Extend QQ16 full-input obstruction to Clifford+T coefficients

research/qq16-clifford-ring-obstruction/PROOF.md

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+# The retained QQ16 obstruction over normalized Clifford+T coefficients
+
+Let `D=Z[1/2]`, let `zeta=exp(pi*i/4)`, and set
+`R=D[zeta]=Z[zeta,1/2]`. This note extends the **coefficient-ring boundary**
+of Nick Petalas's [PR2546](https://github.com/CrocSwap/integer-mult-bounds/pull/2546),
+pinned at `3aff4ccbfaa1005e5093e95535e4d99aa3813e54`. It reuses that work's
+explicit witness. It does not supply a new unread-minor census or read lower bound.
+
+## Statement and fixed observations
+
+Use the unchanged conserved `F0/B_1_2` pair and exactly the two-cut observation
+contract of starship-s's PR2443, with its unread set changed to
+`H=[1,25,3,4,5,6,8,12]`. Grant these complete owner/payload arrays:
+
+1. `x1_A`, `x1_T`, `y1_A`, `y1_T` immediately after complete F0;
+2. the 88 arrays `x1_(U\H)` at that same cut;
+3. `y2_A` and `C=x2_U` immediately after B1, with no intervening modification.
+
+There are 224 granted arrays. Reading and retaining these observations may even
+be free. There is no observation of `y2_U`, original selected inputs, or an
+additional ordinary query. The selected sets A,T, ordinary set U and integer
+query matrix R in the source are fixed. In this note the ring is also named R;
+where a distinction is needed the query matrix is written `R_query`.
+
+**Theorem.** No fixed finite affine-linear circuit whose linear coefficients lie
+in the ring `Z[zeta,1/2]` can recover all 224 original port arrays from these observations
+for arbitrary original inputs. Arbitrary finite auxiliary workspace and affine
+constants do not change the conclusion, when their initial values are the same
+for the two inputs compared. Input-dependent advice or additional observations
+are not part of this contract.
+
+This covers the explicit normalized gate alphabet H, S, T, X, Z, CNOT and SWAP,
+their inverses, permutations, tensor products, direct sums, and scalar linear
+ADDs/dyadic scalings. Here H is the normalized Hadamard, `S=diag(1,i)` and
+`T=diag(1,zeta)`. Fixed phases generated by zeta are included; arbitrary global
+phases are not asserted to belong to this ring. The theorem excludes an even
+larger class of finite R-linear circuits, so no gate synthesis theorem is needed.
+
+## A proper ideal after adjoining the normalizations
+
+The cyclotomic relation is `zeta^4+1=0`; the polynomial is irreducible over Q
+(substitute `x+1` and apply Eisenstein at 2). Consequently every ring element
+has a unique expression `c0+c1*zeta+c2*zeta^2+c3*zeta^3`, with each `cj` dyadic.
+Equivalently the ring is `D[x]/(x^4+1)`.
+
+Let `F25=F5[t]/(t^2-2)`. Two is not a square modulo five (the squares are
+0,1,4), so this quotient is a field. There is a unital ring homomorphism
+
+```text
+phi: D[zeta] -> F25,
+phi(2^(-1))=3,  phi(zeta)=4t=3/t.
+```
+
+Indeed `(4t)^2=2` and `(4t)^4=4=-1`, so the defining polynomial vanishes.
+The integer map extends to D because two maps to a unit. In the unique basis,
+
+```text
+phi(c0+c1*zeta+c2*zeta^2+c3*zeta^3)
+ = (c0+2*c2) + (4*c1+3*c3)*t  (coefficients reduced modulo five).
+```
+
+In particular `phi(1)=1` and `phi(5)=0`. Thus `5R` is proper. We do not claim
+that `5R` is the entire kernel of this particular specialization.
+
+The actual normalizing coefficients lie in the ring:
+
+```text
+i=zeta^2, sqrt(2)=zeta-zeta^3,
+1/sqrt(2)=(zeta-zeta^3)/2, zeta^(-1)=-zeta^3.
+```
+
+Their images are respectively `2,t,3t,2t`. In particular adjoining normalized
+Hadamards has not made five invertible. The coefficient `1/sqrt(2)` has nonzero
+odd basis coefficients, so it is outside the old Gaussian-dyadic subring D[i].
+This is a strict extension of the explicitly excluded class in PR2546.
