiguities are exactly invisible: \(H+E_0\) contributes only a global -phase, while \(-\bar H\) conjugates the propagator entrywise. Omitting -the gap condition fails on the \(C_4\) ring of {[}SM{]} Proposition 20, -whose distinct eigenvalues have degenerate gaps and whose loop phase is -not preserved by diagonal conjugation. The raw frequency set alone -cannot distinguish the non-congruent homometric rulers; reconstruction -uses coefficient-labelled frequency data. +completeness, the real-to-integer transfer (kernel: +\texttt{real\_to\_integer\_surrogate} in +\texttt{OIBridge/TurnpikeScopeTransfer.lean}), congruent reconstruction +and the positive six-mode obstruction remain kernel-proved independently +of it. The listed ambiguities are exactly invisible: \(H+E_0\) +contributes only a global phase, while \(-\bar H\) conjugates the +propagator entrywise. The gap condition is used where coefficient +matching induces a bijection of ordered mode pairs: with degenerate gaps +one frequency's coefficient is a sum over several mode pairs. The +\(C_4\) ring of {[}SM{]} Proposition 20 has distinct eigenvalues, +uniform overlaps and degenerate gaps; its conjugate \(\bar H\) +reproduces every \(|U_{ij}(t)|^2\) and conjugates the loop product +\(H_{01}H_{12}H_{23}H_{30}\), which no diagonal conjugation changes. The +ring therefore does not exhibit a failure, without the gap condition, of +the two-branch form conditional on H-BG: \(\bar H\) lies in the +reflection branch, \(\bar H = -D_s\bar H D_s^\dagger\) with +\(D_s = \mathrm{diag}(1,-1,1,-1)\). The raw frequency set alone cannot +distinguish the non-congruent homometric rulers; reconstruction uses +coefficient-labelled frequency data. \textbf{Corollary (operational antiunitary invariance).} \emph{Transposing the preparation, every operation, and the effect @@ -5115,4 +5125,4 @@ \subsection{References}\label{references}} \emph{JCAP} \textbf{09} (2013) 028; arXiv:1301.4995. \end{document} -% source-sha256: 0f520135a29733945c154e79e82d6460c8cb97286def9eebf980e310731dc4a2 +% source-sha256: c3eeb5e5227f5b135d1359786455cfb50397a17acc48535b0f7e0a4c72e5ede8 diff --git a/verification/lean-manuscript-census.json b/verification/lean-manuscript-census.json index e50b85ffb..cb51f027c 100644 --- a/verification/lean-manuscript-census.json +++ b/verification/lean-manuscript-census.json @@ -164,6 +164,22 @@ { "file": "papers/GR.md", "anchor": "thermodynamic orientation" + }, + { + "file": "book/appendix-b-derivations.md", + "anchor": "Thus the two Hamiltonian branches follow conditional on H-BG" + }, + { + "file": "book/The-Incompleteness-of-Observation-FULL.md", + "anchor": "Thus the two Hamiltonian branches follow conditional on H-BG" + }, + { + "file": "book/appendix-b-derivations.md", + "anchor": "*Conditional proof.* Equality of the probability families" + }, + { + "file": "book/The-Incompleteness-of-Observation-FULL.md", + "anchor": "*Conditional proof.* Equality of the probability families" } ], "note": "GR 3.3: Bohr-frequency completeness, congruent reconstruction, positive six-mode obstruction, antiunitary invariance and thermodynamic orientation are independent kernel results. Full two-branch reconstruction is conditional on the open global sign-aware classification; the unsigned BGIntegerClassification is refuted. The design package landed with #842 is pre-freeze design evidence, not adopted certification."