+
+The specialization is **not assumed to preserve complex conjugation**. For
+example, i maps to 2, while its complex conjugate -i maps to 3; every field
+automorphism fixes these prime-field elements. Gate inverses are instead exact
+ring identities before specialization: `H^2=I`, `S*S^(-1)=I`,
+`T*T^(-1)=I`, and the named permutation gates are involutions. These identities
+remain true under any ring homomorphism. No reduced-unitarity claim is used.
+
+## Source witness, with complete arbitrary-input boundary
+
+For this paragraph write the query matrix as `R_query`. The authenticated packet
+gives `R_A R_A^T=R_T R_T^T=4I`. Put `P_A=R_A^T/4`,
+`z=(1,2,1,1,1,1,3,-2)`, and `q=10e_1` (zero-indexed centre coordinate).
+Direct multiplication gives `(R_query)_H z=q`.
+
+Choose every owner and payload entry of each port array constant, and initialize
+
+```text
+x_U=iota_H z, x_A=-P_A q, x_T=0,
+y_U=0,       y_A= P_A q, y_T=0.
+```
+
+These are legitimate dyadic inputs, including nonzero selected dirt. The actual
+cover maps are permutations of owner/payload addresses; hence they fix these
+constant arrays. This is an input subspace of the original full-array contract,
+not an assertion that arbitrary frames or arbitrary gates preserve constants.
+
+The full conserved F0 equations now give a zero first query and
+
+```text
+x1_U=y1_U=iota_H z,
+x1_A=x1_T=y1_A=y1_T=0.
+```
+
+The completed B1 future logger gives `y2_A=P_A q`, and its ordinary target
+subtraction gives `C=x1_U-y1_U=0`. All 88 read coordinates vanish. In A order,
+
+```text
+y2_A=(5/2,-5/2,0,5/2,0,0,-5/2,0).
+```
+
+Therefore every entry of every granted observation is in `5D`, hence in `5R`.
+The original port `x_1` is the constant array one. These source witness equations
+are credited to PR2546 and recomputed from the pinned packet; they are not new.
+
+Compare this input with the all-zero original input, using identical auxiliary
+workspace and affine constants. The observation difference lies entrywise in
+`5R`, and workspace differences are zero. Any R-linear gate, at any finite width,
+maps vectors in this ideal to vectors in this ideal. This follows directly from
+closure under finite sums and multiplication by R; induction proves it for the
+whole circuit. Injecting/forgetting coordinates or permuting addresses also
+preserves it. Constants cancel in the two executions, regardless of their values.
+The required output difference at every coordinate of original `x_1` is one,
+which cannot lie in `5R` because phi sends it to one. This contradiction proves
+the theorem, without assuming that intermediate values remain constant arrays.
+
+## Positive and failure boundaries
+
+The old unread set `H0=[1,2,3,4,5,6,8,12]` still has a dyadic inverse. Its two
+inverse orders are checked; this note does not obstruct that existing decoder.
+The alternative H is invertible over Q with inverse entry `(0,1)=1/10`. Thus
+the result is a coefficient restriction, not information loss over Q. Allowing
+`1/5` invalidates the proper-ideal argument: `(1/5)*5=1`.
+
+The witness does not obstruct recovery of only the original selected inputs:
+those inputs in this family are already determined dyadically by `y2_A`.
+Nonlinear division, measurement/postselection and normalization, input-dependent
+branching, additional observations, or a changed first pair require separate
+analysis. Merely adjoining these explicit normalized Clifford+T coefficients
+does not repair complete-input recovery at the fixed cut.
+
+No claim binds every available native frame or child operation to this alphabet.
+Physical access, chronological retention, precision, native kernel costs,
+all-size supplier admission and a full controller invoice remain open.

research/qq16-clifford-ring-obstruction/README.md

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+# QQ16 obstruction with normalized Clifford+T coefficients
+
+The alternative unread set from [PR2546](https://github.com/CrocSwap/integer-mult-bounds/pull/2546)
+still cannot support complete-input recovery when its decoder gains normalized
+Hadamards and eighth-root phases. The new result extends that work's explicit
+Gaussian-dyadic boundary to `Z[zeta_8,1/2]`; it reuses and credits its full-input
+witness. No read minimum, minor census, native controller or supplier is claimed.
+
+## Result
+
+Keep the unchanged F0/B1 pair, all arbitrary original inputs, and exactly the
+PR2443 two-cut observations with unread set `H=[1,25,3,4,5,6,8,12]`.
+All 224 granted observation arrays in the inherited witness have every entry
+divisible by five, but the required original port `x_1` is one. The proper ideal
+persists in the larger ring: reduction to `F25=F5[t]/(t^2-2)` sends
+`zeta_8 -> 4t`, `i -> 2`, `sqrt(2) -> t` and `1/sqrt(2) -> 3t`.
+
+[PROOF.md](PROOF.md) proves the ring homomorphism and ideal invariance under
+every fixed finite linear circuit over that ring, including arbitrary finite
+workspace with identical initial values in the two executions. Its explicit
+gate subclass is H, S, T, X, Z, CNOT, SWAP, their inverses, tensor/direct sums,
+permutations and dyadic linear operations. It does not assume the specialization
+commutes with complex conjugation. Gate inverse identities are exact identities
+before reduction.
+
+The original unread set `H0=[1,2,3,4,5,6,8,12]` remains a passing dyadic-inverse
+control. The alternative has a rational inverse with entry `1/10`, so this is
+not an information-loss result over Q. Allowing an odd-denominator gate such
+as `1/5`, nonlinear division, postselection, additional observations or a changed
+first pair lies outside the result. The displayed family does not obstruct
+selected-only recovery. Binding a complete native alphabet, chronological access,
+retention, precision and a favorable total invoice remain separate obligations.
+
+## Exact evidence
+
+The checker independently evaluates the literal constant-array F0/B1 equations
+from the authenticated retained packet, without importing predecessor code.
+[RESULT.json](RESULT.json) retains all 224 original input values
+(112 X and 112 Y), all 224 observation values, both eight-by-eight
+inverse matrices, the field specialization and seven gate-prefix vectors.
+It checks 56 individual inverse basis columns and a complete mixed word return.
+The normalized H prefix actually leaves the earlier Gaussian-dyadic subring.
+
+Six failure boundaries catch a wrong field relation, an odd denominator,
+admission of inverse five, an omitted Hadamard normalization, a wrong phase
+inverse and a changed unread witness. Separate export tests reject changed source
+bytes, typed and semantic receipt changes and a missing receipt; an explicit
+`--emit` mode prints a recomputation without writing package files.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -I -B scripts/research_package.py --package research/qq16-clifford-ring-obstruction
+python3.12 -I -O -B research/qq16-clifford-ring-obstruction/verify.py
+```
+
+Expected primary output starts `PASS: source-bound 224 observations`.
+[VERIFY.json](VERIFY.json) registers the primary checker and four export/control
+groups. [SOURCES.json](SOURCES.json) pins main inputs by Git blob and SHA256 and
+retains exact source READMEs for PR2546 at
+`3aff4ccbfaa1005e5093e95535e4d99aa3813e54` and PR2443 at
+`545c05e17243c962b112b91e7f84b4c1a41de7b8`.
+
+The standalone replay also works in a Git-free export with the listed source
+files. Default replay writes no artifacts and keeps checks active under `-O`.
+The source-bound finite checks support the written proof; they do not replace
+its induction for arbitrary finite circuits. Python uses dynamic allocation and
+arbitrary-size integers, a managed-runtime deviation from fixed allocation.
+Replay dimensions are fixed at 112 ports, eight centre coordinates, four ring
+basis coefficients and eight gate rows; source input sizes are bounded before
+parsing. No floating-point arithmetic or external mathematical library is used.
+
+## Open next questions
+
+Does a concrete proposed native gate or frame leave this ring and actually
+permit the odd denominator? First derive its exact coefficients at the real cut,
+then check whether five remains a nonunit. The current result does not bind every
+native primitive. Alternatively, use the already admissible H0 basis and price
+its complete chronological reads, frame maps, retention and cleanup. Neither
+route follows from scalar invertibility alone. [TASK.json](TASK.json) records
+the changed task scope, overlap coverage and structured handoffs.
+
+## Credit
+
+Nick Petalas's PR2546 supplies the alternative unread set, modular full-input
+witness and prior Gaussian-dyadic obstruction; its explicit normalized-Clifford
+exclusion is the boundary resolved here. starship-s's PR2443 supplies the
+arbitrary-input rational decoder and two-cut contract. Gareth Davidson's PR2432
+supplies the unread-column criterion and H0. Ali Romman and the retained QQ16
+contributors supply the packet, conserved maps and cover semantics. Earlier
+compiler/supplier attribution remains in the unchanged
+[QQ16 notice](../joint-query/NOTICE.md).
+
+Prepared by ywz65 with substantial OpenAI Codex assistance in mathematics, code,
+checking and prose. New work is Apache-2.0. No human or proof-assistant
+certification, native implementation or new kappa is claimed.